Causality Violation in Non-local QFT

1. Causality Violation in Non-local QFT S.D. Joglekar I.I.T. Kanpur Talk given at THEP-I, held at IIT Roorkee from 16/3/05—20/3/05

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. Some all-order generalizations 6. 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).

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 basic idea was to try to avoid “infinities” by assuming a non-local interaction and thus providing a natural cut-off. • 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.

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. 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 • Can be observed also at low momenta • Quantum violations, 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 should 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 • 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

6. 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) .

7. Interpretation of non-local QFT • Another interpretation has also been suggested [SDJ:IJMPA01]: Suppose that standard model arises from a theory of finer constituents as a low energy effective theory. Suppose that the compositeness scale is . 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]. • Renormalization can be understood in a mathematically rigorous manner in this framework [SDJ: JPhyA01].

8. 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. S(g(x) ) at any point • 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)’ . • The input regarding the causality is a very general and basic one.

9. 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:

10. 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

11. 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.

13. Feynman rules • ------------- • ------|------

14. Contributing diagrams

15. 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 ) • 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.

16. 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 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.

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

18. Interpretation of Results • An estimate 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.

19. 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.

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

22. Appendix : Exponential-like growth

23. Action for NLQFT