Causality Violation in Non-local QFT

1. Causality Violation in Non-local QFT S.D. Joglekar I.I.T. Kanpur Talk given at “100 Years After Einstein’s Revolution”: A National Conference to celebrate the World Year of Physics 2005 held at IIT Kanpur from 4-6 November 2005

2. Causality Violation in Non-local QFT PLAN 1. Why non-local QFT’s ? 2. Causality violation: classical and quantum 3. Formulation of causality violation using BS criterion 4. One-loop Calculations 5. Infrared and analyticity properties of causality violating amplitudes 6. Some all-order generalizations 7. Interpretation and Conclusions • References: • Ambar Jain and S.D. Joglekar, Int. Jour. Mod. Phys. A 19, 3409 (2004) i.e.-hep-th/0307208 • Basic works: • G. Kleppe, and R. P. Woodard, Nucl. Phys. B 388, 81 (1992). • 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).

3. Why non-local QFT’s? • Non-local QFT is a QFT that incorporates non-local interaction e.g. ∫d4xd4yd4 zd4w f(x,y,z,w) f(x) f(y) f(z) f(w) • Interest in non-local QFT’s is very old, dating from 1950’s: e.g. • Pais and Uhlenbleck (1950), • Effimov and coworkers (1970-onwards) • Moffat, Woodard and coworkers (1990--) • The interest was motivated by the “infinities” in local QFT’s. These are correlated to the local nature of interaction. • The basic idea was to try to avoid “infinities” by assuming a non-local interaction and thus providing a natural cut-off. • Also, the non-commutative QFT’s, currently being studied, and are a special case of a non-local QFT: The equivalent star product formulation is a non-local interaction. • We shall focus on the last type of non-local theories. These are more desirable compared to the earlier attempts in many ways, to be spelt out later.

4. Causality violation (CV) : Classical & quantum • Interaction Lagrangian is non-local: At a given instant, interaction may take place over a finite region of space: i.e. at points spatially separated. May introduce CV. • Classical violation of causality: For example action-at-a- distance. Such a classical violation of causality is undesirable from the point of view of experience. • For example, consider a system of stationary particles interacting via an action-at-a-distance of range R. These are placed at a distance R each • * * * * * * * * * * * * * * * * * * * * • A signal can instantaneously be communicated to any distance. • Can be observed at relatively larger distances • Quantum violations, (as we shall see) on the other hand are suppressed: g2/16p2 per loop: • Smaller in magnitude • Smaller in range • As we shall see, they are pronounced at larger energies • It is desirable that lowest order does not show CV: This is arranged if the tree order S-matrix is the same as local one.

5. Non-local QFT’s of Kleppe-Woodard type (contd) • There is a systematic procedure to construct a non-local action, given a local action. It involves a regulator function exp[(2 +m2)/2L2] • When the action is constructed, it is an infinite series. We reproduce a first few terms for the lf4theory:

6. Non-local QFT’s of Kleppe-Woodard type • To state briefly, the non-local version of the scalar f4 theory is given in terms of the Feynman rules • ------------- • ------|------ There is onlyone basic vertex, but external lines can be of either variety. X Do not take loops having all shadows lines.

7. Non-local QFT’s of Kleppe-Woodard type: Special Properties • Unlike higher derivative theories and many non-local theories, the asymptotic equation (interaction switched off) is identical to free theory. • No ghosts and no spurious extra solutions: These spoil meaning of quantization, and come in the way of unitarity. • S-matrix same in the lowest order as the free theory: No classical violation of causality • Theory unitary for any finite L. Can be interpreted as a bona-fide physical theory with a space-time/mass scale L(KW91) . • The theory has an equivalent non-local form of any of the local symmetries. • The theory has a quantum violation of causality (KW91) .

8. Interpretation of non-local QFT • Another interpretation has also been suggested [SDJ:IJMPA(01)]: Suppose that standard model arises from a theory of finer constituents as a low energy effective theory. Suppose that the compositeness scale is L. Then, the low energy theory would exhibit nonlocal interactions (via form-factors) of length-scale ~1/L. We thus expect the low energy effective theory to be • non-local, • unitary in the energy range of its validity, and • possessing equivalent of underlying residual symmetries. • On account of composite nature of particles, we expect the symmetries also to involve a non-locality of O[1/L]. The non-local theories under consideration fulfill these criteria. • It is an independent valid question, that starting from a fundamental theory that is causality preserving, whether the low energy “condensed” theory must also be causality-preserving. • Renormalization can be understood in a mathematically rigorous manner in this framework [SDJ: J. Phy. A (01)].

9. A formulation of causality by Bogoliubov and Shirkov • Bogoliubov and Shirkov formulated a condition that S-matrix is causal • (Ref: Quantum field theory: Bogoliubov and Shirkov) • The formulation rests on extremely general principles and does not refer to any particular field theoretic formulation: • The interaction strength ‘g(x)’ is a variable in the intermediate state of formulation • S(g(x) ) : Is an operator acting on the states of the physical system • S(g(x) ) is unitary for a general‘g(x)’: • Causality is preserved only if a disturbance in g(z) at ‘z’ does not affect evolution of state at any point not in the forward light-cone. • Comments on the basic ingredients: • In a QFT, with a Hermitian Interaction Hamiltonian, S-matrix is unitary. This is not altered by a variable g(x)’ . • In a gauge theory, it is easy to construct a BRS invariant action with a variable g(x). • The input regarding the causality is a very general and basic one.

10. A Diagrammatic derivation Stakes a state from –1 to 1 . S† takes a state from 1 to - 1 -1 -------------------------------------------------←----------------------1 S† -1 ----------------------------------------------------------------------- 1 S -1 ---------------------------x----------------------←--------------------1 S† -1 ---------------------------x------------------------------------------- 1 S -1 ---------------------------------------------←------------------------ 1 S† -1 -----------------------------------y-----------------------------------1 S -1 --------------------------x-------------------←------------------------ 1 S† -1 --------------------------x---------y----------------------------------1 S

11. A formulation of causality by Bogoliubov and Shirkov (contd.) • B-S obtained the causality condition: • 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: We do not, of course, observe directly Sn(x1, x2,….., xn ). We observe the integrated versions of these:

12. A formulation of causality violation based on Bogoliubov-Shirkov criterion We take the O(1) and O(g) coefficients from (I) above to find • 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

13. Construction of CV signals • Want to construct quantities that can, in principle, be observed. These must be in terms of Sn: Definition of H1involves coincident points and hence their definition is ambiguous upto a constant counterterm.

14. Feynman rules • ------------- • ------|------

15. Contributing diagrams

16. Results • 22 process: • <H1>= G0[s]+ G0[t]+ G0[u] + an unknown constant counter-term that vanishes asL  ∞ ; with Small s : expand upto s2and use s+t+u = 4m2 = <H1>= • Vanishes as L ∞ • Smaller by an order in (energy2/L2 ): Holds to all orders • Has no infrared or mass-singularity as m 0. No log (m) dependence. Holds to all orders. • Amplitude is real. Holds to all orders. • There are no physical intermediate states in the diagrams.

17. Results (contd.) • On the other hand, for s ~ L2 , G0[t] and G0[u]die off rapidly; while G0[s] increases very rapidly like an exponential. • Thus, CV begins to grow rapidly as energy approaches the scale L of the theory.

18. Results (contd.) • 24 process: • Low s << L2 • expected from power counting • For s ~ L2 , again an exponential-like rise.

19. Generalization of 1-loop results • Many of the above 1-loop results can be generalized to all orders. These are: • Absence of infrared divergences in CV amplitude even as m  0. • Finiteness of CV amplitude • Suppression for 22 process at low energies • Lack of physical intermediate states in cuts. • As far as the structure of the CV for the 4-point function at low energies is concerned, the essential property necessary is the ability to expand G(s,t,u) as a Taylor expansion at least upto O(s,t). This requires that singularities that can lead to s lns, s ln t, and t lnt terms are absent. • An analysis of the singularities that arise from intermediate states and of the nature of mass-singularities of diagrams is needed. Now, consider For a local variation of g(x) g(x)+dg(x). Local variation cannot affect infra-red properties. Hence

20. Generalization of 1-loop results Matrix elements of have a smooth limit as m0, i.e. no mass singularities. Now, the Hn are constructed by real operations from O And hence do not have mass singularities. Some of the required analyticity properties are obtained by noting that O(y) above is a hermitian operator And hence does not develop imaginary part from any physical intermediate states.

21. Interpretation of Results • An estimate/bound of L can be had from precision tests of standard model. Thus, it is not a free parameter; it has to be chosen consistent with data. • Non-local theories with a finite L have been proposed as physically valid theories. • They have (at least) two possible interpretations: • I: 1/ L represents scale of non-locality that determines “granularity” of space-time. Then 1/L is a fixed property of space-time for any theory • II : The non-localtheory represents an effective field theory and the scaleL represents the scale at which the theory has to be replaced by a more fundamental theory. • We can interpret the result in both frameworks, but the meaning attached to it is different.

22. Interpretation of Results • Option I necessarily requires a relatively large causality violation at s ~ L2 . An observation of causality violation at these energies will bolster an interpretation of these theories as a physical theory with first interpretation. • In this picture, for low energies, the De Broglie wavelengh l << the space-time scale of non-locality, and causality violation would go unobserved. On the other hand, for energies ~L, • l ~ h/L, the scale of non-locality. So it is not surprising if CV becomes significant. • As a side remark, we note that in the classical limit, h  0, l  0 even for small momenta. CV is observed even for small KE. • Option II leaves the possibility that as s ≤ L2 , the non-local theory becomes less and less valid; because then we should have to use the underlying theory to calculate quantities. In this case, the large CV obtained by calculation would be an artifact of approximation that replaces the more fundamental theory by an effective non-local theory.

24. Appendix : Exponential-like growth

25. Action for NLQFT