Causality Violation in Non-local QFT

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Causality Violation in Non-local QFT

• S.D. Joglekar
• I.I.T. Kanpur

Talk given at 100 Years After Einsteins
Revolution A National Conference to celebrate
the World Year of Physics 2005 held at IIT
Kanpur from 4-6 November 2005
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Causality Violation in Non-local QFT

• PLAN
• 1. Why non-local QFTs ?
• 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).

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Why non-local QFTs?

• Non-local QFT is a QFT that incorporates
  non-local interaction
• e.g. ?d4xd4y d4 zd4w f(x,y,z,w) f(x) f(y) f(z)
  f(w)
• Interest in non-local QFTs is very old, dating
  from 1950s 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 QFTs. 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 QFTs, 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.

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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.

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Non-local QFTs 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
  lf4 theory

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Non-local QFTs 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 only one basic vertex, but external
lines can be of either variety. X Do not take
loops having all shadows lines.
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Non-local QFTs 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) .

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Interpretation of non-local QFT

• Another interpretation has also been suggested
  SDJIJMPA(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 O1/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).

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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.

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A Diagrammatic derivation

• S takes 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

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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
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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 xlty, H 2(x,y,z) 0 xlty, 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

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Construction of CV signals

• Want to construct quantities that can, in
  principle, be observed. These must be in terms of
  Sn

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

• -------------
• ------------

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Contributing diagrams
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Results

• 2?2 process
• ltH1gt G0 s G0t G0u an unknown constant
  counter-term that vanishes as L ? 8 with

Small s expand upto s2 and use stu 4m2
ltH1gt

• Vanishes as L ?8
• 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.

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Results (contd.)

• On the other hand, for s L2 , G0t and
  G0u 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.

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Results (contd.)

• 2?4 process
• Low s ltlt L2

• expected from power counting
• For s L2 , again an exponential-like rise.

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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
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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.
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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-local theory represents an effective
  field theory and the scale L 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.

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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 ltlt 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.

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Appendix Exponential-like growth
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Action for NLQFT