space-time vortices and geometric algebrajec/spacetimevortices3.pdf · space-time vortices and...

© John E.Carroll Monday, 24 June 2002

SPACE-TIME VORTICES and GEOMETRIC ALGEBRA

J.E.Carroll, Dept. of Engineering, Trumpington Street, University of

Cambridge, CB2 1PZ

[email protected]

Abstract: The paper examines a theory of three dimensions of time, guided by multi-

vector Geometric Algebra, that is consistent with Special Relativity, Maxwell’s

Equations, Klein-Gordon and Dirac equations. Rest mass appears as an operator in

‘transverse time’. A reason why div B = 0 but div E = ρ is provided. Advanced as

well as retarded solutions to the wave-equation are retained to give circulations of

signals in space and time: space-time vortices. This new theory uses temporal

boundary conditions to ensure causality and is the first theory to quantise photons

without using already quantised harmonic oscillators. A new explanation is provided

of the Einstein-Podolsky-Rosen paradox.

INDEX: SPACE-TIME VORTICES and GEOMETRIC ALGEBRA 1

1. Introduction 31.1 General 3

1.2 Three dimensional time 3

1.3 Using the algebra of multi-dimensional symmetry 6

2. Outline of Paper 7

3. 3+3 Geometric Algebra 83.1 1-vectors and derivatives 8

3.2 Passive Rotations 10

3.3 The multi-vectors 11

4. Transverse Time 124.1 Preliminaries 12

4.2 Complex multi-vectors 13

4.3 Dirac Equation 14

4.4 Conjugation 17

5. Maxwells EquationS 185.1 The Potentials 18

© John E.Carroll Monday, Friday, 26 April 2002

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5.2 The Fields E and B 19

5.4 Sources 21

5.4 RTT invariant power 22

5.5 Normal Modes 23

5.6 Stored energy and energy transfer 24

5.7 Forward, reverse, retarded and advanced waves 25

5.6 Promotion/demotion/annihilation 26

6. Space-time vortices 286.1 Concept of space-time vortices 28

6.2 Vortex excitation and detection 29

6.3 Quantisation 32

6.4 Causality, Localisation and Uncertainty 33

7. Chirality 34

8. Polarisation 358.1 Stokes Parameters 35

8.2 ‘Instantaneous’ knowledge of polarisation around the vortex 36

9. Entanglement 36

10. Conclusions 37Acknowledgements 39

Appendix 1. 39

Fourier Analysis & Annihilation of Analytic Signals [32, 38] 39

Appendix 2. Lorentz Trans-formations 41

References 43


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1. INTRODUCTION

1.1 General

Cramer [1] provides an interpretation of quantum theory where a transaction is

said to occur between the observer and observed object so that ψ represents a signal

travelling forward in time and ψ* travelling backward in time completing a

‘handshake’ between the observer and the observed. Another way of looking at this is

to say that signals are circulating in time. However in fluid flow, a circulation or

vortex is a ‘peculiar and characteristic glory’ of three-dimensions according to

Truesdell [2]. A fluid circulation cannot be supported within a single dimension but

has to have a minimum of three. The combination of Cramer’s concepts on signals

and Truesdell’s insight into fluid flow suggests that three-dimensional time needs

exploring to see how signals may circulate in time. Another motivation comes from

noting that, with inviscid fluids, whirlpools or vortices persist in whole units and are

created or destroyed as they interact with boundaries or with other vortices. This leads

to a speculation that space-time vortices might form a foundation for quantisation.

Although Maxwell’s equations have spatial curl (eg spatial vorticity) the single

dimension of time prevents any temporal vorticity.

1.2 Three dimensional time

This paper considers three-dimensional (3-d) time and shows how photons can

be identified as space-time vortices (STVs) with circulations of fields in space and

time. This new theory is not just a new interpretation of quantum theory [3] but

actually derives quantisation for photons from temporal boundary conditions. There is

no appeal to the theory of quantised harmonic oscillators although, to save work,

appeal is made to the associated operator algebra [4] . In the theory presented here,

time has three orthogonal components. ‘Principal’ time t3 (or t) is the conventional

measured time, while t1 and t2 forms a two dimensional ‘transverse time’. New

insights will be offered in four areas.

(a) Rest mass in the Dirac and Klein-Gordon equations, as well as zero mass in

Maxwell’s equations, will be shown to consistent with rest mass being linked to

variations in ‘transverse’ time.


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(b) Charge and current in Maxwell’s equations will be linked to variations in

transverse time.

(c) The lack of magnetic sources is a natural outcome of one of the requirements for

the inherent space-time symmetry of the magnetic field in this 3-d space and 3-d time.

(d) Einstein-Podolsky-Rosen entanglement [5, 6, 7, 8] and the associated correlation

of polarisation are explained without violating causality.

Three dimensonal (3-d) time came under considerable scrutiny as a theory that

could potentially enlighten the physics of tachyons [9, 10, 11, 12, 13 14, 15 ] .

However criticisms had already been voiced [16] and difficulties were frequently

discussed [17, 18, 19, 20, 21]. There were three main areas for concern.

(i) 3-d time implies also a 3-d reciprocal time which suggests 3-d energy. Reference

16 in particular discusses how this could lead to unphysical particle decay processes.

(ii) Some theories of 3-d time were at variance with known experimental results by

failing to allow for the transverse Doppler effect or the Thomas precession.

(iii) 3-d time ‘has not been observed’. The inverted commas are deliberate because it

depends on what one expects to observe.

The first two areas of concern arise from the assumption that any 3-d temporal

space would have the same homogeneity as its spatial 3-d counterpart, where there is

no preferred choice of reference axes. Without any preferred choice of axes,

reciprocal temporal space would give rise to three components of energy which could

be freely interchanged by rotation of the temporal axes. This was at the heart of the

concerns in reference 16. Cole and Buchanan prevented a free rotation of the axes of

time by arguing that changes in direction of the main temporal axis required energy

[22]. In the present work, there is a preferred ‘transverse’ temporal plane Ot1t2 so that

a 6-dimensional wave equation becomes a ‘Klein-Gordon’ equation:

– (∂s12 +∂s2

2 +∂s32)Φ +Mo

2Φ + ∂t3 2Φ = 0 ; (1)

where Mo2Φ = (∂t1

2 + ∂t22)Φ ; (c = = 1) so that mass appears as a temporal

operator.

The usual homogeneity of space is retained in the work here so that there is no

restriction on any particular orientation for the orthogonal spatial axes Osn.

However, in temporal space, there is a ‘principal’ time, which is convenient to label as

Ot3 (or Ot). This temporal direction has to remain normal to the temporal plane Ot1t2 .


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The homogeneity of space means that a velocity v can, for example, have its spatial

changes taken along Os3 and its temporal changes will always be directed along Ot3.

Then Ot1, Ot2, Os1 and Os2 are all transverse to v for a standard Lorentz transformation

and therefore remain Lorentz invariant. This ensures the Lorentz invariance of the

temporal operator Mo2. Hence Mo in equ. (1) can be identified with rest mass .

It could be argued that there is already circumstantial evidence for more than

one dimension of reciprocal time because free energy and rest mass are both

observed. (Free energy)2 is associated with the principal temporal operator, – ∂t32

(=1), and oscillating waves in t. The transverse temporal operator Mo2 in equ. 1 is

associated with (rest mass energy)2 and the sign will require evanescent waves in t1

and t2. The periodic wave and evanescent wave cannot be freely rotated into one

another so that rest mass and free energy can not be simply interchanged by re-

labelling the temporal axes. This ‘prevention’ of rotating the ‘axis’ of principal time

can be compared with reference 22 where energy is required.

A major difficulty with 3-d time is that the spectacular successes of Maxwell’s

equations provide little motivation to consider any new theory that modifies these

equations. It is not surprising to find that all the proposals concerning 3-d time recover

Maxwell’s equations in some limit. However there is no consensus on the proposed

changes to Maxwell’s equations as a result of 3-d time. Cole has a credible starting

point. The usual electromagnetic vector potential A is the spatial part of the potential

and the temporal part now has a 3-component ‘temporal vector’ ΦΦΦΦ [23, 24]. These

potentials lead to a 9-component E field in a 3 x 3 space-time array along with a new

3-component ‘magnetic’ field W that augments the usual 3-component B field. Fields

similar to W have been proposed with only |ΦΦΦΦ| and |A| being alleged to be measurable

[25, 26]. Teli, links each dimension of time directly with a dimension in space and

arrives at highly symmetric forms of Maxwell’s equations [ 27, 28]. Barashenkov and

Yurlev [29] refer to Cole’s approach and have both positive and negative energy

solutions with causality limiting the solutions to those with positive energies. While

Cole’s starting point is essentially followed here, there are significant differences in

the subsequent development.


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1.3 Using the algebra of multi-dimensional symmetry

The importance of making the correct formulation for Maxwell’s equations in

3+3 space-time suggests a study of ‘Geometric Algebra’ (GA). GA is the algebra of

multi-dimensional symmetry. For over three decades, GA has been brilliantly

promoted by David Hestenes along with others [30,31,32,33,34, 35,36]. Space-Time

Algebra (STA) is the name that is reserved for a branch of GA which specialises in

3d-space +1d-time i.e. the standard space-time. Maxwell’s equations emerge as

essential features of multi-vector symmetry in STA [39,40,37,38, 39]. Assuming one

is going to explore 3-d time, then the success of STA suggest that a ‘3+3’ space-time

GA should be examined. It will be found that Maxwell’s equations can be derived

from ‘3+3’ GA in much the same way as Maxwell’s equations can be derived in STA.

The new arguments supersede those of the author’s previous work [40]. The postulate

of having a preferred ‘transverse’ temporal plane is found to recover the classical

Maxwell’ equations for E and B fields, except that now each of these fields turn out to

be ordered pairs: e.g. E = E1 , E2. Here it is possible to associate the spatial vector

fields E1(2) with directions Ot1(2) . Using conventional complex notation, it is then

possible to write E = E1 + iE2 and B = B1 + iB2 . These complex fields satisfy the

conventional Maxwell’s equations. However the normal modes now have the

following features:

(1) separate modes for positive and negative frequencies;

(2) separate modes with advanced and retarded solutions;

(3) separate modes with ‘left’ and ‘right’ handed temporal chiralities, a term to be

explained more fully later on.

(4) Two orthogonal polarisations for each mode, as in classical theory.

All these modes play a role so that, unlike classical Maxwell theory, the advanced

modes are not immediately discarded. In essence there are more degrees of freedom

in the ‘3+3 Maxwell equations’ than in the ‘STA Maxwell equations’. What will be

discovered is that creating an excitation cannot fully define that excitation. It is only

during the detection of the excitation that sufficient boundary conditions are specified

to determine the outcome.

The ‘complex’ quantity i ( i2 = –1), while behaving for the most part like the

conventional well known i, has here a unique geometrical interpretation of causing a


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90o rotation in the ‘transverse’ temporal plane so that exp(i θ) represents a rotation by

an angle θ within the Ot1t2 plane. Two conditions are imposed:

(A) The orientation of any field in transverse time is postulated to be unobservable by

classical measurements. This will mean that terms like ψ = ψο exp(i θ) are

unobservable in classical measurements but terms like ψ*ψ will be observable.

Power, energy and polarisation will all be formulated in terms of ‘observable’

quantities.

(B) The net power flowing backwards in time is postulated to be zero. This will yield

unidirectional power flow, causality and also quantisation of the e.m. fields.

2. OUTLINE OF PAPERSection 3 examines a Geometric Algebra (GA) in 3-d space and 3-d time.

Although GA has gained wide acceptance, it is noticeable that many papers on topics

in GA still believe it to be essential to have a brief introduction. This trend has not

been followed here. However in an attempt to increase accessibility to the work and

put the algebra into familiar terms, the full panoply of GA notation is not used in this

paper. The theory is developed directly from the commutation rules for orthogonal

GA ‘vectors’. In particular, special features of three-dimensions [e.g. vector products

and curl] are retained rather than making use of the wedge notation of GA. Instead of

a 2-d GA of transverse time, section 4 uses a ‘standard’ complex algebra .

Rest mass now appears as an operator in transverse time in the Klein-Gordon

and Dirac equations. Although potentially this means that eigen values for mass might

be found, such eigen values have not yet been studied so that unspecified values of

Mo2 are introduced as in equation (1). However this is no worse than the arbitrary

introduction of a mass Mo into the classic Dirac and Klein-Gordon equations, but has

the benefit of offering a future line of research. The Maxwell equations and charge

appear in section 5 with charge also being related to an operator in transverse time.

Although the new Maxwell fields are superficially identical to the classical

equations, the new fields are now complex: i.e. they form ordered pairs that ‘rotate’

with a rotation of the transverse temporal axes. The complex notation reduces the

algebra and is shown to be fully compatible with GA.

Combinations of the complex fields lead to normal modes that are analytic

complex functions with the properties (1) - (3) listed in section 1. Promotion,


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demotion and annihilation operations arise now from complex variable theory and not

from quantum theory. Section 6 considers concepts of space-time vortices and vortex

number conservation. These concept lead on to quantisation and causality. Standard

introduction to quantisation of the Maxwell field relies on the theory of quantised

harmonic oscillators. This is not used here, though it is convenient to make use of the

standard operator algebra.

Discussions on temporal chirality, polarisation and entanglement round off the

work with a final discussion indicating the potential of the work.

3. 3+3 GEOMETRIC ALGEBRA

3.1 1-vectors and derivatives

Excellent tutorials for ‘3-d’ GA and ‘3+1’ STA are available from both books

and the web [30,32,41,42,43]. It is therefore probably otiose to add an introductory

material except in so far as modifications are made to the notation required to cover

‘3d space+3d time’ GA (‘3+3’ GA). However importance is placed in this work on

three dimensions so that the concepts of curl (or vorticity) and vector products are

retained rather than using the notation of the GA wedge product. GA places

considerable stress on co-ordinate free representations, but here the fact that the

transverse temporal plane has to be singled out suggests that there is no shame in

embedding the theory in a coordinate system at the start and referring to this plane as

Ot1t2.

In‘3+3’ GA , the spatial and temporal unit 1-vectors are set as γγγγs = γs1, γs2, γs3

and γγγγt = γt1, γt2, γt3 respectively. The co-ordinate vector is taken as X = x, t where:

x = (x1, x2, x3) ; t = (t1, t2, t3) ; with

X = (x.γγγγs+ t .γ.γ.γ.γt) = Σn (xnγsn +tnγtn) (2)

Bold quantities (like x or t) represent the components of 3-component ‘vectors’ but

underlined bold versions (like X) denote a GA multi-vector. Context usually shows

which is a GA multivector and which is a conventional 3-component vector. A time-

like metric is assumed : t12 + t2

2 + t32 – x1

2 – x22 – x3

2 in line with the time-like metric

of STA. Normalisation and orthogonality of the unit vectors requires:


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γtm γtn = – γtn γtm (n ≠ m); γtm2 = 1 ;

γsm γsn = – γsn γsm (n ≠ m); γsm2 = –1 ;

γsm γtn = – γtn γsm (all m and n)

Consider two arbitrary 1-vectors: V = γγγγs....Ps + γγγγt.Π.Π.Π.Πt and W = γγγγs....Qs + γγγγt.Θ.Θ.Θ.Θt .

Then the commutation rules of equs.(3) yield a geometric product:

V W = –Ps ....Qs – ϕs γγγγs ....(Ps x Qs) + ΠΠΠΠt .Θ.Θ.Θ.Θt + ϕt γγγγt ....(ΠΠΠΠt x ΘΘΘΘt) + γγγγs.γ.γ.γ.γt. (. (. (. (Ps ΘΘΘΘt – ΠΠΠΠt Qs)

(4)

Here ϕs = γs1 γs2γs2 and ϕt = γt1 γt2γt2 are respectively spatial and temporal 3-vectors

with ϕs2 = 1 and ϕt

2 = –1. The full 6-d pseudo-scalar (or 6-vector) is S :

S = ϕt ϕs ; S2 = 1; (5)

Pseudo 1-vectors, i.e. 5-vectors, are given from

γsnS = – Sγsn ; γtnS = – Sγtn. Similarly γsnγtnS = Sγsnγtn gives a pseudo 2-vector or 4-

vector while a pseudo 3-vector remains a 3-vector: e.g ϕs S = – S ϕs = ϕt .

Temporal and spatial derivatives are given from

∂∂∂∂t = (∂t1, ∂t2, ∂t3), ∂∂∂∂s = (∂s1, ∂s2, ∂s3)

which in turn give co and contra-variant space-time derivatives:

= (– ∂∂∂∂s. γγγγs + ∂∂∂∂t.γ.γ.γ.γt) ; = ( ∂∂∂∂s. γγγγs + ∂∂∂∂t.γ.γ.γ.γt) (6)

Here one finds that

X =(– ∂∂∂∂s. γγγγs + ∂∂∂∂t.γ.γ.γ.γt )( x. γγγγs +t .γ.γ.γ.γt) = 6 (7)

where 6 is the invariant dimensionality of 3+3 space. Even more important in the

context of this paper is that

X =( ∂∂∂∂s. γγγγs + ∂∂∂∂t.γ.γ.γ.γt )( x. γγγγs +t .γ.γ.γ.γt) = 0 (8)

Differentiating a general 1-vector with gives

V = ( ∂∂∂∂s. γγγγs + ∂∂∂∂t.γ.γ.γ.γt )( γγγγs....Ps + γγγγt.Π .Π .Π .Π t)

= − ∂∂∂∂s.Ps − ϕsγγγγs . (∂∂∂∂s x Ps) + ∂∂∂∂t.ΠΠΠΠt + ϕt γγγγt . (∂∂∂∂t x ΠΠΠΠt) + γγγγt .γ.γ.γ.γs....(∂∂∂∂t Ps +∂∂∂∂s ΠΠΠΠt) (9)

Retention of a curl operator in 3-dimensions highlights temporal vorticity (∂∂∂∂t x ΠΠΠΠt) in

complete analogy with spatial vorticity (∂∂∂∂s x Ps) but with the three dimensions of time

replacing the three dimensions of space. Space-time vorticity is formed from the 9

)3(


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elements of (∂∂∂∂t Ps + ∂∂∂∂s ΠΠΠΠt). The analysis will be restricted to what will be termed as

‘analytic 1-vectors’ where, similar to (8), the first order derivatives vanish: V = 0.

3.2 Passive Rotations

Rotations are formed in GA by ‘rotors’ such as RS = exp (½θ γs1 γs3). A Taylor

expansion using (γs1 γs3)2 = –1 shows that RS = cos(½θ) + γs1 γs3 sin(½θ) . The

reversion of RS, denoted as RS~, changes the ordering of all the 1-vectors so that

RS~ = exp(½θ γs3 γs1). Then Vrot = RS VRS~ rotates V in the γs1 γs3 plane by an angle

of θ . The two sided operation with the half angle in each rotor are key features of

rotor algebra. The commutation properties (equ.3) show that γs2 and all γtnare

unaltered by this rotation RS. The important point to observe is that rotations in GA

are not characterised by an axis but by a plane. This is very evident on noting that

there are 4 axes that are ‘normal’ to the plane γs3 γs1 so that it can only be rotation

within the plane that has a clear meaning. Now the homogeneity of space means that

there is no a priori reason to prefer any one spatial direction so that one chosen set of

orthogonal 1-vectors γs1, γs2 and γs3 can be passively rotated into another set without

changing the physics.

Rotors RTT = exp (½ µ γt1 γt2) rotate the axes Ot1 and Ot2 within the transverse

plane Ot1t2 by an angle µ . GA vectors V rotate into VTT = RTT VRTT~ . Again it is

assumed that there is homogeneity in the transverse temporal plane so that there is be

no preferred choice of direction for γt1 or γt2 within the plane Ot1t2. This is one of the

postulates for the theory.

Lorentz transformations follow closely to those in STA. These are given by

‘rotors’ RL = exp (½α γt3 γs3). The factor of ½ still appears and the change of a multi-

vector V into VL is given from VL = RL VRL~ . The principle direction of time, γt3,

is always taken to be normal to the temporal plane Ot1t2. The use of rotors like RS can

re-label any arbitrary spatial direction as γs3 so the velocity v can always be taken to

lie along the space-time direction γs3 and γt3, while |v| determines the boost parameter

α as in classic relativity. For the Lorentz rotor RL given above, (γt3 γs3)2 = 1 so that

exp (½α γt3 γs3) = cosh ½α + γt3 γs3 sinh ½α partially interchanges principal time γt3

and γs3 No real value of α can completely interchange γt3 and γs3 unlike a rotation.


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However the concept that the rotor only rotates 1-vectors that lie in the relevant plane

still holds. Hence the transverse temporal and transverse spatial 1-vectors γt1, γt2

and γs1, γs2 are invariant under rotors such as RL . Hence the transverse temporal

plane Ot1t2 remains transverse to principal time and space.

Now as already proposed, the operator (∂t12 + ∂t2

2) is to be linked to (mass

energy)2 = Mo2 with evanescent waves in t1 and t2 . Observe that RL (γt1∂t1 + γt2∂t2)RL~

is Lorentz invariant and hence if Mo is consistent with representing mass then it has to

be consistent with rest mass. Free energy is to be linked with oscillating waves in t3

in the classical way so that rotors like RT = exp (½φ γt1 γt3), that change transverse

time into principal time, must in general be disallowed. Passive rotations through

rotors RST = exp (α γtM γsN), (M =1, 2 ; N=1- 3) should for similar reasons be

disallowed because some combinations of RST and RL would mix evanescent fields

and oscillating fields in transverse time. While Cole suggested that rotation of the

main temporal axis (principal temporal axis in this work) required energy [22], this

present work is saying that mass energy and free energy are not freely interchangeable

because oscillatory waves cannot change into evanescent waves. A discussion of the

circumstance in which such an interchange could occur lies outside the scope of this

paper.

Insisting on a preferred plane Ot1t2 implies a ‘non-integrable’ field theory. In

other words there are certain paths where integrals would be dependent on the choice

of path. Never the less in the conventional 3+1 space-time, the theory remains an

integrable field theory. The arrow of time [44, 45] would then become the arrow of

principal time.

3.3 The multi-vectors

Following the discussion after equ. 5, there are: one scalar (0-vector) plus one

pseudo-scalar (6-vector), six 1-vectors plus six pseudo 1-vectors (5-vectors), fifteen 2-

vectors plus fifteen pseudo 2-vectors (4-vectors) , plus twenty 3-vectors giving 64

elements in total. To reduce the algebra, n-vectors and pseudo-n-vectors will be

combined so that ‘scalar’ will refer to both scalar plus pseudo-scalar, ‘1-vector’ refers

to 1-vector plus pseudo 1-vector etc. Ordering of symbols will matter because the


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pseudo-scalar S anti-commutes with all 1-vectors as pointed out shortly. With the

above caveat, a general multi-vector may be formed in the following way:

Ψ (1 of 0-vector) + γγγγs.Ps+ γγγγt.Π.Π.Π.Πt(6 of 1-vectors) + ϕt γγγγs.Fs + ϕt γγγγt.Φ.Φ.Φ.Φt + γγγγt .γγγγs .Gst (15 of 2-vectors)

+ ϕtΞ + ϕt γγγγt .γγγγs.Xst(20 of 3-vectors) (10)

where Ψ = ψ +S ψ/; Ps = ps+S p/s ; ΠΠΠΠt = ππππt+S ππππ/

t ; Fs = fs +S f/s ;

ΦΦΦΦt = φφφφt +S φφφφ/t ; Gst = gst+ S g/

st ; ΞΞΞΞ = ξ +S ξ/ ; Xst = xst +S x/st ;

[ps = ps1, ps2, ps3 ; φφφφt = φt1, φt3, φt3; gst = gsn tm n = 1-3 , m = 1-3 etc. ]

Three points need emphasis.

(1) Selecting Ps by way of an example, then ps gives amplitudes in the spatial 1-vector

γγγγs.ps while p/s gives amplitudes in the spatial pseudo-1-vector γγγγs.S p/

s. Both ps and p/s

are real; complex amplitudes have yet to be considered. Similarly for all other

components.

(2) Equ. (9) is not a unique form.

(3) Having decided on this form, the order of the symbols matters. For example

γγγγs.Ps = γγγγs.ps + γγγγs .(S p/s) = ps.γγγγs – (S p/

s).γγγγs ≠ Ps.γγγγs

Rather than discuss the full 6-d space-time without restriction, the analysis now makes

the important restriction that there is a preferred transverse temporal plane.

4. TRANSVERSE TIME

4.1 Preliminaries

The preferred transverse plane is defined from the 1-vectors γt1 and γt2 . The bi-

vector γt1γt2 = i defines i with i2 = –1 and γt1i = –i γt1 = γt2 . The temporal gradient

operator ∂∂∂∂t.γγγγt can be written as :

∂∂∂∂t.γγγγt = γγγγt .∂∂∂∂t = (∂t1 – i∂t2) γt1 + ∂t3 γt3 = γt1 (∂t1 + i∂t2) + γt3 ∂t3 (11)

The differential transverse temporal operator, ∇∇∇∇ T = (∂t1 – i ∂t2), gives ∇∇∇∇ T (t1 + i t2) = 2

(the number of transverse temporal dimensions) while ∇∇∇∇ T (t1 + i t2) = 0. It is helpful

to emphasise that it is possible to have a field F such that ∇∇∇∇ T*F = 0 while ∇∇∇∇ TF ≠ 0,

(or indeed vice-versa). However ∇∇∇∇ T∇∇∇∇ T* = ∇∇∇∇ T*∇∇∇∇ T = M02 as in equ. 1. If either

∇∇∇∇ T*F = 0 or ∇∇∇∇ TF = 0 then M0 = 0. Now because i does not commute with γt1 or γt2

separately:


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∂∂∂∂t.γγγγt = ∇∇∇∇ T γt1 + ∂t3 γt3 = γt1∇∇∇∇ T* + γt3 ∂t3 (12)

However i commutes with itself and 1-vectors γt3, γs1, γs2, γs3 and hence any

combination of these. Transverse temporal components (γt1 at1+ γt2 at2) can then be

represented (as in an Argand diagram) by a single complex component AT :

γt1 AT = γt1 at1 + γt2 at2 = γt1 (at1 + i at2) = (at1 − i at2)γt1 (13)

The commutation properties of i permit a single sided operation exp(i θ) to

represent a rotation by θ in the Ot1t2 plane without having the usual half angle formula

of rotors. Hence as already stated i represents a rotation by 90o in the Ot1t2 plane. The

‘d’Alembertian’ operator used in equ. (8) can then be written in the form

= (∂∂∂∂s.γγγγs +∇∇∇∇ T γt1+ ∂t3 γt3) = (∂∂∂∂s.γγγγs + γt1∇∇∇∇ T* + ∂t3 γt3) (14)

If Φ is such that Φ = 0, then

Φ = – ∂∂∂∂s....∂∂∂∂s Φ +M02

Φ + ∂t3 ∂t3 Φ = 0 (15)

where M02 = ∇∇∇∇ T*∇∇∇∇ T = ∇∇∇∇ T ∇∇∇∇ T* means that all such Φ satisfy a Klein-Gordon

equation.

4.2 Complex multi-vectors

Using the notation of section 4.1, a general multivector V may now be re-written as:

V = Ψ + γγγγs.Ps+ γt3Πt3 + γt1ΠΠΠΠT + γt3 γγγγs. iFs + i Φt3 – γt3 γt1 i ΦΦΦΦT + γt3 γγγγs .Gst3 + γt1 γγγγs . GsT

+ γt3 i Ξ + γγγγs. i Xst3 − γt3 γt1 γγγγs. i XsT (16)

A complex notation may be defined as in table 1 allowing equ. (16) to be grouped into

even and odd grade multi-vectors with subscripts e and o respectively.

TABLE 1

(Ps+ i Xst3) → AosC ; (Πt3 +i Ξ) → ΦoC ; Π Π Π ΠT → ΦΦΦΦoT ; i XsT → AosT ;

(Ψ + i Φt3) → ΦeC ; (Gst3 +iFs) → AesC ; –i ΦΦΦΦT → ΦΦΦΦeT ; GsT → –AesT ;

Notice that i S = S i so, for example, iAosC or i S AosC remains of the generic form

(p//s+ i x//

st) + S (p///s+i x///

st). Obviously care has to be taken in observing the order of

γγγγs. AosC as opposed to AosC.γγγγs for example and in evaluating terms like (AosC)2 and in


14

separating out the pseudo-vector components before reaching any conclusions.

Similar remarks apply to all the other terms regardless of whether they are even or

odd grade. This ability to combine the vectors, complex vectors and pseudo vectors is

found to reduce the amount of algebra.

The multivector V then splits into even and odd grade complex components:

V = Ve + Vo (17)

where

Ve = [ΦeC + γt3 γγγγs.AesC + γt3 γt1 ΦeT + γγγγs.γt1AesT] ;

Vo = [γt3 ΦoC+ γγγγs.AosC + γt1ΦoT + γt3 γγγγs.γt1AosT] ;

Note that γt1Vo= [ΦoT + γt3 γγγγs.AosT –γt3 γt1ΦoC – γγγγs. γt1AosC ] has a ‘similar’ structure to

that of Ve. However there is an important distinction in that the even-grade multi-

vectors form a ‘sub-algebra’. In other words given even-grade multi-vectors Ve and

We, then the geometric product Ve We is also an even-grade multi-vector. This sub-

algebra is known as the spinor algebra and has, in some important work, been

considered to be closely related to the Dirac equation [46, 47] with the Dirac wave-

function ‘Ψ’ considered to be a rotor determining the properties of the electron.

The product of two odd grade multivectors Vo Wo is also an even grade

multivector to that the odd grade multivectors cannot form a similar sub-group.

A Lorentz rotor RL = exp (α γt3 γsN) will not change ΦeC, γt3 γsN AesNC , γt1ΦΦΦΦoT ,

γt3 γsN γt1AosNT , γsMAosMC or γsM γt1AesMT (M ≠ N). General Lorentz rotors (including

spatial rotations) will transform terms γγγγs.AosC + γt3ΦoC and γγγγs. γt1AesT + γt3γt1ΦeT

like conventional ‘3+1’ space-time 4-vectors even though AesT , AosC etc. are

necessarily complex.

4.3 Dirac Equation

Although the Dirac equation is subsidiary to the main thrust of the paper, it

leads into Maxwell’s equations and forms a cornerstone in demonstrating a consistent

connection of rest mass with transverse time. The treatment here differs from the well

considered treatments by Patty and Smalley [48] and by Boyling and Cole [49] who

both have a traditional scalar mass [50, 51]. Here following equ.(1) the transverse

temporal operator is assumed to be related to mass. Now the assumption is made that


15

the first derivative of any even-grade multi-vector is zero [similar to equ.(8)] – the

justification for this comes a posteriori.

( ∂∂∂∂s. γγγγs + ∂t3γt3 +∇∇∇∇ T γt1) [ ΦeC +γt3 γγγγs. AesC + γt3 γt1 ΦΦΦΦeT+ γγγγs.γt1AesT] = 0 (18)

Subdividing the terms

γt3 ( ∂t3γt3 – ∂∂∂∂s. γγγγs)[γt3 ΦeC + γγγγs. AesC ] – ∇∇∇∇ T [ ΦΦΦΦeT + γt3 γγγγs .AesT] = 0 (19)

γt1 (∂t3γt3 + ∂∂∂∂s. γγγγs)[ γt3 ΦΦΦΦeT+ γγγγs.AesT] + ∇∇∇∇ T* [ ΦeC + γt3γγγγs. AesC ] = 0 (20)

Select terms in equ.(19) noting that γt3 ϕs γγγγs = γγγγs i S:

γt3 [∂t3 ΦeC + ∂∂∂∂s.AesC – ∇∇∇∇ T ΦΦΦΦeT] = 0 (21)

γγγγs.[∂t3AesC + ∂∂∂∂sΦeC – ∇∇∇∇ TAesT + i S(∂∂∂∂s x AesC)s ] = 0 (22)

Comparison of equs. (19) and (20) lead to the related equations:

γt1 [∂t3 ΦeT − ∂∂∂∂s. AesT +∇∇∇∇ T* ΦeC ] = 0 (23)

γt1γt3γγγγs. [ ∂t3AesT − ∂∂∂∂sΦeT +∇∇∇∇ T *AesC − i S (∂∂∂∂s x AesT)s] = 0 (24)

Gauge transformations that leave the equations unaltered are given from

AesC →→→→AesC + ∂∂∂∂s χC ; ΦeC →→→→ ΦeC – ∂t3 χC + ∇∇∇∇ T χT (25)

AesT →→→→AesT + ∂∂∂∂s χT; ΦeT →→→→ ΦeT + ∂t3 χT + ∇∇∇∇ T* χC (26)

provided that

∂2t3 χC/T – ∂∂∂∂s. ∂∂∂∂s χC/T = –∇∇∇∇ T∇∇∇∇ T* χC/T (27)

As noted, AesT/C contain terms in i, S and iS so that i SAesT/C are re-arrangements of

the coefficients in AesT/C and do not invalidate equations 22 and 24 where iS appears

as an operator on the fields. It turns out that it is more helpful to associate the term i S

with ΦeC/T and the operator ∂∂∂∂s so as to re-arrange the equations 21-24 as:

γt3[∂t3(– i SΦeC) −i S ∂∂∂∂s.AesC –∇∇∇∇ T (i S ΦΦΦΦeT)] = 0 (28)

γγγγs.[∂t3AesC + i S∂∂∂∂s (– i S ΦeC) – ∇∇∇∇ TAesT + i S(∂∂∂∂s x AesC)s ] = 0 (29)

γt1 [∂t3(– i SΦeT) + iS∂∂∂∂s.AesT + ∇∇∇∇ T* (– iSΦeC)] = 0 (30)

γt1γt3γγγγs.[∂t3AesT − i S ∂∂∂∂s (– i SΦeT) + ∇∇∇∇ T *AesC − i S (∂∂∂∂s x AesT)s] = 0 (31)

Given an arbitrary four component vector F1,F2,F3,ψc it is possible to follow a

technique given by Greiner [52] and write:-


16

i

∂+∂−∂∂+∂−∂∂+∂−∂∂−∂−∂−

C

C

C

Ψ)FF(Ψ)FF(Ψ)FF(.F.F.F

31221

23113

12332

332211

= 1

000000

000000

∂

−

−

ii

ii

+ 2

000000

000000

∂

−

−

ii

ii

+

∂

−

−

3

000000000

000

ii

ii

ψ

3

2

1

FFF

C

= ξ1 ∂1+ ξ2∂2 + ξ3∂3 = ξξξξ.∂

ψ

3

2

1

FFF

C

(32)

where the three matrices ξ1 , ξ2 , ξ3 may be written as partitioned matrices:

ξξξξ = ξ1 , ξ2 , ξ3 =

σσ

2

2

00 ,

σσ−

00

3

3

ii ,

σσ−

00

1

1

ii

Here 2 x 2 hermitian matrices σn are the conventional Pauli matrices. The three 4 x 4

hermitian matrices ξn have the same commutation properties amongst themselves as

the three Pauli matrices σn. Consequently the ξn matrices, like the Pauli matrices, can

be considered as spin matrices with eigen values of +/– 1. The extra dimensions

simply recognise the extra dimensions of 3+3 space-time. Now define

Ae4-C =

Φ−

Ces

Ces

Ces

AAAiS

3

2

1

eC

; Ae4-T =

Φ−

Tes

Tes

Tes

AAAiS

3

2

1

eT

; (33)

Equ. 28 to 31 combine to give the coupled pair of ‘vector’ equations:

∂t3 Ae4-C + ∂∂∂∂s. ξξξξ SAe4-C = ∇∇∇∇ T Ae4-T ;

∂t3 Ae4-T − ∂∂∂∂s. ξξξξ SAe4-T = –∇∇∇∇ T*Ae4-C ;

It also is seen that each individual component of either Ae4-T or Ae4-C satisfies a Klein-

Gordon equation:

∂2t3 Ae4-C/T – ∂∂∂∂s. ∂∂∂∂s Ae4-C/T = –M0

2 Ae4-C/T (35)

Define an 8 component vector Ψ Ψ Ψ Ψ = [ ]CT

−−

e4Ae4A and 8 x 8 matrices ααααn = S

− nn

ξξ0

0 ,

along with a 4x4 identity matrix Σ0 and 8 x 8 matrix ββββMo where

ββββ Mo =

Σ∂−Σ∂Σ∂−Σ∂−

00

21

21

otot

otot

ii

. (36)

Normalise so that ββββ2 2 2 2 = 1 with Mo2 = ∇∇∇∇ T∇∇∇∇ T* = ∇ ∇ ∇ ∇ T*∇∇∇∇ T consistent with equation (1).

Note that β αβ αβ αβ αn = – ααααn β β β β and re-write equs. (34) to look like a conventional Dirac

equation with a scalar mass [51]:

i ∂t3 β Ψβ Ψβ Ψβ Ψ + i (∂∂∂∂s. βαβαβαβα) ΨΨΨΨ – Mo ΨΨΨΨ = 0 (37)

)34(


17

The matrices ββββ and β αβ αβ αβ αn in equ. 37 commute in the same manner as the equivalent

matrices in the conventional Dirac equation. Hence any discussion on Lorentz

transformations still follow the usual procedures : see for example Ref. 51 (p.547).

The fact that there are now 8 complex elements in Ψ, instead of 4 is a consequence of

the added dimensions of transverse time. However these extra elements do not make

any key difference to Lorentz transformations. The matrices ααααn are Hermitian and so

this ensures that the usual continuity equation holds:

∂t3(ΨΨΨΨΨΨΨΨ) + ∂∂∂∂s. (ΨΨΨΨ αααα ΨΨΨΨ) = 0 (38)

The interaction between e.m. fields and electrons, using this formalism, has not yet

been developed and any discussion is omitted here.

The ‘wave-vector’ Ψ Ψ Ψ Ψ in equ. 37 is in effect a selection of complex amplitudes of

even-grade multi-vectors. If these are the amplitudes of the elements of different

rotors [46,47] then, prima facie, there could be adverse consequences. A rotor

exp (θ γt3γt1) = cos θ + γt3γt1 sin θ interchanges principal time (γt3) and transverse time

(γt1) while exp (φ γsnγt1) = cosh φ + γsnγt1 sinh φ interchanges transverse time (γt1) and

space γsn. As already noted, such rotations might lead to difficulties in inter-changing

free energy and mass energy in unphysical ways. On these grounds it might be

expected that the coefficients of γt3γt1, γt3γt2 , γsnγt1 ,γsnγt2 should all be zero and

consequently Ae4-T should be zero. However in considering equs. (37), the αααα matrix

is constructed in such a way that a non-zero Ae4-T does not directly rotate free energy

into rest mass energy and appears to have no direct effect in space and principal time,

other than changing the amplitudes of ΨΨΨΨ ααααn ΨΨΨΨ. The extra temporal dimensions

therefore identify an ‘internal spin-space’ without directly interchanging free energy

and mass energy in unacceptable ways.

4.4 Conjugation

One conventional use of i occurs with temporal variations exp (– iωt) [cf.

engineering texts with exp (jωt)]. This avoids separate equations in cos(ωt) and

sin(ωt) by the complex addition of the two phases. This conventional use does not

have a geometrical significance. However in this paper, conjugation changing i into –

i is equivalent to interchanging γt1 and γt2 and reverses the ‘handedness’ or chirality

of time. Conjugation therefore has a geometric interpretation. Features of temporal


18

chirality are discussed further after discussing modal solutions to Maxwell’s

equations.

With 3d-time in this work it is postulated that the orientation of transverse

time cannot be classically observed. Now exp iθ ΨΨΨΨ is equivalent to ΨΨΨΨ experiencing a

‘rotation in transverse time’ (RTT) but ΨΨΨΨΨ Ψ Ψ Ψ is independent of any such rotation (RTT

invariant). The quantity ΨΨΨΨ ααααn ΨΨΨΨ.is also invariant to rotations in transverse time.

Consequently one may speculate that quantum theory may have cognisance of

transverse time but its observables, available for measurement in classical space, have

no such knowledge. Given a property with a quantum Hermitian operator A =A*, the

observable is <A> = <ψ|Α|ψ> is RTT invariant and this is considered here as

circumstantial evidence for the non-observability of transverse time.

5. MAXWELLS EQUATIONS

5.1 The Potentials

In STA, the Maxwell potentials are formed from V = γt3 Φ + γγγγs.A where

Φ and A are real. With 3+3 GA with the preferred transverse temporal plane, there are

two similar possible assumptions. One assumption is that V = γt3 γt1ΦeT + γγγγs.γt1AeT .

With γt1 and γt2 and i all invariant with spatial rotations and Lorentz transformations, it

follows that AeT and ΦeT still transform as the A and Φ of STA except that AeT and

ΦeT are ‘complex’, eg: γt1AeT = γt1A1e+ γt2A2e = γt1(A1e+ iA2e). In other words there

are two sets of 3d-space –1d-time vectors: (Ae1 , Φe1) and (Ae2 , Φe2). The first is

associated with the directions γt1 and the second with γt2.

Alternatively, following STA more closely, assume that V = γt3 ΦoC + γγγγs.AoC

where ΦoC = ΦoR + i ΦoI ; AoC = AoR +i AoI . Again spatial rotations and Lorentz

transformations transform the complex ΦoC and AoC in the same way as Φ and A in

STA. If the transverse temporal axes are rotated by an angle θ then the complex

fields will change by a phase factor exp iθ. Which of the two assumptions is correct

is not clear at the present stage of this theory but because both assumptions lead to

similar conclusions it is this latter assumption that is made here because it is closer to

the assumptions of STA. The analysis follows that for the Dirac equations but is

modified slightly to account for taking the odd grade components, rather than even

grade components. Similar to the system of equations for the Dirac equation, it is


19

again proposed that using the differential, as suggested by equation 8 along with the

concept of an analytic multi-vector, the first order derivatives are set to zero :

( ∂∂∂∂s. γγγγs + ∂t3γt3 +∇∇∇∇ T γt1) (γt1ΦoT + γt3 γγγγs.γt1AosT + γt3 ΦoC+ γγγγs.AosC ) = 0 (39)

This leads to the set of four equations:

−γt1 γt3[∂t3 ΦoT + ∂∂∂∂s.AosT – ∇∇∇∇ T *ΦΦΦΦoC] = 0 (40)

γt1γγγγs.[∂t3AosT + ∂∂∂∂sΦoT – ∇∇∇∇ T*AosC + i S(∂∂∂∂s x AosT)s ] = 0 (41)

[∂t3 ΦoC − ∂∂∂∂s. AosC +∇∇∇∇ T ΦoT ] = 0 (42)

γt3γγγγs. [ ∂t3AosC − ∂∂∂∂sΦoC +∇∇∇∇ T AosT − i S (∂∂∂∂s x AosC)s] = 0 (43)

Noting that E and B fields in STA may be combined into a single bivector γt3γγγγs.F , it

is surmised that equ. (42) and (43) should be the cornerstones in defining E and B

fields in ‘3+3’ GA. A first attempt might become more transparent by writing

ΦoC = φ with AosC = − ΑΑΑΑ (44)

and then suggest that

−∇∇∇∇ T AosT = [ − ∂t3A − ∂∂∂∂sφ + i S (∂∂∂∂s x A)s] = E + i S B = F (45)

This view will be modified shortly but gives the right direction for the argument.

5.2 The Fields E and B

Now setting t3 = t (remembering that c = 1 in the units here) and removing the explicit

grades of vectors, the equations 40-43, with the suggested changes of equs.(44 - 45),

are re-written as

∂t3 ΦoT + ∂∂∂∂s.AosT = ∇∇∇∇ T *φ = GT (46)

γt1γγγγs.[∂t3AosT + ∂∂∂∂sΦoT + + i S(∂∂∂∂s x AosT)s ] = −∇∇∇∇ T*A = FT (47)

∂t3φ + ∂∂∂∂s. A = −∇∇∇∇ T ΦoT = GC (48)

−∂t3A − ∂∂∂∂sφ + i S (∂∂∂∂s x A)s = −∇∇∇∇ T AosT = F (49)

Because photons are massless, ∇∇∇∇ T∇∇∇∇ T* = ∇∇∇∇ T*∇∇∇∇ T = 0. However this does not

mean that both the operators ∇∇∇∇ T and ∇∇∇∇ T* must return zero. In order to obtain non-

zero sources and fields, GC= – ∇∇∇∇ T ΦoT and F = – ∇∇∇∇ TAosT are retained as non-zero.

However for zero photon mass, ∇∇∇∇ T*∇∇∇∇ T ΦoT = ∇∇∇∇ T*∇∇∇∇ T AosT = 0 and ∇∇∇∇ T∇∇∇∇ T* φ =

∇∇∇∇ T∇∇∇∇ T*A = 0 so that GT = ∇∇∇∇ T* φ = 0 and FT =∇∇∇∇ T*A =0 . Then operation on equ. (47)


20

and (48) with ∇∇∇∇ T* can be seen to make both sides of these equations zero. Using the

definitions of GC and F in equs. (48) and (49), equs. (46) and (47) imply that:

∂t GC + ∂∂∂∂s. F = 0 (50)

∂tF + ∂∂∂∂s GC + i S(∂∂∂∂s x F)s = 0 (51)

Then in the conventional 3+1 space time, it is ∂t GC that appears as the source for the

fields ∂∂∂∂s.F while ∂∂∂∂s GC appears to be the source for the fields [∂tF + i S(∂∂∂∂s x F)s].

Now, the theory so far has been arranged so that in general one could and

should write (GC +S GC/ ) instead of just GC and of course GC and GC

/ are complex.

Similar results hold for all the components (see immediately after equ. 9). In 3+3

space-time, the pseudo-multi-vector is taken to complement the multi-vector. With all

the pseudo-vector components made explicit, equations 50-51 now split into 4

equations:

∂∂∂∂s. F = − ∂t GC ; (52)

∂∂∂∂s. F/ = − ∂t G/

C (53)

∂tF + ∂∂∂∂s GC = − i (∂∂∂∂s x F/)s ; (54)

∂tF/ + ∂∂∂∂s G/

C = − i (∂∂∂∂s x F)s (55)

The excitation of the fields in the equivalent 3+1 space-time is fundamentally

determined by the two complex terms (GC, GC/) so that with appropriate complex

multipliers χ and ξ , it is possible to combine these two excitations into a mixed entity

with a magnitude Q where Q is found from:

χGC +ξGC/ = 0 ; ξGC + χGC

/ = Q ; (56)

with a normalisation given as ξ2 − χ2 = 1 ; χ has the ‘dimensions’ of S while ξ has the

dimensions of ‘unity’ : i.e. ξGC and χGC/ are the same grade of multi-vector as Gc

while χGC and ξGC/ are the same grade of multi-vector as GC

/ . The new equations are

then formed by writing

(ξF + χ F/)= E ; (χF + ξ F/) = i B (57)

∂∂∂∂s. E = − ∂t Q ; ∂∂∂∂s.B = 0 (58)

∂tE + ∂∂∂∂s Q = (∂∂∂∂s x B)s ; ∂t B = − (∂∂∂∂s x E)s (59)

The combined equivalent multivector in 3+1 space is then


21

γt3γγγγs.Fm = γt3γγγγs.(− E + iS B) (60)

Here then the ‘E’ field is a bivector and B field is the pseudo-bivector or 4 vector (not

to be confused with the 4-vectors of 3+1 space-time). The question of why there are

no sources for the B-field in Maxwell’s equations is now answered in this work by the

statement that the ‘B-field is that part of the electromagnetic field which does not

have sources for its divergence’. In other words, the lack of sources is part of the

definition of the B-field and helps to distinguish it from the E-field.

5.4 Sources

Equation (66) and (68) define sources: ‘current’ density J = ∂∂∂∂s Q and ‘charge’

density ρ = −−−− ∂t Q. The continuity equation requires that div J + ∂tρ = 0 and this holds

because Q satisfies a classical massless wave equation. It might be argued that

because electrons have mass, the charge excitation Q should have ‘mass’. However Q

is simply an excitation and not a particle, so there is no a priori reason for Q to have

mass. Reversing time and space reverses ρ and J in keeping with PCT [53].

The next problem is that ρ = −−−− ∂t Q does not look like some spatial density.

This is reconciled by noting that the interpretation of these equations will shortly be in

terms of exciting units of space-time vortices. The average excitation over a space-

time (3+1) volume Ω T is given from ρav = (−−−− ∂t Q) dV dt /Ω T which will give an

average temporal frequency of the number of centres of vortex excitation. An average

spatial frequency of the number of these excitations is similarly given from

Jav = ( ∂∂∂∂sQ) dV dt /Ω T . It follows that 3+3 space-time ties both charge and mass

into the structure of the fields in transverse time, opening up future lines of research

with the potential for a more direct test of this theory, perhaps through mass to charge

ratio. In particular, the theory suggests that the most fundamental particles described

by these fields must be such that zero mass automatically means zero charge, though

zero charge does not mean zero mass. The theory would then support the statement

that the neutrino, with zero charge and not being a combination of more fundamental

particles, must have zero mass. There is no necessary conflict with the fact that

massive particles can have zero charge provided that such massive particles are

composites of fundamental particles. The net (positive plus negative) charges of the


22

contributory fundamental particles could be zero while the mass of course would

combine.

The 3+3 space time formalism has allowed Maxwell’s equations to be fully

recovered along with charge, current and an explanation of why there are no magnetic

sources. The key change from the classical version is that the equations are in general

complex; real fields are a special or simplified case. The proposed new arguments,

following STA as closely as possible, differ from ref. [40] where the two components

of the spatial E field were associated with direction γt1 and direction γt2 respectively.

The complex fields may at first sight appear embarrassing because we measure

only real power and real energy. Repeating the postulate that classical measurements

cannot distinguish the orientation of transverse time, it will be terms like E*.E that

are observable etc and these will form the energy with related terms for power flow.

In GA terms, it is noted E E = (γt1E1+ γt2E2)(γt1E1+ γt2E2) = E1.E1+ E2.E2

5.4 RTT invariant power

The real world of power and energy can be related to the complex world through

standard techniques for Poynting’s theorem [54] giving:

V div [(E* x B) + (E x B*)] dV + V [(E*. J) + (E . J*)] dV

= – V ∂t [(E*. E) + (B*. B)] dV (61)

This is interpreted in the standard way. The divergence term gives the outward power

flow [(E* x B) + (E x B*)] from a surface Su enclosing a volume V . The current term

gives the energy lost within the enclosed volume V. The sum of these two terms

equals the rate of loss of the stored energy. This form of Poynting’s theorem is RTT

invariant e.g. E*. E = [exp (iθ) E]*[exp (iθ) E] etc.. Hence energy and power flow

are RTT ‘observables’.

This theory will be applied to E and B fields with a k-vector k. We shall be

concerned with propagation of energy along the direction +/− k so that in source free

regions the zero divergence of the field requires that k.E = k.B = 0 . Taking then a

unit vector n || k , then n.E = n.B = n.E* = n.B* = 0. In source free regions equ. (61)

may be written for this special case as

V ∂∂∂∂s.n [(E*xB) + (ExB*)].n dV = – V ∂t [(E*.E) + (B*.B)] dV (62)


23

A complementary equation, derived from the same basic Maxwell’s equations is

V ∂t [(E* x B) + (E x B*)].n dV = − V ∂∂∂∂s.n (E*.E +B*.B) dV (63)

In Appendix 2, it is shown that for changes of velocity along the direction n, then both

(E*.E +B*.B) and [(E* x B) + (E x B*)].n are Lorentz invariants. This is not true for

(E.E +B.B) and [(E x B) + (E x B)].n .

5.5 Normal Modes

Initially only one relative orientation of the three temporal axes γt1 γt2 and γt3

are considered: this will be called positive temporal chirality. The additional

complications of chirality are then for the moment ignored. The field pair E , B can

be treated in the same manner as Cohen-Tannoudji et al. [55] to find normal modes

with real wave-vectors k where k.k = k2 ; k > 0. Even though the bivector i not

implies a rotation of the transverse temporal axes Ot1 and Ot2 by 90o one may still

write cos kx = ½(exp –i kx + exp i kx) and sin kx = ½ i (exp –i kx − exp i kx). The

standard Fourier analysis E^(k, t) = E(x, t) exp (– i k.r) dx can still be used but of

course E(x, t) is complex so that E^(k, t)* ≠ E^(–k, t) . This is the distinct difference

from reference 55 with real fields. Because we shall deal with separate k-components

we write:

E (r, t ) = Σk E^(k, t) exp(i k.r) (64)

The time varying amplitudes E^ [or B^] Fourier analyse the fields E (r, t) [or B(r, t)].

The summation may be replaced with an integral if more appropriate but at the

moment only single k-vector is now considered. From div E = 0 and div B = 0 :-

k.E^(k, t) = 0 ; k.B^(k, t) = 0 ; (65)

From vector identities: curl curl E = k2 E. Hence, with k > 0,

k (k E) = ∂t (curl B) ; k (curl B) = ∂t (k E) ; (66)

i ∂t α α α α − k α α α α = 0 ; k α α α α = ( k E + i curl B) ; (67)

i ∂t β β β β + k ββββ = 0 ; k ββββ = ( k E – i curl B) ; (68)

The α α α α modes have only ‘positive’ frequencies, while β β β β modes have only ‘negative’

frequencies, where ‘positive’ implies the variation exp(– ikt); k > 0. Complex E and

B means that αααα* ≠ ββββ unlike ref. 55 with real E and B.


24

Any complex signal with only one sign of frequency, creates an analytic signal

[56] in the principal time t . As discussed in Appendix 1, the groupings in equs. (67)

and (68) create complex vectors αααα and ββββ which are analytic in a complex time t = t +i τ.

Although it has to be admitted that the original geometrical significance of i is no

longer clear when dealing with the term i τ , the fact that i2 = –1 must allow the

variations exp(i k t) with k > 0 to follow the formal algebra of complex analytic

functions. The analytic normal modes, normalised by some factor N, yet to be

determined , can now be expressed as:-

αααα (k) = Ν −½ [E^(k, t) – (k/k) x B^(k, t)] (69)

ββββ (k) = Ν −½ [E^(k, t) + (k/k) x B^(k, t)] (70)

The normal modes are the outcome of Fourier analysis of space and consequently are

local in k-space but non-local in space. To localise spatially the energy in these modes

it will be necessary to discuss a superposition of k-vectors.

From equ.(65), both E^ and B^ are normal to k and hence k.αααα = k.β β β β = 0.

Consequently αααα and ββββ each have two independent spatial components normal to the

vector k. We order these components and write the scalar product αααα(k)*.α.α.α.α(k) as

αααα(k) αααα(k) [similarly for ββββ(k)].

5.6 Stored energy and energy transfer

The stored ‘energy’ U and ‘energy transfer’ P are given in a normalised form

as:-

U(k, t) = < ½(E^*....E^ + B^*....B^) >

= = = = (1/4)Ν < αααα(k)αααα(k) + ββββ(k)ββββ(k) > (71)

P(k, t).(k/k) = < ½ (k/k).[E^ x B^* + E^* x B^] >

= (1/4)Ν < αααα(k) αααα(k) − ββββ(k) ββββ(k) > (72)

where < > denotes integration over some volume, say V formed by an area A [with a

vector normal in space of (k/k)] and length L [with L lying along the direction (k/k)].

In the classical problem αααα* = β β β β so that over the volume V, containing no soruces, the

energy transfer P is zero. This is equivalent to saying that inflow of energy that

carried by the normal modes at one end, is the same as the outflow of energy at the

other end of V.


25

5.7 Forward, reverse, retarded and advanced waves

The volume V described above, can be compared with a classical resonant

cavity where waves, varying as exp[ j(k.x – k t)] and exp[ j(– k.x – k t)], travel in the

forward and reverse directions [57] along the cavity (volume V). In this work, these

waves are the α(±k) modes with positive frequencies k > 0 and are travelling forward

in time like conventional classical waves and can be referred to as ‘retarded forward’

waves and ‘retarded reverse’ waves. If the excitation is at x = 0, then the reverse

wave exp[ j(– k.x – k t)] can be thought of as the reflection of the forward wave

exp[ j(k.x – k t)] from the ‘far’ end of the cavity. Standing waves form with nodes

spaced ½ wavelength apart. The forward and reverse waves can then be considered to

form a series of loops. The number of loops within the length of the resonant cavity is

a relativistic invariant.

Now within the volume V, unlike the classical resonant cavity, there are waves

with negative frequencies. These are the β(±k) modes varying as exp[ jk.x – j(– k) t]

and exp[ –jk.x – j(– k) t] and are said to have negative frequencies (– k) < 0.

Although the wave exp[ j(– k.x + k t)] is travelling in the forward spatial direction k,

the wave is said to be ‘advanced forward’ wave travelling backwards in time.

Similarly the wave exp[ j(k.x + k t)] is an ‘advanced reverse’ wave. Traditionally,

advanced waves are discarded but form an important part in the theories of Cramer [1]

and of Wheeler and Feynman [58] . These waves are also retained here.

Now in this work there is a need for clarity about distributed energy. The fields

with a given k-vector are non-local and consequently the associated distributed energy

U (equ.65) is also non-local: ie distributed over the whole of space just as the field

with a single value of k. Only a superposition of different k-vectors can localise these

fields and so localise the energy U. Now the quantity P (equ.66) determines the

energy flux at the boundaries and this is then localised to the sources and sinks that

generate the energy. Conventional wisdom might argue that positive frequencies

(α−modes in this work) should contribute positively to the energy while negative

frequencies (β−modes in this work) should contribute negatively. Equ. 66 shows this

to be correct for localised energy flux P at the boundaries. However equ. 65 shows

that both α- and β- modes contribute positively to the non-localised energy U. If α-

and β- modes both increase in energy then there is a non-local increase in U which


26

would violate energy conservation and causality. Local excitations of energy require

increases of energy in the α-modes to be compensated by decreases in the energy of

the β- modes (or vice-versa) so that U does not change. This problem is revisited

presently.

5.6 Promotion/demotion/annihilation

Only one wave-vector value k is considered for the moment. Define

Θ(k, k) = exp[i(k.r – kt)] (73)

The normal modes can be designated as a ‘retarded’ forward mode αααα(k) = Ak Θ(k, k)

and an ‘advanced’ forward mode ββββ(–k) = B–k Θ(–k, –k). The positive frequency mode

travels in the direction of positive time and direction k. The negative frequency mode

is regarded as travelling backwards in time along the same path as αααα(k). The

amplitudes Ak and B–k are to be normalised later. There are also a ‘retarded’ reverse

mode αααα(k) = A–k Θ(–k, k) and an ‘advanced’ reverse mode ββββ(k) = Bk Θ(k, –k) where

the fields travel in the –k direction.

Now consider a ‘modulator’ aκ = Aκ Θ(κκκκ, κ), (κ > 0, κ κ κ κ || k) as an analytic

signal or rotor which modulates αααα(k) and thereby generates (or promotes) a higher

frequency analytic mode as follows:-

αααα(k + κ κ κ κ) = aκ αααα(k) (74)

where αααα(k+ κ κ κ κ) = A(k+κ) Θ(k+κκκκ, k+κ) with (k+κ) > 0. The normalisation again will be

determined later. Similarly a ‘rotor’ aκ = A/κ Θ(κκκκ, κ)–1 (κ > 0, κ κ κ κ || k), can modulate

the fields and demote αααα(k) to a new lower frequency mode:

α α α α(k κκκκ) = aκ αααα(k) (75)

where αααα(k– κ κ κ κ) = A(k–κ) Θ(kκκκκ, k–κ) : (k–κ) > 0

Once again, normalisation will be determined later.

As discussed, Fourier analysis in space gives non-local amplitudes so that the

normal modes αααα(k) and ββββ(k) are non-local. Changing αααα(k) to αααα(k+/– κ κ κ κ) gives

frequency modulation resulting in a new non-local modal amplitude. There are now

two points to make. First, if the energy is to be localised then there has to be a group

of such modes. Modulation is the start of producing a group of relevant non-local


27

modes. Second, if the mode αααα(k+/– κ κ κ κ) has a different energy from the mode αααα(k) then

this is a non-local change of energy. Such a change by itself would give rise to a

potentially boundless increase in energy, dependant on the extent of space over which

the αααα mode was defined. This points to the idea that, there must be a sufficient

number of normal modes appropriately modulated at any one point of space so that

the global (non-local) energy remains constant.

Τhis process of modulation by a signal aκ may look trivial but αααα is a complex

analytic function of time with positive frequencies. Multiplying analytic functions

together (modulation) has special rules [56]. If the modulation process attempts to

make (k–κ) ≤ 0 then, from the Appendix 1, aκ αααα(k) = 0 . This result for analytic

signals states that attempts at reversing the sign of the frequency in an analytic signal,

through modulation, results in making that signal zero: i.e. causes annihilation. The

real and imaginary components in the modulated signal are cancelled by the

modulation process. This is standard complex signal analysis; it does not require

quantum theory. However, just as in quantum theory, promotion and demotion

operations can result in ‘ladders’ of frequencies k +/–Nκ with corresponding wave-

vectors. The frequency κ is not at present restricted to any particular value but care

will have to be taken when ‘annihilation’ occurs.

Frequency reversal caused by modulation annihilates a mode rather than

change the sign of the frequency. Frequency reversal can be accomplished in a

different way e.g. via conjugation. From Appendix 1, αααα is analytic in the lower half

complex plane of t while αααα* is analytic in the upper half plane. The demotion

(promotion) operator for αααα* is equivalent to the promotion (demotion) operator for αααα

but still any reversal of frequency by modulation leads to anni-hilation. It follows that

for the conjugated normalised fields one may write

αααα(k + κ κ κ κ) = αααα(k) aκ; αααα(k – κ κ κ κ) = αααα(k)

aκ (76)

Now in this case the modulation operators or rotors aκ and aκ are to operate

backwards on the modal vectors αααα(k). Again it is essential that (k–κ) > 0 if

annihilation is to be avoided.

Similar promotion and demotion operations arise for ββββ(k). These are bκ =

Bκ Θ(κκκκ, –κ) and bκ = B/κ Θ(κκκκ, –κ)–1 with the same rule for analytic signals that


28

annihilation occurs whenever one attempts to force (k–κ) ≤ 0. Indeed ββββ*(–k) and

αααα(k) have identical analytic properties. These are key steps. Complex Maxwell’s

equations provide analytic modes that where modulation can promote, demote and

annihilate analytic signals with ladders of frequencies. At present no specific ladder

has not been established. Annihilation does not require any appeal to quantised

harmonic oscillators or minimum energy states.

6. SPACE-TIME VORTICES

6.1 Concept of space-time vortices

Traditionally the advanced solutions are discarded but here they are retained as

an essential part of the theory. Equs. (69 - 72), for a given k (k > 0), contain both

retarded and advanced waves. The same equations can be repeated with k replaced by

–k (with the same k > 0 as before). Now ββββ(–k) = Bk Θ(–k, –k) is the forward wave

while αααα(–k) = Ak Θ(–k, k) is the reverse wave. This suggests that, for the positive

temporal chirality that is being considered here, the energy is transported through the

agency of two forward waves αααα(k), ββββ(–k) and two reverse waves αααα(–k), ββββ(k). Here

the positive frequency α-waves are the classical retarded waves with the β-waves the

classically discarded advanced waves. Now all of these waves contribute positively to

the energy [U(k, t) + U(–k, t)] within any volume V. The simultaneous appearance of

positive and negative frequencies is required in any temporal Fourier analysis where

any change of field δF occurs at a definite time so that δF(t) = 0 for t < to [59].

Feynman used this argument to justify the need for anti-particles [60]. Similarly any

change of field δF occurring at a definite plane of space requires both +/–k wave-

vectors.

Define the term ‘direction’ to mean either k/k or –k/k (k > 0) without regard

to sign. Then the concept of space-time vortices (STV) is that with a single direction

of wave-vector, both advanced and retarded waves with both positive and negative

frequencies are excited. That is four waves appear with wave-vectors +/– k and

frequencies +/–k (k > 0) for a given k. This is equivalent to an assertion that signals

circulate in time and space. Such an STV may not look like a conventional whirlpool

because attention is confined to one spatial direction and to k-space rather than space.

Attention is turned once again to a lossless resonant microwave cavity with waves in

the direction +/– k/k . In such a cavity there are standing waves with nodes at every


29

half-wavelength. The distance between nodes in the present paper is called a ‘loop’

because it is possible to think of a forward wave and a reverse wave completing the

circuit between the nodes in a loop. The number of such loops within a fixed length

in a resonant cavity is a relativistic invariant and is proportional to the frequency. The

vortex concept is analogous to such a cavity with the number of loops being the

number of vortices. The positive and negative frequencies for a given k form standing

waves which then give ‘loops’ formed by the retarded and advanced waves. The

concept then is that there is a fixed energy per loop for a given type of excitation. The

energy per vortex loop must be proportional to the frequency for compatibility with

special relativity. The difference between the vortex loops and the classical theory of

the resonant cavity is that the vortex loop has distinct α and β type modes which

contribute differently to the power flow.

6.2 Vortex excitation and detection

It is now supposed that vortices already exist. For example, it is possible to

envisage that the quantum mechanical ground state is a ground state of vortices but

further discussion is provided shortly. The problem then is how to excite the vortices

to higher energy states and then detect them. This excitation and detection must give a

uni-directional flow of energy without violation of either causality or special

relativity. More work needs to be done on the interaction between ‘electrons’ and the

‘electromagnetic’ field within this theory so that in the present state of development

only an outline can be given. Equ. 61 suggests that observable energy can be excited

through J*.E + J.E*. An oscillating current J (= ∂∂∂∂sQ in equ. 59) is regarded as the

source of the modulators aκ = Aκ Θ(κκκκ, κ) and similarly aκ. It is recognised that in

geometric algebra, writing a complex field αααα = ααααt1 +i ααααt2 is a short cut a geometrical

interpretation. Here for example, γt1αααα = γt1ααααt1 +γt2 ααααt2 . Then noting that i γt1 = – γt1i ,

the rules of GA show that the order of modulation matters:

aκ a

κ γt1αααα = γt1 (αααα a2κ) or aκγt1αααα a

κ= γt1 αααα (77)

It can be seen how the ‘same’ double modulation can give either a double change in

the frequency of a mode, or leave the frequency unaltered.

As already noted the fields, αααα(k) and ββββ(k), are local to k-space but are non-

local mode-amplitudes in space. Frequency modulation of αααα(k) and ββββ(k) is therefore a


30

non local excitation. Consequently if one modulates such a non-local field and then

asserts that vortex loops represent energy, there has to be a non-local change of

energy! Such a change, taken in isolation, would then be at variance with all concepts

of causality and special relativity. However by modulating all four waves of the

original space-time vortex, a method is found of conservation of the non-local energy

but yet obtaining local energy transfer.

Modulate the forward retarded mode αααα(k) → αααα(k + κκκκ) and the reverse retarded

mode αααα(–k) → αααα(–k – κκκκ) . This makes sense because at the excitation one does not

know if the energy will be detected on the ‘left’ or the ‘right’ so that both options are

kept open by exciting both of these waves. These two waves together will form vortex

loops (c.f. standing waves as in a resonant cavity) with increased numbers of α vortex

loops from the original ‘standing waves’ αααα(k) and αααα(–k). At the point of excitation,

there is no knowledge as to whether the excitation will be detected in the +k direction

or the –k direction. Consequently energy is given to both αααα(k) and αααα(–k). The

increase in numbers of vortex loops increases the global energy and must be

compensated by a simultaneous modulation of the advanced mode ββββ(k) → ββββ(k κκκκ)

and retarded mode ββββ(–k) → ββββ(–k + κκκκ) to give a reduction in the number of β vortex

loops so that the global energy remains constant. It will be a requirement that there is

sufficient energy in the ββββ-waves and this will have to be discussed shortly.

There would then be no net non-local change in the number of vortex loops

and thus no additional non-local change of energy U. However the power flow

changes from the original quantity [P(k) + P(–k)] . The changes would give an

inward flow of energy in the +k fields compensated by an outward flow of energy in

the –k fields at the excitation.

Detection of these excited vortices is now considered to be an entirely separate

process from excitation and requires a further interaction with the electronic charges.

There is another non-local modulation of the non-local waves, but now by the detector

discerns the difference between advanced and retarded waves in the forward and

reverse directions. What will this second modulation achieve?

First the detection process decides from the geometry whether is it detecting a wave

on the ‘left’ or the ‘right’ – remember that both excitations were made available.

Assume therefore that we are detecting the energy in the forward wave at a later time


31

than the excitation of this wave. Consequently we are detecting the retarded wave in

the +k direction, the sign now being of importance. Second, because the reverse wave

is not being detected one expects that energy change in this retarded reverse wave is

restored to its unmeasured value so that αααα(–k– κκκκ) → αααα(–k – κ κ κ κ + κκκκ) = αααα(–k). Now the

same rule must apply that there cannot be a non-local global change of the stored

energy. In other words the total number of vortex loops over an extensive non-local

region shall not change ensuring zero non-local change in the vortex energy from the

numbers of vortex loops. The proposed modulation for the detector of the forward

retarded wave then makes simultaneously the changes: (i) αααα(k + κκκκ) to αααα(k +2κκκκ) (ii)

αααα(–k– κκκκ) to αααα(–k – κκκκ+ κκκκ) (iii) ββββ(k–κκκκ) → ββββ(k – κ κ κ κ + κκκκ) (iv) ββββ(– k + κκκκ) → ββββ(– k + 2κκκκ).

Considering the interference of the advanced and retarded waves in any length L, the

number of αααα-vortex loops remain as (k +κ)L while the number of ββββ-vortex loops

remain as (k – κ)L. Non-local loop numbers are conserved. However the final energy

balance must be given from:

U =(1/2)Ν <[αααα αααα(k+2κκκκ) + ββββ ββββ(–k +2κκκκ)]forward + [αααα αααα(–k) + ββββ ββββ(k)]reverse > (78)

P =(1/2)Ν <[αααα αααα(k+2κκκκ) – ββββ ββββ(–k +2κκκκ)]forward + [αααα αααα(–k) – ββββ ββββ(k)]reverse > (79)

Here the modes have been re-grouped into retarded (or forward waves) and advanced

(or reverse waves). The energy transfer in the reverse direction [αααα αααα(–k) – ββββ ββββ(k)]reverse

balances to zero with the normalisation for αααα and ββββ modes being the same. It was

already noted that ββββ(– k + 2κκκκ)* has the same analytic properties as αααα( k 2κκκκ) so with

normalisation, the scalar ββββ ββββ(– k + 2κκκκ) may be numerically calculated from αααα αααα(k 2κκκκ)

giving the forward energy in positive frequency modes as:

Uf =½Ν <[αααα αααα(k + 2κκκκ) + αααα αααα(k 2κκκκ)]> (80)

Pf =½Ν <[αααα αααα(k + 2κκκκ) – αααα αααα(k 2κκκκ)]> (81)

Now use the promotion and demotion operators of section 4, but with a frequency of

2κ, instead of κ, to give:

Uf =½Ν <[ααααk a2κ a2κ

ααααk + αααα ka2κ

a2κ ααααk]> (82)

Pf =½Ν <[ααααk a2κ a2κ

ααααk – αααα ka2κ

a2κ ααααk]> = Uo (83)

Here Uo is the smallest unit of energy transfer associated with the frequency of

modulation given from the excitation frequency 2κ. The observable promotion and


32

demotion operators only come as a2κ and a2κ and it is only these that have to be

evaluated.

The two distinct stages, excitation and detection, are now seen as essential

features of this new theory drawing on a ‘3+3’ GA. The minimum energy transferred

per interaction is the energy associated with 2 vortex loops (eg energy in one whole

wavelength of the 2κ modulation). This final energy transfer Uo is connected with the

forward waves with a group velocity c so that the detection process cannot succeed

until the energy has arrived. There is no violation of causality in spite of non-local

fields.

Now return to the question of whether or not there will always be sufficient

energy in the β-fields to allow the envisaged interaction. If experiments could be

devised to ensure no energy in the β-modes then interaction should not be possible.

On a speculative note, one may recall that Cole predicted dark matter from his version

of 6-dimensional relativity[61]. Energy contained in the β-modes would remain dark

and so perhaps dark energy and β-modes may be linked.

6.3 Quantisation

Assume that equations 82-85 hold for all k-vectors accessible by promotion,

demotion and annihilation using the operators a2κ. Then these equations are identical

to the equations of the harmonic oscillator with a frequency 2κ. The standard operator

algebra can now lead to the same ladder of quantised frequencies, wave-vectors and

modes as in the harmonic oscillator with k = kN = (2N+1) κκκκ. However it is only the

algebra that is being used and the route to quantisation here is through: (i) the

properties of analytic signals; (ii) a concept of conservation of vortex loops ; (iii) that

energy is transferred by the retarded mode alone. Making use of standard operator

algebra one then finds:

< αααα(kN) a2κ a2καααα(kN) > = (N+1)Uo ; (84)

< αααα(kN) aκ aκ αααα(kN) > = NUo (85)

Consistency requires that k0 = κκκκ the ground state before annihilation with Uo ∝ 2κ so

that, as already noted, the energy per loop is linked with frequency and this frequency

is now shown to be the excitation rotor or modulator frequency.


33

Another point of consistency comes with considering the ground state (wave-

vector κκκκ) which is never observed. In a lossless resonant e.m. cavity, it can be argued

that the fields have been excited but not detected. Any ‘detection’ of the standing

waves is only by sampling the fields. The vortex scheme then argues that the ground

state mode αααα(κκκκ) is modulated by the excitation component aκ changing αααα(κκκκ) to

αααα(2κκκκ). The second modulation for detection does not apply because the cavity is

lossless. The longest wavelength of any such trapped observable α α α α mode is then

correctly given from λ = π/κ, as measured in a resonant cavity.

6.4 Causality, Localisation and Uncertainty

The scale of a space-time vortex loop is determined by 1/|k|||| . Modulation of

the fields adds or removes vortex loops. With an STV, the scale of a vortex loop

1/|k| | | | can in principle be decreased indefinitely while retaining the same normalised

minimum energy per whole loop (keeping the excitation frequency κ constant). To

excite the minimum energy over any spatial length ∆x , one looks at vortices with

scales in the range |kN|∆x ~ 2π and requires an interaction changing frequencies kN to

kN+1 increasing the field energy. To remove a unit of energy, there must be changes in

the frequencies kN+1 into kN. The uncertainty principle of course arises from standard

Fourier requirements where an increasing spread of values of |kN| is required for a

smaller spread in spatial positions [56].

Causality follows from the fact that it is the forward or ‘retarded’ modes that

carry the detected energy while the energies transported by the ‘advanced’ fields

cancel. Although modulation with a single k-value shows how quantisation occurs, a

group of k-vectors must be considered to see that energy has to be carried at the group

velocity. Because only energy in the forward wave is detected this wave determines

the group velocity and speed of energy transfer.

An unpublished preliminary statistical treatment of excitation has considered

how many ways additional loop pairs can be added into a sea of vortex loops

following classic arguments such as that given in Huang (p192 et seq)[62]. This

suggests that there is justification for using boson statistics in discussing such loops,

thereby supporting this speculative new approach.


34

7. CHIRALITYAt the end of section 4, the concept of temporal chirality was introduced. At present

γt1,γt2 and γt3 have been assumed to form a ‘right hand’ set of temporal axes with γs1,γs2

and γs3: a right handed set of spatial axes. Interchanging γt1 and γt2 changes

i = γt1γt2 into – i and changes right handed temporal chirality into left handed

temporal chirality. It is argued here that the two chiralities are independent. For

example if a ‘right handed’ system has fields E+ + iS B+ = [(Er+iEi) + iS(Br+iBi)] and

‘left handed’ fields have E – iS B = [(E/r–iE/

i) – iS(B/r–iB/

i)] then these fields may be

combined to give a new complex field of mixed chirality:

E +iS B = (Er+iEi +E/r–iE/

i) + iS (Br+iBi – B/r+iB/

i ) (86)

With classical Maxwell equations there are only a real fields E and B. Although

(E++ iS B+)* has left handed chirality, it is not necessarily equal to (E– iSB). The

additional freedom by using chirality has important consequences .

Positive frequency modes with ‘positive’ temporal chirality are written with a

subscript + as α α α α ++++= (k E++++ + i curl B++++). The chirality can be changed but to keep a

positive frequency for all α-modes, the α and β modes are simultaneously

interchanged. Hence the mode with positive frequency and ‘negative’ temporal

chirality is written as α α α α − − − − = (k E−−−− + i curl B−−−−) . The whole system of promotion,

demotion, annihilation and quantisation works for the these independent α α α α − − − − (and ββββ−)

modes in exactly the same way as for α α α α + + + + (and ββββ+).

The independence of α α α α + + + + and αααα−−−− can be further emphasised by requiring that

α α α α ++++(k) α α α α −−−−(k) = α α α α −−−−(k) α α α α ++++(k) = 0 . (87)

then

αααα(k) αααα(k) = α α α α ++++(k) α α α α ++++(k) + αααα−−−−

(k) α α α α −−−−(k). (88)

Similarly

ββββ(k) ββββ(k) = β β β β ++++(k) β β β β ++++(k) + ββββ−−−−(k) β β β β −−−−(k). (89)

The energies of the two temporal chiralities of the electromagnetic field have to be

added together with these energies independent. Equ. 49 is shown next to lead to

correlated polarisations in the 2 independent modal chiralities.


35

8. POLARISATION

8.1 Stokes Parameters

With fields which are fundamentally complex there can be a lack of physical

clarity as to what is meant by polarisation. Stokes parameters [63] can determine real

physical polarisations from complex fields through the following formalism:

K 0 = α x * α x + α y * α y

K 1 = α x * α y + α y * α x

K 2 = iα y * α x − iα x * α y

K 3 = α x * α x − α y * α y

By normalising K0 to one, K = K1, K2, K3 forms a vector in a 3-d

polarisation space. Subscripts + or – are added to indicate which temporal chirality is

being considered.

Now define matrices Pauli matrices ππππ = π1 , π2 , π3 but without any

implications of spin etc. :

π1 =

0110 ; π2 =

−0

0i

i ; π3 =

−1001 (91)

Reduce the normal modes as before to their orthogonal components perpendicular to

the vector direction k. For positive temporal chirality, define a 3-component array K++++

= K++++n with ππππ = πn so that.

αααα++++ (k) αααα++++(k) =1 + K+.π π π π (92)

A similar relation holds for the negative temporal chirality αααα−−−− and K−−−−.... Then consider

αααα+(k) αααα+(k)αααα−(k) αααα−(k)

= (1 + K+.ππππ)(1 + K−.ππππ) = 1 + K+.K− + i π.π.π.π.(K+ x K−) (93)

Equation 93 has used the commutation properties of Pauli matrices. Equ. 87 now

requires that equ.93 is zero. This means that K+ is ‘anti-parallel’ to K−.making K+.K−

= – 1 and K+ x K− = 0. In other words the two chiralities have correlated

polarisations. Table 2 below interprets this correlation between the two temporal

chiralities. For example

)90(


36

Table 2 Correlation of Polarisation

K1+ = 1 α x = 1/√2 α y = 1/√2 linear: +45;

K1 = –1 α x = –1/√2 α y = 1/√2 linear: –45;

K2+ = 1 α x = 1/√2 α y = i/√2 circular; +;

K2 = –1 α x = 1/√2 α y = −i/√2 circular: –;

K3+ = 1 α x = 1 α y = 0 linear; vert;

K3 = –1 α x =0 α y = 1 linear; horiz.;

8.2 Instantaneous knowledge of polarisation around the vortex

Excitation of STVs creates a non-local web of vortices from normal modes

with +/–k vectors and +/– k frequencies defined from ααααk and ββββk. Like Fourier

analysis of local excitations, the spatial Fourier components are the same everywhere.

Again comparison may be made between an STV and a lossless resonant microwave

cavity. Information about polarisation is known over the whole cavity and similarly

over the whole STV. Causality is not violated by the instantaneous non–local

knowledge within the STV because energy transfer requires the co-operation of

adjacent k-vectors. Differences in these k-vectors have to travel with cg , the group

velocity of light. The proposed vortex system provides instantaneous knowledge

about chirality and polarisation over space but energy extraction occurs locally and

causally as determined by cg.

9. ENTANGLEMENTEinstein first highlighted the problems of entanglement [5] with slightly

abstract discussions about position and momentum. Real experiments have shown

how ‘Photon entanglement’ occurs where measurements of polarisation on two (or

more) photons are always correlated even though the photons are too far apart to be

able to communicate at the speed of light in the time available between the

measurements [64]. Mermin emphasises the role of correlations in QT [65]. An

explanation using non-local vortex concepts could be as follows.


37

Suppose that at the origin, vortices (photons) L and R are excited. The

excitation occurs as described in section 6.2. There are waves propagating in both +/–

k directions with appropriate excitations. These k-waves are non-local, their spatial

extent will be as far as the Fourier analysis extends over space. There is no change in

this non-local energy because of an increase in the number of α vortices and a

decrease in the number of β vortices. Because two photons are produced together, the

energy of the two excitations is split between the positive and negative temporal

chiralities which have correlated polarisations as in table 2. Without any measurement

or preparation of the state of the photons, there are too many degrees of freedom to

determine which chirality and polarisation belongs to L or R. Now comes the

detection process. The boundary conditions for the detection of the ‘R’ photon in the

+k direction will determine what polarisation can be detected at ‘R’ and with what

chirality. The detection process cancels the modal energy in that particular chirality

with the appropriate polarisation. ‘R’ is detected and the energy in the excited

vortices, that are associated with ‘R’, is removed. The only energy remaining is in the

other chirality and hence the polarisation of ‘L’ is determined. ‘L’ will be detected

only by a correctly polarised detector. There is no communication between ‘L’ and

‘R’ because the correlation of their polarisations was determined at the outset as part

of the global space-time vortex excitation. However no use can be made of this

correlation of polarisations as the energy has not arrived at ‘L’. Advanced waves in

the +k direction now become retarded waves when viewed in the –k direction. The

detection of ‘L’ via retarded waves then removes the energy and has to have the

polarisation correlated with ‘R’. There is no ‘spooky action’ at a distance.

It is envisaged that in a one dimensional spatial system, such as a ‘lossless’

optical fibre, the vortices extend over the whole of the fibre. With a non-focussed

system in space, just taking +/– k with a single k-vector is no longer appropriate. The

spherical waves require superpositions of k-vectors which will effectively weaken in

their probability of interaction as one moves away from the source. This is not

considered here. The role of source coherence is also a factor that may play a

significant role.

10. CONCLUSIONSA new interpretation of Maxwell’s equations is explored in which vorticity in

time and space are balanced against each other in a way consistent with Special


38

Relativity. Although 3 dimensions are introduced into time, a preferred ‘transverse’

temporal plane is introduced. Rest mass is interpreted from variations within this

plane. The concept was shown to be consistent with Klein-Gordon and Dirac

equations and also with Maxwell’s classic equations now including terms for charge.

Maxwell’s equations remain intact except now they have additional degrees of

freedom determined by spatial fields attached to each direction of transverse time.

These additional fields can be treated as a single complex field and lead to analytic

modes. Promotion, demotion and annihilation emerge as features of the complex

analysis and do not have to appeal to quantum theory of harmonic oscillators – other

than to make convenient use of the algebra.

A concept of conservation in the number of vortex loops is employed in

separate processes of excitation and detection. With a ‘single’ k-vector, fields

circulate back and forth in space and time. However to remove/add energy at the

boundaries requires careful balancing of vortex loops to ensure global energy is not

altered in an unphysical way. Unidirectional energy flow and quantisation follows.

The space-time vortex concept shows how it is possible to combine local

power extraction with the non-local properties of instant ‘communication’ around

space-time vortex loops and yet retain causality. These features should be of

considerable value in a better understanding of ‘entanglement’ . One of the key points

in this work is that detection is every much as part of the boundary conditions as

excitation. Until the detection process is completed, not everything has been decided –

there are unresolved degrees of freedom that only are resolved by detection.

So much has been written on the foundations and philosophy of quantum

theory [refs 66, 67, 68, 69, 70 give a minute sample] that to review these in relation to

this preliminary account of space-time vortex theory would be too lengthy and

premature. Although a review remains as future work, accounts by Cramer [1] and

Mermin [64] must be acknowledged as key elements in shaping the author’s views. It

could also be claimed that there are elements of Everett’s work [69] in having many

different solutions that are compatible with the observable boundary conditions.

It could also be argued that the random electrodynamics of Marshal and others is

relevant [71, 72, 73, 74]. The additional degrees of freedom with complex Maxwell

equations, combined with the possibility of only observing terms like E*E mean that


39

there are a multiplicity of differently phased solutions for any given E*E. The

interference of all these differently phased fields could lead to a random field and so

lead to random electrodynamics. However a major difference here from previous

philosophical accounts is that STV derives quantisation of the em field: other work

interprets quantum theory.

STVs offer new routes for future exploration, notably: (i) calculation of particle

masses from Beltrami like equations involving fields in transverse time or perhaps

charge/mass ratios; (ii) better understanding of entanglement with importance for

quantum computing; (iii) higher order interactions that might even provide new if

speculative ideas about unifying gravity and quantum theory . Acceptance of vortex

theory could at last give a ‘crisp definition’ [75] of the difference between classical

and quantum measurements. Quantum measurements reconcile the boundary

conditions set by transverse time and space-time vortices. Classical measurements

have no knowledge of either. Perhaps all this at last points towards a ‘completion’ that

was sought by Einstein [5].

Acknowledgements

Dr.E.A.B.Cole is thanked for his continued encouragement.

APPENDIX 1.FOURIER ANALYSIS & ANNIHILATION OF ANALYTIC SIGNALS [32, 38]

Although there are two chiralities for all the modes, the fact that these behave the

same in the respect of topic in this appendix means that the subscripts +/– are omitted.

Concentrating on α α α α modes, combinations of real fields

[E1(k, t) + i E2(k, t)] and [B1(k, t) + i B2(k, t)] create complex functions of time t,

αααα(k, t) . Concentrating only on the temporal variations, αααα(t) is synthesised from :-

αααα(t) = ∞

0αααα^(k) exp (–i k t) dk/2π A.1

This converges over the lower half of the complex plane t = t + iτ ; τ < 0 leaving αααα(t)

analytic in t. The spectral analysis can be determined from a contour integral over the

upper half complex plane:-


40

αααα^(k) = Γ upperαααα(t) exp (i k t) dt k > 0 A.2

With t = (t + i τ ), τ > 0 and k > 0 , Jordan’s lemma allows A.2 to be written as:-

αααα^(k) = ∞

∞−αααα(t) exp (i k t) dtk > 0 A.3

If k < 0 , then the lower half complex plane has to be used

αααα^(k) = Γ lowerαααα(t) exp (i k t) dt k < 0 A.4

αααα(t) is analytic within Γlower so there are no poles. Hence αααα^(k) = 0 for k < 0.

Now modulate αααα(t) by exp (–i K t) to form ααααP(t) = [exp (–i K t) αααα(t)] with K > 0

αααα^(k')P = Γ upperααααP(t) exp (i k' t) dt A.5

For k' > K > 0, A.5 (like A.3) can be limited to the real time axis to show:-

αααα^(k')P = αααα^(k'K) A.6

The frequency has been ‘promoted’ by K : i.e. given an upper frequency ko in the

initial system, the highest frequency in the promoted frequency is k' = ko + K.

If αααα(t) is now modulated to form ααααD(t) =

[exp (–i K t) αααα(t)] with K > 0 , the frequency is demoted. The spatial spectrum is

pushed down: i.e. given an upper frequency ko in the initial system, the highest

frequency in the demoted frequency is k' = ko – K. If K > ko then there are no

positive frequencies components and the signal is annihilated. This analysis of

annihilation can be put in terms of contour integrals but for brevity a pictorial view is

given in Figure 1. Given a limited range of frequencies in the positive spectrum for

the initial signal, demotion that attempts to reverse the sign of the frequency in all

these components will annihilate the analytic signal.


41

The same arguments apply, interchanging the

upper and lower contours in the discussion, for

ββββ(t). However preserving the clockwise path of the

contour integration, the integration along the real

time axis for the ββββ modes is in the opposite sense

for that part of the integration with αααα modes: i.e. ββββ

modes can be said to have an opposite sense of

increasing time to the αααα modes. The rule however

remains that any temporal modulation forcing

frequency reversal, annihilates the mode.

Conjugation also changes the sign of the

frequency. However if αααα is analytic in the lower

half plane, αααα* is analytic in the upper half plane

and the change of sign in the frequency is fully

consistent with the analytic regions of complex

time. Frequency reversal by conjugation does not

annihilate the signal.

Finally because αααα is analytic over the lower half of the complex t plane while ββββ is

analytic over the upper half, a linear combination such as a αααα+ b ββββ is not in general

analytic. It is for this reason that the fields E(r t) and B(r t) are not analytic.

APPENDIX 2. LORENTZ TRANS-FORMATIONS

Let the boost parameter be α and the velocity be along the spatial direction n

(|n| = 1). Define: ch = cosh ½α ; sh = sinh ½α ;

Ch = cosh α ; Sh = sinh α

RL= exp(½αγt3n.γγγγs) = ch + γt3 n.γγγγs sh ; RL~

= (ch + n.γγγγs γt3 sh) B.1

Apply these rotors to the multivector z = t γt3 +x.γγγγs

zL = RLz RL~= (ch + γt3 γγγγs.n sh )(t γt3 +x.γγγγs)(ch + γγγγsγt3 .n sh) B.2

The algebra can be considerably simplified if we write x = x|| n +x⊥ m

(ch + γt3 γγγγs.n sh )(x⊥ m.γγγγs)(ch + γγγγsγt3 .n sh) = (x⊥ m.γγγγs)

(ch + γt3 γγγγs.n sh )(x|| n.γγγγs)(ch + γγγγsγt3 .n sh) = (ch + γt3 γγγγs.n sh )2(x|| n.γγγγs)

= (Ch + γt3 γγγγs.n Sh )(x|| n.γγγγs) =Ch x|| n.γγγγs − − − − γt3 Sh x|| B.3

ko

k>0

spectrum

k<0

ko+K

promotion

k'>0

k'<0

(ko–K)<0

annihilation

k>0

k'<0

Figure 5

Promotion/Annihilation (for

fields with only positive k)


42

(ch + γt3 γγγγs.n sh )(t γt3(ch + γγγγsγt3 .n sh) = (Ch + γt3 γγγγs.n Sh ) t γt3

= (Ch t γt3 − t γγγγs.n Sh ) B.4

zL = γt3(Ch t − − − − Sh x|| )+ γγγγs.(Ch x|| n − t n Sh + x⊥ m ) B.5

Consider the effect on the electromagnetic x.γγγγs fields.

Write F = –E + iSB , then RL(γt3F. γγγγs) RL~= (γt3FL. γγγγs) gives the general Lorentz

transformation. Once again the technique of creating terms parallel and perpendicular

to n simplifies the algebra. F = F|| n +F⊥ m then noting that γt3 γγγγs.n commutes with

γt3 γγγγs.n but anti-commutes with γt3 γγγγs.m so that

γt3γγγγs.FL = [ch + γt3 γγγγs.n sh ]γt3 γγγγs.(F|| n +F⊥ m)[ch + γγγγsγt3 .n sh]

= γt3γγγγs. F|| n + [Ch + γt3 γγγγs.n Sh ] γt3γγγγs.F⊥ m

= γt3γγγγs. F|| n + Ch γt3γγγγs.F⊥ m − γγγγs.n Sh γγγγs.m F⊥

= γt3γγγγs.F|| n + Ch γt3γγγγs.F⊥ m +ϕs γ γ γ γs.(n x m) Sh F⊥

[ϕ t ϕt = − 1 ; ϕtϕs = S ; ϕt = γt3 i = i γt3 ; ϕs = − i γt3 S]

= γt3 γγγγs.F|| n + Ch γt3γγγγs.F⊥ m − i γt3 S γ γ γ γs. (n x m) Sh F⊥

= γt3 γγγγs.F|| n − − − − Ch γt3 γγγγs.F|| n + Ch γt3γγγγs.F − i γt3 S γ γ γ γs. (n x F) Sh

(γt3FL. γγγγs) = γt3γγγγs. [Ch F − i S (n x F) Sh + F.n (1 − − − − Ch)] B.6

Setting F = E + iS B then

EL = [Ch E + (n x B) Sh + E.n (1 − − − − Ch)] B.7

BL = [Ch B − (n x E) Sh + B.n (1 − − − − Ch)] B.8

This checks with Jackson p. 552 but using hyperbolic functions to effect the ‘Lorentz

rotation’.

Now consider the RTT invariant multi-vector product :

F*F = γtγγγγs.F*. γtγγγγs.F = F*.F − iS γt γγγγs.(F*x F) B.9

Hence F*.F is also Lorentz invariant and (F* x F) transforms like F

F*F = (E*.E +B*.B) +iS (B*.E −−−−E*.B) + γt γγγγs. [iS(E*xE +B*xB) + (ExB* ++++E*xB)]

B.10


43

Separating the different grades of multivector give the Lorentz invariants

(E*.E +B*.B) and (E.B* −−−−E*.B), while ΠΠΠΠ = [(ExB* ++++E*xB)] +iS[(E*xE +B*xB)]

B.11

transforms like F in that

γt3 γγγγs.Π || n + Ch γt3 γγγγs.Π⊥ m − i γt3 S γ γ γ γs.(n x m) Sh Π⊥ B.12

For changes of velocity where Π⊥ = 0 , [(ExB* ++++E*xB) +iS(E*xE +B*xB)].n is

Lorentz invariant.

However if the electromagnetic power flow is in a direction p such that p.F =

0, then p x ΠΠΠΠ = 0 so that for any vector p⊥ perpendicular to p then p⊥ . ΠΠΠΠ = 0 . Hence

for surface integrals with normals that are parallel to p⊥ there is no contribution from

either (ExB* ++++E*xB) or (E*xE +B*xB). Although these vectors partially

interchange under a Lorentz transformation along the direction p⊥ , there is no

contribution to the power flow.

REFERENCES

1 J.G. CRAMER, The Transactional Interpretation Of Quantum Mechanics, Rev.Mod. Phys. 58, (1986) 647– 687.

2 C. TRUESDELL, The Kinematics of Vorticity Indiana Press, Bloomington (1954)3 C.A.FUCHS and A.PERES, Quantum Theory Needs No Interpretation, Physics Today 53 (2000) 70–714 R.LOUDON, The Quantum Theory of Light (3rd edition ) Oxford UniversityPress (Oxford) (2000)5 A. EINSTEIN, B. PODOLSKY and N. ROSEN Can quantum-mechanicaldescription of physical reality be considered complete? Phys. Rev. 47, 777-780 (1935)6 S.J. FREEDMAN and J.F. CLAUSER Experimental test of local hidden-variabletheories Phys. Rev. Lett. 28, 938-941 (1972)7 A. ASPECT, J.DALIBARD, G. ROGER: "Experimental test of Bell's inequalitiesusing time- varying analyzers" Physical Review Letters 49 , (1982) 1804 .8 .A. ASPECT, P. GRANGIER, G. ROGER, Experimental realization of Einstein-Podolsky-Rosen-Bohm gedanken experiment; a new violation of Bell's inequalities Physical Review Letters 49 (1082), 91.9 A.F. ANTIPPA and A.E.EVERETT , Tachyons, causality, and rotational invariance,Phys. Rev. D, .8, (1973).2352-60.10 P.DEMERS, Symmetrization of length and time in a C3 Lorentz space in linearalgebra with application to a trichromatic theory of colours, Canadian Jnl. Phys.,53,(1975),1687-8.


44

11 E.A.B.COLE “Superluminal transformations using either complex space-time orreal space-time symmetry” Nuovo Cimento .40A, (1977).171-80.12 P.T.PAPPAS, Physics in six dimensions: an axiomatic formulation, Lett. NuovoCimento, 22, (1978), 601-7.13 E.A.B.COLE , Emission and absorption of tachyons in six-dimensional relativity,Phys Let., 75A, (1-2), (1979).29-30.14 1981, PAVSIC, M., “Unified Kinematics Of Bradyons And Tachyons In 6-Dimensional Space-Time” Jnl. Phys. A. 14 (12): pp 3217-322815 1983, MACCARRONE G.D, PAVSIC M, RECAMI E “Formal And Physical-Properties Of The Generalized (Subluminal And Superluminal) LorentzTransformations” Nuovo Cimento 73B (1): 91-11116 1970, DORLING J. “The dimensionality of time” Am. Jnl. Phys. 38, pp.539-40.17 1979, SPINELLI G. “Against the necessity of a three-dimensional time”, Lett. Nuovo Cimento,26, (9),pp.282-4.18 1980 WEINBERG, N.N. ‘On some generalisations of the LorentzTransformations’Phys.Lett. 80A 102-10419 1980, COLE EAB. ‘Particle decay in six-dimensional relativity’. Jnl. Phys. A,13, pp.109-15.20 1981, ZIINO G. “Three-dimensional time and Thomas precession” Lett. Nuovo Cimento, 31, pp.629-32.21 1983, STRNAD J. “Experimental evidence against a three-dimensional time” Phys. Lett. A, 96A, (5) pp.231-2.22 1982, COLE E.A.B.. BUCHANAN SA. “Space-time transformations in six-dimensional special relativity” Jnl. Phys. A 15,L255-7.23 1980, COLE E.A.B.. “New electromagnetic fields in six-dimensional special relativity.” Nuovo Cimento.60A, pp.1-12.24 1985, COLE E.A.B., “Generation of new electromagnetic fields in six-dimensional special relativity”. Nuovo Cimento 85B, (1) pp.105-17.25 1978, DATTOLI G. MIGNANI R. “Formulation of electromagnetism in a sixdimensional space-time” Lett. Al Nuovo Cimento, 22, pp.65-8.26 1978, VYSIN V. “Approach to tachyon monopoles in R/sup 6/ space” Lett. AlNuovo Cimento, 22, pp.76-80.27 1984, TELI MT. “Electromagnetic-field equations in six-dimensional space-time with monopoles.” Nuovo Cimento 82B, pp.225-34.28 1984, TELI MT. PALASKAR D. “Electromagnetic-field equations in the six-dimensional space-time R6”


45

Lett. Nuovo Cimento, 40, pp.121-529 1997, BARASHENKOV V.S., YUR'IEV M.Z.,. “Electromagnetic waves inspace with three-dimensional time” Nuovo Cimento B-112B, pp.17-22.30 HESTENES, D. 1966 Spacetime algebra New York Gordon and Breach31 1982, HESTENES D “Space-time structure of weak and electromagneticinteractions” Found. Phys.12, pp.153-68..32 HESTENES, D. 1985 New Foundations for Classical Mechanics Dordrecht Reidel33 HESTENES, D. 1985 Quantum Mechanics from self interaction Foundations ofPhysics 15 63-8734 LASENBY, A., DORAN, C. AND GULL, S. Gravity, gauge theories andgeometric algebraPhil Trans. R.Soc. Lond. A (1998), 356 , 487-58235 GULL S.F. LASENBY A.N. & DORAN C.J.1993 Imaginary Numbers are not real– the geometric algebra of spacetime Foundations of Physics 23 1175-120136 1997, LOUNESTO, P., Clifford Algebras and Spinors, CUP Cambridge LondonMath. Soc. Lecture Notes 23937 1985, SALINGAROS N. “Invariants of the electromagnetic field and electromagnetic waves” Am. Jnl. Phys.53, pp.361-3.38 1993, JANCEWICZ B. “A Hilbert space for the classical electromagnetic field”Found. Phys.23, . pp.1405-21.39 2000, SHAARAWI AM. “Clifford algebra formulation of an electromagnetic charge-current wave theory” Found. Phys.30, pp.1911-41.40 2001 CARROLL J.E. Proceedings of CQ08 Rochester; Conference proceedingsto be published41 http://www.mrao.cam.ac.uk/~clifford/ptIIIcourse/course99/42 http://www.mrao.cam.ac.uk/~clifford/publications/abstracts/imag_numbs.html43 http://carol.wins.uva.nl/~leo/clifford/talknew.ps44 1990 MURASKIN M. “The arrow of time”. Physics Essays, vol.3, no.4, Dec., pp.448-52. Canada.45 1993 MURASKIN M. “Wave packet solution and mathematical aesthetics” Physics Essays, vol.6, no.3, pp.409-1946 1971 HESTENES, D., ‘Vectors, Spinors and Complex Numbers in Classical andQuantum Physics’ Am.J.Phys. 39, 1013-102747 1975 HESTENES,D. ‘Observables, operators and complex numbers in the Diractheory’ J.Math.Phys. 16 556-57248 PATTY C.E., SMALLEY L.L., ‘Dirac-Equation In A 6 Dimensional Spacetime -Temporal Polarization For Subluminal Interactions Physical Review D 32: (4) 891-897 198549 BOYLING J.B, COLE E.A.B ‘6-Dimensional Dirac-Equation’ InternationalJournal Of Theoretical Physics 32: (5) 801-812 May 199350 BJORKEN, J.D. AND DRELL, S.W.1964; ‘Relativistic Quantum Mechanics’McGraw-Hill51 BAYM, G , ‘Lectures on Quantum Mechanics’ W.A.Benjamin Reading Mass196952 GREINER W. Relativistic Quantum Mechanics Wave Equation p 105 –6Springer 3rd edition 2000 Berlin


46

53 STREATER R.F., WIGHTMAN A.S. 1964 PCT, Spin And Statistics And AllThat Benjamin Cumming Reading & Massachusetts54 STRATTON J.A. 1941 Electromagnetic Theory Chapter 2 Macgraw HillNew York55 COHEN-TANNOUDJI, C. DUPONT-ROC J. AND GRYNBERG, G 1989Photons and Atoms J.Wiley New York (originally in French Photons et Atomes1987 Inter-editions et Editions du CNRS)56 COHEN, L. 1995 Time Frequency Analysis (Chapter 2) Prenctice Hall .57 RAMO, S. WHINNERY J.R. AND VAN DUZER, T. 1994 Fields and Waves inCommunication Electronics (3rd Edition) Wiley58 WHEELER J.A AND FEYNMAN R.P. 1945 Interaction With The Absorber AsThe Mechanism Of Radiation Rev. Mod. Phys. 17, 157-18059 BRACEWELL R.N. 1978 The Fourier Transform and its Applications, McGrawHill (2nded.)60 FEYNMAN R.P. 1987 The Reason For Anti-Particles In Feynman R.P. AndWeinberg, S. Elementary Particles And The Laws Of Physics C.U.P. Cambridge61 E.A.B.Cole . Prediction of dark matter using six-dimensional special relativity.Nuovo Cimento B .115B,pp.1149-58, 200062 K. Huang Statistical Mechanics Wiley New York 196363 E. HECHT, Optics 2nd Edition p.321 Addison Wesley 198764 D.BOUWMEESTER, A.EKERT AND A.ZEILINGER, 2000 The Physics ofQuantum Information, (Springer).65 MERMIN N D, What is quantum mechanics trying to tell us, Am.J. Phys, 66 ,753-767.66 WHEELER J A AND ZUREK W H 1983 Quantum theory and measurement(Princeton University Press)67 PERES A 1993 Quantum Theory: Concepts and Methods (Kluwer)68 ISHAM C.J. 1995 Lectures on Quantum Theory (Imperial College Press)69 BUB, J. 1997 Interpreting the Quantum World (Cambridge University Press)70 EVERETT H 1957 `Relative State Formulation of quantum mechanics', Reviewof Modern Physics 29, . 454-462;71 MARSHALL T.W. 1963 Random Electrodynamics Proc.Roy.Soc. A276, 475–491.72 BOYER T.H. 1975 Random electrodynamics: The theory of classicalelectrodynamics with classical electromagnetic zero–point radiation Phys.Rev. D11, 790–830.73 DE LA PENA L., CETTO A.M. 1986 Physics of Stochastic Electrodynamics ,Il Nuovo Cimento 92, 189–217.74 L. DE LA PENA_AUERBACH and A.M. CETTO 1978 Quantum mechanicsderived from stochastic electrodynamics Foundations of Physics 8, ( 1978) 191–210.75 ZUREK W.H. 1991 Decoherence And The Transition From Quantum ToClassical Physics Today 10, 36-44.

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