15th International Conference onPseudo-Hermitian Hamiltonians of Quantum

Physics PHHQP-15 Palermo 18-23.05.2015

RODIONOV V.N.

1

In 1956 P.A.M. Dirac in his report at the conference in Russia in the Joint Institute of Nuclear Researches of the Dubna has formulated a number of the concrete problems, which in his opinion, will excite of future physicists. Seven problems were written by Dirac, part of which is relevant and to our times.

2

Here we can see the ‘genuine comments of Dirac which been written at the blackboard in the conference hall of Dubna

3

This is comment of Dirac in more visual form:

1) infinities2) fine structure constant3) nucleons and mesons4) ratio of the masses of

elementary particles5) fundamental length6)gravitation7) neutrino

4

Further we assume in more detail to consider the questions, which are numbered as 4, 5 and 7 with the point of view of the contemporary development of Pseudo-Hermitian representations of the Quantum Physics.

Now it is well-known fact, that the reality of the spectrum in models with a non-Hermitian Hamiltonian is a consequence of PT -invariance of the theory, i.e. a combination of spatial and temporary parity of the Hamiltonian:

[H,PT ]ψ = 0. It should be noted that the solution of this

problem was initiated by observations C. Bender and S. Boettcher (1998).

5

When the PT-symmetry of Hamiltonian systems are unbroken, the spectrum of energies in the quantum theory is real.

This surprising results explain the growing interest in this problem. For the past a few years has been studied a lot of new non-Hermitian PT–invariant systems.

I intend to focus on one of them.

6

As it is well known, the idea about the existence of a maximal mass in a mass spectrum of elementary particles which was equal to the Planck mass, was suggested by Moisey Markov in 1965. This new particle was called by the author the Maximon:

Closely with this concept was introduced and the notion of the Fundamental length

. The question is how we can characterize the

state of affairs in the study of the concepts of Maximal mass and Fundamental length in the history of the development of contemporary physics?

GeVMM PlankMaximon1910

cml f3310

7

THE MYSTERY OF MASS What causes particles to have masses? Why do the masses of fundamental particles differ so en’ormously, for example: the top-quark more than 300.000 times heavier then the electron? A solution has been proposed by Peter Higgs: that the whole of space is ‘permeated by a field. Interaction of particles with this field and gives rise to their mass.

Higgs proposed

a field to explain the

origin of mass

Einstein discovered that mass is equivalent to energy

Newton related mass and weight

i

8

For decades we have ‘witnessed the impressive success of Standard Model (SM) in explaining properties and regularities observed in experiments with elementary particles.

In agreement with the contemporary experimental data this structure has been disclosed for any fundamental particles of SM, up to distances of the order of ten in minus seventeen ‘centim’eters

).10( 17 cm

9

Intuitively it is clear that an elementary particle should carry small enough portions of different "charges" and "spins". However, for the particle of mass m this quantity is in the contemporary QFT may have arbitrary values in the interval 0 ≤ m < ∞.

10

In this situation the question naturally arises: up to what mass values one may apply the concept of a local quantum field?

Formally, the standard QFT remains a logically imp’eccable sch’eme even when in the elementary act of interaction of the particles there are masses which are comparable, say, with the masses of cars or a buses.

11

Indeed, interaction of the particles with the initial energies, which are comparable with the masses of "cars", in a result of interaction may to generate a "bus" when will be obtained the sufficiently high energy !

12

Under all the absurdity of the situation considered above, however we should emphasize, that modern QFT does not pro’hibit such physically meaningless results of this extrapolation!

13

Thus, one should think about how may be modified theory to eliminate these physically meaningless violations of QFT. The variant such program was suggested by Kadyshevsky and his colleagues (see, for example [1] – [3]).

[1] V. G.Kadyshevsky, Phys. Part. Nucl., 29, 227-231 (1998). [2] V. G.Kadyshevsky and M. D.Mateev, Nuovo Cimento A, 87,

324–349 (1985). [3] V.G.Kadyshevsky, M. D. Mateev, V. N.Rodionov, A.S.Sorin

Doklady Physics 51, p.287 (2006), e-Print:hep-ph/0512332. Towards a maximal mass model. CERN TH/2007-150;

hep-ph/0708.4205.

14

11

The key idea of this approach is the following hypothesis: the mass spectrum of elementary particles, i.e. the objects, which are described by local fields, has to be limited by a certain value M = Mmax:

m ≤ M, which was named by the Maximal mass of mass spectrum of elementary particles. This statement has to be accepted, as a new fundamental principle of Nature, which similarly to the relativistic and quantum postulates should underlie modified QFT.

15

The question is - how to realize this hypothesis? In the papers [1]-[3] the value M is considered in a pure geometric way as the radius of the five-dimensional hyperboloid whose surface is a realization of a curved momentum four-space or the space of anti-de Sitter

where the new operator of the Klein–Gordon may be written in the form

Here also the following notation is used

and is the fifth component of the momentum of particle m.

,225

220 Mppp

).cos(2),( 5 MpMMpKG

22 /1cos Mm5p

16

It is vary important that in the fermion sector geometrical theory of Kadyshevsky[1-3] we can see the emergence of new terms in the modified Dirac operator

and then one can obtain corresponding PT-symmetric γ5-depended non-Hermitian Hamiltonians

)2/sin(2)(),( 55 MMppMpD

),(),( MpHMpH 17

Another interesting approach was developed by Miloslav Znojil for the case of the non-relativistic problem of the introduction of a Fundamental Length [4,5]. In this articles also as in the first case (see[1-3]) for the introduction of Minimal Length in quantum theories was used apparatus of PT-symmetric Non-Hermitian Hamiltonians in quantum physics. [4]M.Znojil, Phys. Rev. D 78, 025026 (2008) [5] M.Znojil, Phys. Rev. D 80,045022 (2009)

18

It should be noted if we use the notation of Fundamental length

and a Maximal mass M we can write the simple expression , which is demonstrating the direct connection between articles [1–3] and [4,5].

Ml f /1

19

fl

It is very important that PT-symmetric approaches are common for this two models! It should be noted that the use of the non-Hermitian PT - symmetric Hamiltonians may be a very fruitful environment for the considering of new physics beyond the SM.

The question is - could one ‘implement and in the first relativistic case, which previously considered with geometric positions [1-3], a purely algebraic non-Hermitian approach to the construction of the model with restriction mass of the fermions? 20

The answer on this question is the following!

Let us now consider the receipt of the modified Dirac equations for free massive

particles m using the simple algebraic transformations of the ordinary Klein-Gordon operator.

In this case the approach is similar to the known reception Dirac in which we can represent the Klein-Gordon operator in the form of a product of two commuting matrix operators.

However now we have deal with non-Hermitian PT - symmetric operators:

21

where the physical mass of particles

is expressed through new two parameters and (see [6],[7]).

[6] Rodionov V.N., Kravtsova G.A SSN 0027-1349. Moscow University Physics Bulletin, 2014, Vol. 69, No. 3, pp. 223-229.

[7] Rodionv V.N., Kravtsova G.A. Theoretical and Mathematical Physics, 182(1), 100-113 (2015).

))(( 25125122 mmimmim

22

21

2 mmm

1m 2m

22

For so the function would obeyed to the equations of Klein-Gordon

one can demand that it also satisfies one of equations of the first order

0),(~)( 22 txm

0),(~)( 251 txmmi

0),(~)( 251 txmmi

23

Modified equations of Dirac (MD) of course, are less common than KG equations, and although every solution of one of the MD satisfies to KG, reverse a’pproval has not designated.

It is also obvious that the Hamiltonians, associated with this equations, are non-Hermitian, because in it there are γ5-dependent mass components ( ):

24

HH

)( 251 mmpH

)( 251 mmpH

Here the following notations are used β = γ0, γ5 = −iγ0γ1γ2γ3. It is easy to see from that the mass m is real, when the inequality is accomplished. The last inequality is identical to the condition previously considered by C. Bender and his ‘colleagues [8]: [8] C.M. Bender, H.F. Jones, R.J. Rivers : Phys. Lett. B 625, 333 (2005)

22

21 mm

25

Note that in this model if to leave

only condition one can find infinite many solutions associated with the appearance of two different mass parameters m1 and m2, which are corresponding to the single physical mass m.

However a feature of the model with γ5-mass contribution is that it may contain another restriction of mass parameters in addition to the known.

.

26

21 mm

Indeed besides the last inequality

need take into account the rules of conformity of this parameters in the Hermitian limit m2 → 0.

Without this assumption the development of Non-Hermitian models no may be adequate. With this purpose one can determine an additional mass scale, which may more complicated way depend on the parameters m1, m2.

27

In particular with this parameter,

can would put an upper bound on the mass spectrum of particles by analogy with geometrical theory [1]-[3].

These considerations can help find in the frame of the stringent restrictions of m ≤ m1, the existence of more complicated non-linear dependence on limiting mass value m ≤ M(m1,m2).

28

This expression meets the requirements of the principle of conformity: for all ordinary fermions , when M → ∞ we should obtain ordinary Hermitian theory. In this sense the principle of conformity identical to the transition to the “flat limit” in the geometrical model (see [1-3]).

Explicit expression for M(m1,m2), may be obtained from the simple mathematical theorem. Really, from the inequality expressing the comparison of the ‘arithm’etical mean and the geometrical average of two positive numbers

we have

),( 212 2

21 mmMm m

m 29

22

22

)( 22

2

mmmm

Using the last expression we can obtained the following pictures. There are dependence of m/m1 and m/M on the parameter x=2m2/m1=m1/M.

1)red curve - , 2)green curve- , x=2m2/m1=m1/M We can see that the second curve has the maximum under

4

2

11 x

mm

4

2

1 xMm x

30

21 2mm

Thus, the solutions of the system of equations

relative to the parameters m1 and m2 may be represented in the form

It is easy to verify that the obtained values of the mass parameters satisfy to a number of conditions regardless of which the sign will be chosen. However these formulas only in the case of the upper sign (-) is agreed in the limit

with the ordinary Dirac equations that is corresponded to the transition to Hermitian limit

, .

221 /112 MmMm

)/11( 222 MmMm

M

mm 1 02 m

31

222

21 mmm

Mmm 221 2/

But when the lower sign (+) is chosen this transition to the Hermitian Limit is absent.

Thus in the considered model appear yet unknown particles are described by new equations which has no ‘counterpart in the SM.

Should be noted that analogical particles appeared earlier in the geometrical theory [1-3] and where they were named as "exotic particles" .

32

Thus we have absolutely different dependence of on the parameter x=m/M. Another words, for each of ordinary particle may be exist the new partner, which possesses the same mass and possibly has a number of another similar properties.

It is very important that new particles arise together with the advent of restriction of mass of elementary particles and hence thanks to the development of non-Hermitian approach to quantum theory.

21 ,mm

33

34

This is Figure demonstrates the dependence of the parameters , on the . We can see here the lowest curves, which is corresponding to the and (ordinary particles). At the same time, the upper curves show analogical dependencies of parameters and , which is corresponding to the case of the exotic particles.

1m

2mMm /

1m 2m

1m2m

It is important that the appearance of "exotic particles", is closely related with the change of widely known boundaries of the PT-symmetry of the considered task with γ-5 mass extension .

In the case of ordinary particles, we have a narrower area of the bounding undisturbed region of the mass parameters . In the second area, we are dealing with yet unknown type of “exotic particles”, which are bounded by

35

2/0 12 mm

121 2/ mmm

120 mm

This leads to a model with two different modified Dirac operators. New approach is responsible as for ordinary particles, for which after increasing of M to infinity we have traditional description of physical processes. And also can be used for the so-called "exotic particles” for which there are no the Hermitian limit and therefore they have not an’alogue in SM.

For example, if ordinary massless particles are determined by the parameters

, whereas for ‘anal’ogical determination of massless "exotic particles" we have ,

Where is the parameter of Maximal mass, which limits spectrum mass of fermions.

36

0and 0 21 mm

M mMm 2and 2 21 M

We will touch upon also the question since describing the motion of modified Dirac particles in the magnetic field, if their own magnetic moment is different from the Bohr magneton.

As it was shown by Schwinger, that the Dirac equation of particles in the external electromagnetic field can be expressed in the form:

37

,0)()|,()()( dyyAyxxmP ext

where is the mass operator of fermion in external field .

Here the following notation also is used .

By means of expansion of the mass operator in series according to with precision not over then linear field terms one can obtain the modified equation.

exteApP

)|,( extAyx

exteA

38

extA

We also will take into account the mass extension and the interaction of charges and AMM of fermions with magnetic field . In a result we have the modified equation of Dirac-Pauli [9,10]

Here the following notations are used

and also µ is magnetic moment of a fermion, g – fermion

gyromagnetic factor, µ0 - the Bohr magneton, [9]V.N.Rodionov International Journal of Theoretical

Physics.04.11.2014. DOI 10.1007/s10773-014-2410-4 [10] V.N.Rodionov Physica Scripta. Phys. Scr. 90 (2015) 045302.

F

0)(~)( 2251 xFmmP

2/)2()( 00 g)(2/ i

251 mmm

39

This equation preserves the

relativistic covariance and consistent with the phenomenological equation of Pauli obtained in his early papers.

However the Hamiltonian corresponding to this equation has a pseudo-Hermitian form.

40

Indeed, the Hamiltonian form of this equation in the homogeneous magnetic field is the following [9],[10]

where

It is clear that

),(~),(~ trHtri t

).()( 251 HmmPH

HH

41

It should be noted that now the operator projection of the fermion spin at the direction of its movement

is not commute with the Hamiltonian and hence it is not the integral of motion.

For this aim we find the operator μ3, which characterizes the projection of fermion polarization on the direction of magnetic field . The operator μ3 commutes with the Hamiltonian and can be represented in the matrix form

P

121

121

211

211

3

00

00

00

00

miPP

miPP

PiPm

iPPm

42

H

For this operator we can obtain

where is characterized the projection of fermion spin at the direction of the magnetic field

For the Hamilton operator one can write

43

,~ 2 ~ 213 nm

1

,~~ EH

and the matrix form of this operator has the form

HmmPiPP

HmiPPmP

mPiPPHm

iPPmPHm

H

1231

1123

2311

21231

0

0

0

0

44

Performing calculations, in a result for modified Dirac-Pauli equation one can find the exact solution for energy spectrum (see [9,10]):

where are characterized the projection of fermion spin at the direction of the magnetic field H

It should be noted that the last formula is a valid not only for charged fermions, but and for the neutral particles possessing AMM.

221

22

23 2),( HnmmpHE

,...2,1,0n eH22

0 g

1

45

In this case one must simply replace

the value of quantized transverse momentum of a charged particle in a magnetic field on the ordinary value

Let us now turn to a more detailed

consideration energy of the fermions in the ground state in the external magnetic field.

222

212 pppn

46

,...2,1,0n

Eigenvalues of energy not trivial depends on the parameter x=m/M and the projection of fermion spin at the direction of the magnetic field H. Thus for the case of ultra cold polarized ordinary electronic neutrino (x << 1) with precision not over then linear field terms we can write

However, in the case of “exotic electronic neutrino”

we have the different value The main difference of the second formula is

consists in appearance of the multiplier, which is equal to the relation of the Maximal mass to the mass of neutrinos. It is clearly that this factor may be extremely large!

GsmE em

HH

c

13c 1041.4H where;1

2

0

mM

HH

c cmMmHHE 2

0 01)1/,/,1(

47

At this picture we can see the distribution of energetic levels as a function of parameter m/M for different values of quantum numbers n=0,1,2,3,4 and when the orientation of spin fermion along the direction of the magnetic field.

48

At this Figure we can see the distribution of energetic levels as a function of parameter m/M for different values of quantum numbers n=0,1,2, 3 and when the orientation of spin fermion directed against the direction of the magnetic field. In this case the partly disturbed PT-symmetry of Hamiltonian and breaks of levels take place. 49

Thus we considered modified Dirac–

Pauli equations that are entered using γ5-mass factorization of an ordinary Dirac operators.

We also considered the interacion of fermions with intensive uniform magnetic fields, focusing on their (g − 2) gyromagnetic factor. Due to effective research proc’edures, we derive the exact solutions of the energy spectra of pseudo-Hermitian Hamiltonians, taking into account the spin effects of the fermions.

251 mmm

50

The basic research results are the elucidation of the new border areas of the unbroken PT-symmetry of non-Hermitian Hamiltonians.

In particular, it is shown that the reality energy spectrum of fermions at rest can be expressed by limiting the intensity of the magnetic field

, where Δμ is an anomalous magnetic moment of particles.

12

max 2/ mmHH 12

max 2/ mmHH 12

max 2/ mmHH

51

Hence, in the intensive magnetic fields, accounting of the vacuum magnetic moment of particles can lead to a substantial change of borders of PT- symmetry in the considered model

mmHm 12

52

Recall, that a considerable increasing of am’endments, which are defined by magnetic field are connected with the possible existence of the so-called “exotic particles”.

In turn, the appearance of these particles is a direct consequence development of the Pseudo-Hermitian approach to this problem.

It is very not trivial that the fermions different types may be significantly separated with the help of the intensive magnetic field already at low energies of fermions.

53

If the exotic particles are really exist, their detection with the help of magnetic field may be used as indirect proof of the existence of the Maximal mass.

This circumstance can drastically change the basic paradigm of contemporary experimental physics.

54

Indeed, now the basic thesis of the search the limit of existence of the mass spectrum of elementary particles is closely connected with increasing of the energy of existing and construction of new accelerators.

However one may go by another way with the help of using magnetic fields, the intensity of which is achieved in contemporary laboratories under limited energies of fermionic beams.

55

And if we take into account that

in contemporary SM the limit of maximal mass may tends to infinity, then thanks to the non trivial dependence of the fermion energy in the magnetic field one can obtain the “upper border” of restriction mass spectrum of elementary particles!

56

It is very important that the appearance of "exotic particles", is closely related to description of this

phenomena as consequence of development of Non-Hermitian approach in quantum physics!

Now it is time for experimentation! Thank you

for your attention!

57

Summary

1) We develop original methods for studying relativistic pseudo-Hermitian Hamiltonians in intensive electromagnetic fields. One can name this approach as the modified picture of Furry which used in electromagnetic fields without perturbation theory. Thus, a number of the physical phenomenon in the intensive magnetic field investigated by us with the help of gamma-5-mass extension.

2) It was shown that based on the modified approach, the restrictions in the mass spectrum of fermions may be obtained.

3) We also have accomplished calculations of energy spectra of fermions without using of perturbation theory taking into account the interaction of their anomalous magnetic moments with the intensive magnetic fields. Here it was shown that when the orientation of fermionic spin against the direction of the magnetic field there are disturbances of PT-symmetry of the Hamiltonian.

4) Existence of the restriction in the mass spectrum of fermions causes the appearance in pseudo-Hermitian model of the so-called "exotic particles" having very non-trivial properties.

5) If presence of the "exotic particles" will be really discovered, then the values of the upper limit of Maximal mass may be obtained even under low energies of fermions. Currently, according to our estimates may be considered the question about the experimental measurement of value upper limit in the mass spectrum of neutrinos at the level of the values of the Plank mass.

6) We will be grateful if you find mistakes in our calculations.

58

3) We also have accomplished calculations of energy spectra of fermions without using of perturbation theory taking into account the interaction of their anomalous magnetic moments with the intensive magnetic fields. Here it was shown that when the orientation of fermionic spin against the direction of the magnetic field there are disturbances of PT-symmetry of the Hamiltonian.

4) Existence of the restriction in the

mass spectrum of fermions causes the appearance in pseudo-Hermitian model of the so-called "exotic particles" having very non-trivial properties.

5) If presence of the "exotic particles" will be really discovered, then the values of the upper limit of Maximal mass may be obtained even under low energies of fermions. Currently, according to our estimates may be considered the question about the experimental measurement of value upper limit in the mass spectrum of neutrinos at the level of the values of the Plank mass.

6) We will be grateful if you find mistakes in our calculations.

59

4) Existence of the restriction in the mass spectrum of fermions causes the appearance in pseudo-Hermitian model of the so-called "exotic particles" having very non-trivial properties.

5) If presence of the "exotic particles" will be really discovered, then the values of the upper limit of Maximal mass may be obtained even under low energies of fermions. Currently, according to our estimates may be considered the question about the experimental measurement of value upper limit in the mass spectrum of neutrinos at the level of the values of the Plank mass.

6) We will be grateful if you find mistakes in our calculations.

60

5) If presence of the "exotic particles" will be really discovered, then the values of the upper limit of Maximal mass may be obtained even under low energies of fermions. Currently, according to our estimates may be considered the question about the experimental measurement of value upper limit in the mass spectrum of neutrinos at the level of the values of the Plank mass.

6) We will be grateful if you find mistakes in our calculations. 61