Causality Violation in Causality Violation in Non-local QFTNon-local QFT

S.D. JoglekarS.D. Joglekar

I.I.T. KanpurI.I.T. Kanpur

Talk given at THEP-I, held at IIT Roorkee from 16/3/05—Talk given at THEP-I, held at IIT Roorkee from 16/3/05—20/3/0520/3/05

Causality Violation in Causality Violation in Non-local QFTNon-local QFTPLANPLAN1. Why non-local QFT’s ?1. Why non-local QFT’s ?2. Causality violation: classical and quantum2. Causality violation: classical and quantum3. Formulation of causality violation using BS 3. Formulation of causality violation using BS criterioncriterion4. One-loop Calculations4. One-loop Calculations5. Some all-order generalizations5. Some all-order generalizations6. Interpretation and Conclusions6. Interpretation and Conclusions

References: Ambar Jain and S.D. Joglekar, Int. Jour. Mod. Phys. A 19, 3409 (2004)Basic works: G. Kleppe, and R. P. Woodard, Annals Phys. 221, 106-164 (1993).N. N. Bogoliubov, and D. V. Shirkov, Introduction to the theory of quantized fields (John Wiley, New York, 1980).

Why non-local QFT’s?Why non-local QFT’s? Non-local QFT is a QFT that incorporates non-local Non-local QFT is a QFT that incorporates non-local

interactioninteraction e.g. ∫de.g. ∫d44xdxd44yy dd4 4 zdzd44w w ff(x,y,z,w) (x,y,z,w) (x)(x)(y)(y)(z)(z)(w)(w) Interest in non-local QFT’s is very old, dating from Interest in non-local QFT’s is very old, dating from

1950’s: e.g.1950’s: e.g. Pais and Uhlenbleck (1950),Pais and Uhlenbleck (1950), Effimov and coworkers (1970-onwards) Effimov and coworkers (1970-onwards) Moffat, Woodard and coworkers (1990--) Moffat, Woodard and coworkers (1990--) The basic idea was to try to avoid “infinities” by The basic idea was to try to avoid “infinities” by

assuming a non-local interaction and thus providing a assuming a non-local interaction and thus providing a natural cut-off.natural cut-off.

Non-commutative QFT’s, currently being studied, and Non-commutative QFT’s, currently being studied, and are a special case of a non-local QFT: The equivalent are a special case of a non-local QFT: The equivalent star product formulation is a non-local interaction.star product formulation is a non-local interaction.

We shall focus on the last type of non-local theories. We shall focus on the last type of non-local theories. These are more desirable compared to the earlier These are more desirable compared to the earlier attempts in many ways.attempts in many ways.

Causality violation (CV) : Classical & Causality violation (CV) : Classical & quantumquantum Interaction Lagrangian is non-local: At a given instant, Interaction Lagrangian is non-local: At a given instant,

interaction may take place over a finite region of space: i.e. interaction may take place over a finite region of space: i.e. points spatially separated. May introduce CV.points spatially separated. May introduce CV.

Classical violation of causality: For example action-at-a- Classical violation of causality: For example action-at-a- distance. Such a classical violation of causality is distance. Such a classical violation of causality is undesirable from the point of view of experience.undesirable from the point of view of experience.

For example, consider a system of stationary particles For example, consider a system of stationary particles interacting via an action-at-a-distance of range R. These are interacting via an action-at-a-distance of range R. These are placed at a distance R eachplaced at a distance R each

A signal can instantaneously be communicated to any A signal can instantaneously be communicated to any

distance. distance. Can be observed at relatively larger distancesCan be observed at relatively larger distances Can be observed also at low momentaCan be observed also at low momenta Quantum violations, on the other hand are suppressed: Quantum violations, on the other hand are suppressed:

gg22/16/16per loop:per loop: Smaller in magnitudeSmaller in magnitude Smaller in rangeSmaller in range As we shall see, they are pronounced at larger energiesAs we shall see, they are pronounced at larger energies It is desirable that lowest order should It is desirable that lowest order should notnot show CV: This is show CV: This is

arranged if the tree order S-matrix is the same as local one.arranged if the tree order S-matrix is the same as local one.

Non-local QFT’s of Kleppe-Non-local QFT’s of Kleppe-Woodard typeWoodard type

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To state briefly, the non-local version of the scalar To state briefly, the non-local version of the scalar theory is given in terms of the Feynman rulestheory is given in terms of the Feynman rules

There is only one basic vertex, but external lines can be

of either variety. X

Non-local QFT’s of Kleppe-Non-local QFT’s of Kleppe-Woodard type: Special Woodard type: Special PropertiesProperties Unlike higher derivative theories and many Unlike higher derivative theories and many

non-local theories, the asymptotic equation non-local theories, the asymptotic equation (interaction switched off) is identical to free (interaction switched off) is identical to free theory.theory.

No ghosts and no spurious extra solutions: No ghosts and no spurious extra solutions: These spoil meaning of quantization, and These spoil meaning of quantization, and come in the way of unitarity.come in the way of unitarity.

S-matrix same in the lowest order as the free S-matrix same in the lowest order as the free theory: theory: No No classical violationclassical violation of causality of causality

Theory unitary for any finite Theory unitary for any finite Can be Can be interpreted as a bona-fide physical theory interpreted as a bona-fide physical theory with a space-time/mass scale with a space-time/mass scale (KW91) (KW91) ..

The theory has an equivalent non-local form of any The theory has an equivalent non-local form of any of the local symmetries.of the local symmetries.

The theory has a quantum violation of causality The theory has a quantum violation of causality (KW91) .(KW91) .

Interpretation of non-Interpretation of non-local QFTlocal QFT Another interpretation has also been suggested Another interpretation has also been suggested

[SDJ:IJMPA01]: Suppose that standard model arises [SDJ:IJMPA01]: Suppose that standard model arises from a theory of finer constituents as a low energy from a theory of finer constituents as a low energy effective theory. Suppose that the compositeness effective theory. Suppose that the compositeness scale is scale is Then, the low energy theory would exhibit Then, the low energy theory would exhibit nonlocal interactions (via form-factors) of length-scale nonlocal interactions (via form-factors) of length-scale ~1/~1/We thus expect the low energy effective theory We thus expect the low energy effective theory to be non-local, unitary in the energy range of its to be non-local, unitary in the energy range of its validity, and possessing equivalent of underlying validity, and possessing equivalent of underlying residual symmetries. On account of composite nature residual symmetries. On account of composite nature of particles, we expect the symmetries also to involve of particles, we expect the symmetries also to involve a non-locality of a non-locality of OO[1/[1/

Renormalization can be understood in a Renormalization can be understood in a mathematically rigorous manner in this framework mathematically rigorous manner in this framework [SDJ: JPhyA01]. [SDJ: JPhyA01].

A formulation of causality by A formulation of causality by Bogoliubov and ShirkovBogoliubov and Shirkov

Bogoliubov and Shirkov formulated a condition that S-matrix is Bogoliubov and Shirkov formulated a condition that S-matrix is causalcausal

Ref: Quantum field theory: Bogoliubov and ShirkovRef: Quantum field theory: Bogoliubov and Shirkov The formulation rests on extremely general principles and The formulation rests on extremely general principles and

does not refer to any particular field theoretic formulation:does not refer to any particular field theoretic formulation: The interaction strength The interaction strength ‘g(x)’‘g(x)’ is a variable in the is a variable in the

intermediate state of formulationintermediate state of formulation S(g(x) ) : Is an operator acting on the states of the physical S(g(x) ) : Is an operator acting on the states of the physical

systemsystem S(g(x) ) is S(g(x) ) is unitaryunitary for a for a generalgeneral ‘g(x)’‘g(x)’: : Causality is preserved Causality is preserved only if only if a disturbance in g a disturbance in g(z)(z) at at ‘z’ ‘z’ does does

not affect evolution of state at any point not in the forward not affect evolution of state at any point not in the forward light-cone. S(g(x) ) at any point light-cone. S(g(x) ) at any point

Comments on the basic ingredients:Comments on the basic ingredients: In a QFT, with a Hermitian Interaction Hamiltonian, S-matrix is In a QFT, with a Hermitian Interaction Hamiltonian, S-matrix is

unitary. This is not altered by a variable unitary. This is not altered by a variable g(x)’g(x)’ . . The input regarding the causality is a very general and basic The input regarding the causality is a very general and basic

one.one.

† †S S SS I

A formulation of causality by A formulation of causality by Bogoliubov and Shirkov Bogoliubov and Shirkov (contd.) (contd.) B-S obtained the causality condition:B-S obtained the causality condition:

† 0, ~S g

S g x y Ig x g y

This is a necessary condition for causality to be preserved. Any violation of this condition necessarily implies causality violation (CV) in the QFT.

The above equation can be given a perturbative expression using the unitarity condition along with the perturbative expansion:

4 4 41 2 1 2 1 2

1

11 d d .... d ...... , ,.....,

! n n n nn

S x x x g x g x g x S x x xn

We do not, of course, observe directly Sn(x1, x2,….., xn ). We observe the integrated versions of these:

4 4 41 2 1 2

1 1

1 d d .... d , ,....., 1! !

n n

n n n nn n

g gS x x x S x x x S

n n

A formulation of causality A formulation of causality violation based on Bogoliubov-violation based on Bogoliubov-Shirkov criterionShirkov criterion

1 2 2 1 1

† † †2 3 1 2 2 1 2 1

3 1 2 2 1 2 1

1 1 1 1 1 1

, , ,

, , , , , , ,

, , , , ,

H x y iS x y iS x y iS x S y

H x y z iS x y z iS x S y z iS x y S z iS x z S y

iS x y z iS x S y z iS x y S z iS x z S y

iS x S y S z iS x S z S y

Causality condition (I) necessarily implies in particular:

H1(x,y) =0 x<~y, H 2(x,y,z) =0 x<~y, z

Thus, CV can be formulated in terms of H1(x,y), H 2(x,y,z), ….etc which contain perturbative expansion terms of the S-matrix. We can convert these in terms of observable quantities Sn ‘s

We take the O(1) and O(g) coefficients from (I) above to find

Construction of CV Construction of CV signalssignals Want to construct quantities that can, in principle, be Want to construct quantities that can, in principle, be

observed. These must be in terms of observed. These must be in terms of SSnn::

4 41 1 1

4 42 1 1

† † †2 3 1 2 2 1 2 1

3 1 2 2 1 2 1

1 1 1 1 1 1

4 4 42 2

d d ) , ) ,

d d ,

, , , , , , ,

, , , , ,

d d d ) )

H x y x y H x y y x H y x

x y iS x y iT S x S y

H x y z iS x y z iS x S y z iS x y S z iS x z S y

iS x y z iS x S y z iS x y S z iS x z S y

iS x S y S z iS x S z S y

H x y z x y y z H

, , 5x y z symmetric terms

Definition of H1 involves coincident points and hence their definition is ambiguous upto a constant counterterm.

Feynman rulesFeynman rules

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Contributing diagramsContributing diagrams

ResultsResults 222 2 processprocess: <: <HH11>> = = [s]+ [s]+ [t]+ [t]+ [u] [u] + + an unknown an unknown

constant counter-term that vanishesconstant counter-term that vanishes as as ∞ ; ; withwith

Small s : expand upto s2 and use s+t+u = 4m2

= = <<HH11>> =

Vanishes as ∞

Smaller by an order in (energy2/ )

Has no infrared or mass-singularity as m 0. No log (m) dependence.

Amplitude is real.

There are no physical intermediate states in the diagrams.

Results (contd.)Results (contd.) The above results can be generalized to all orders [JJ04]:

Requires an analysis of analytic properties in s, t, u and dependence on m. Uses the fact that the basic operator

On the other hand, for On the other hand, for s ~ s ~ [t] and [t] and [u][u] die off rapidly; die off rapidly;

while while [s] [s] increases very rapidly like an exponentialincreases very rapidly like an exponential..

Thus, CV begins to grow rapidly as energy approaches the Thus, CV begins to grow rapidly as energy approaches the scale scale of the theory. of the theory.

1

9;

4 2 1 !

ns ss A A A

n n

Results (contd.) Results (contd.)

224 process:4 process: Low Low s <<s << ( )CV a quadratic function of momenta

Total

expected from power counting

For For s ~ s ~ , again an exponential-like rise

Interpretation of ResultsInterpretation of Results An estimate of An estimate of can be had from precision tests of can be had from precision tests of

standard model. Thus, it is not a standard model. Thus, it is not a free parameter; free parameter; it has it has to be chosen consistent with data.to be chosen consistent with data.

Non-local theories with a Non-local theories with a finite finite have been proposed have been proposed as physically valid theories. as physically valid theories.

They have (at least) two possible interpretations:They have (at least) two possible interpretations: II: 1/ : 1/ represents scale of non-locality that determines represents scale of non-locality that determines

“granularity” of space-time. Then 1/“granularity” of space-time. Then 1/ is a fixed is a fixed property of space-time for any theoryproperty of space-time for any theory

IIII : The non-local : The non-local theory represents an effective field theory represents an effective field theory and the scaletheory and the scale represents the scale at which represents the scale at which the theory has to be replaced by a more fundamental the theory has to be replaced by a more fundamental theory.theory.

We can interpret the result in both frameworks, but We can interpret the result in both frameworks, but the meaning attached to it is different.the meaning attached to it is different.

Interpretation of Interpretation of ResultsResults Option I necessarily Option I necessarily requires a relatively large causality requires a relatively large causality

violation at s ~ violation at s ~ An observation of causality violation An observation of causality violation at these energies will bolster an interpretation of these at these energies will bolster an interpretation of these theories as a physical theory with first interpretation.theories as a physical theory with first interpretation.

In this picture, for low energies, the De Broglie wavelengh In this picture, for low energies, the De Broglie wavelengh the space-time scale of non-locality, and causality the space-time scale of non-locality, and causality violation would go unobserved. On the other hand, for violation would go unobserved. On the other hand, for energies ~energies ~

hhthe scale of non-locality. So it is not surprising if the scale of non-locality. So it is not surprising if CV becomes significant. CV becomes significant.

As a side remark, we note that in the classical limit, As a side remark, we note that in the classical limit, h h 0, 0, even for small momenta. CV is observed even even for small momenta. CV is observed even for small KE.for small KE.

Option II leaves the possibility that as Option II leaves the possibility that as s ≤ s ≤ the non-the non-local theory becomes less and less valid; because then local theory becomes less and less valid; because then we should have to use the underlying theory to calculate we should have to use the underlying theory to calculate quantities. In this case, the large CV obtained by quantities. In this case, the large CV obtained by calculation would be an artifact of approximation that calculation would be an artifact of approximation that replaces the more fundamental theory by an effective replaces the more fundamental theory by an effective non-local theory.non-local theory.

General ResultsGeneral Results

Infrared properties of the CV amplitude: It can be Infrared properties of the CV amplitude: It can be shown that as shown that as m m 0, 0, we do not produce a logarithmic we do not produce a logarithmic singularity. We consider the operatorsingularity. We consider the operator

†[ , ],

S g mO y i S g m

g y

That occurs in the CV amplitude. It is easy to show that O [y] is a hermitian operator. So the diagonal elements of O [y], viz. O y

Appendix : Exponential-like Appendix : Exponential-like growthgrowth

Action for NLQFT