microcausality and locality

Question

There is this thing I got confused:

Microcausality is the statement that spacelike separated local field variables commute so that we can specify field variables on a spatial slice as a complete base. It is usually referred to as a statement about locality---if microcausality is broken then the "local" operators are not that "local".

There is another statement about the notion of "locality" in S-matrix language---an S-matrix have poles corresponding to particle exchange, and the residue factorizes into S-matrices of sub scattering processes in the limit that these processes happen far from each other. It is in the line of cluster decomposition principle.

So my question is: do these two statements somehow have connections to each other, or even are equivalent? Or they are simply two very different statement and not connected at all?

This post imported from StackExchange Physics at 2014-05-04 11:31 (UCT), posted by SE-user user106592

Answer 1

My answer is somewhat complementary to Arnold Neumaier's. The cluster decomposition principle for the S-matrix can be derived from microcausality and the relativistic spectrum condition that defines the vacuum state (Lorentz invariance is actually not necessary), assuming that the vacuum is a pure state (which amounts to saying it is the only translation invariant vector state in the vacuum Hilbert space). More precisely, in this case one can show by means of the Jost-Lehmann-Dyson representation for truncated correlation functions that truncated vacuum expectation values must have exponential (Yukawa-like) decay in spacelike directions for theories with a mass gap; the same argument leads to a power-law (Coulomb-like) decay in spacelike directions in the absence of a mass gap. When you plug this result into the LSZ reduction formula, you get the corresponding cluster decomposition for the S-matrix.

More importantly, the above cluster property of truncated vacuum expectation values is crucial for the derivation of the LSZ reduction formula when the theory has a mass gap (see, for instance, H. Araki, "Mathematical Theory of Quantum Fields", Oxford (2000) or M. Reed and B. Simon, "Methods of Modern Mathematical Physics III - Scattering Theory", Academic Press (1979), section XI.16). If there is no mass gap, stricly speaking the formula no longer holds due to infrared divergences (the Coulomb-like decay is too slow for the formula to hold, as it happens in non-relativistic quantum mechanics with long-range potentials like Coulomb's). In this case one must "fix" the reduction formula by resorting to a "zero-recoil" approximation for soft processes (Bloch-Nordsieck, etc.), which forces us to restrict to the domain of physical validity of this approximation, i.e. infrared-safe processes. A mathematically rigorous justification of this approximation is to date still lacking, though.

Comments

I'm quite curious, how is “assuming that the vacuum is a pure state” equivalent to "saying it is the only translation invariant vector state in the vacuum Hilbert space" ? Does the "pure state" have the same meaning as in quantum statistical mechanics?

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"Pure" means as an algebraic state on the field algebra (i.e. a positive, normalized linear functional defined by taking expectation values of polynomials of field operators). This means it cannot be written as a nontrivial convex linear combination of two other algebraic states. It can be shown that a vacuum state being pure is equivalent to uniqueness of the vacuum vector state in the above sense (see, for instance, Araki's book quoted in my answer). So, in a sense, yes, it's just like in quantum statistical mechanics.
 

Answer 2

This is the topic of Chapter 4 (''The Cluster Decomposition Principle'') of Volume 1 of  Weinberg's QFT book. Your second notion of locality is just cluster decomposition, whereas your first notion of locality is the microcausality condition on field operators. Section 4.4 shows that under reasonable assumptions, the only way to ensure cluster decomposition is to have an interaction Hamiltonian of a special form (4.4.1) involving creation and annihilation operators satisfying CCR/CAR (4.2.5), which makes the interaction density and certain other field operators constructed from it satisfy microcausality.

As shown by the discussion in Section 3.5, microcausality is also needed to ensure the Lorentz invariance of the S-matrix, which enters Weinberg's argument about the cluster decomposition.