In QFT in Minkowski Spacetime it is usual to link the one-particle states to unitary representations of the Poincaré group.

The argument, which can be seen in Weinberg's QFT book, is roughly as follows: Minkowski spacetime has the Poincaré group as its isometry group. By the postulates of special relativity we then expect that the Poincare group $\mathfrak{P}(1,3)$ be a symmetry group for a quantum theory compatible with special relativity.

This leads, by Wigner's theorem, that if $\mathfrak{H}$ is the Hilbert space of a quantum theory with such symmetries, there be an unitary linear projective representation $\pi : \mathfrak{P}(1,3)\to GL(\mathfrak{H}).$

Weinberg then assumes that $\pi$ is actually a representation and not a projective one (implying the phase is zero).

In that setting, he shows that the presence of this representation leads to the existence of the momenta and angular momenta operators $P^\mu, J^{\mu\nu}$.

So from the presence of Poincare symmetry the one-particle state space has the observables we would expect to describe particles. Then in the standard way to quantize the fields, we build the necessary Fock space over this one particle space.

Non in QFT in curved spacetimes, following the algebraic route, we have on $\ast$-algebra $\mathfrak{A}$ and states are positive normalized functionals $\omega : \mathfrak{A}\to \mathbb{C}$. Each states generate a Hilbert space representation through the GNS construction.

It turns out now that given a so-called gaussian state $\omega$, the GNS representation generated has as its Hilbert space a Fock space and $\omega$ is mapped into the vacuum. The ones from Minkowski Spacetime QFT can then be seen as special cases of this.

But there is a caveat here: in my understanding, a Fock space representation suggests a "variable number of particles" picture. Now in the Minkowski spacetime case, the Poincare symmetry of the background, when required of the representation, leads to a Fock space over a one-particle space which has momenta and angular momenta observables naturally.

Now in this general gaussian state picture in curved spacetimes we have a Fock space. But I cannot see how the one-particle space has momenta and angular momenta observables naturally. Without these how could we interpret the states as particle states?

So my question here is: how do we interpret the Fock space picture derived from gaussian states in curved spacetime QFT if we lack Poincare symmetry? How does momenta and angular momenta appear as observables in the related one-particle state space, if not as the generators of translations and rotations?

  • $\begingroup$ Why do you say we "lack Poincare symmetry"? In curved space it just becomes a local symmetry. $\endgroup$
    – Kosm
    Mar 17, 2018 at 15:17

1 Answer 1


The same problem appears already in Quantum Field Theory (QFT) in flat spacetime but in presence of time-dependent external potentials (i.e. classical fields coupling to the quantum field). When they break the Poincare symmetry it does not make much sense to speak about particles (particle states) locally as their notion is no more unique (as before, related to the symmetries of the background). Instead, one tries to give them meaning asymptotically, e.g. in in and/or out representations (of the operator algebra, e.g. canonical (anti)commutation relations), for $t\rightarrow\pm\infty$, where the external fields are switched-off. This workaround is very popular in physics, however, not completely satisfactory, as e.g. some long-range interactions, like Coulomb fields, do not vanish asymptotically sufficiently fast.


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