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This question refers to the differences between usual formalism of ordinary quantum mechanics (QM) and usual formulation of QFT. Speciffically, there are three questions I would like to know:

  1. The Hilbert space of a relativistic QFT must carry a unitary representation of the universal cover of the Poincaré group. But it seems this a mathematical difficult problem for the case of interacting fields. What are the mathematical difficulties in finding such a space? How does whether they are free or interacting fields affect the choice of the relevant Hilbert space?

  2. I have read here that functorial QFT (FQFT) is an approach that extends the Schördinger's picture to QFT. But what would be its evolution equation equivalent to the Schrödinger equation? (See here for discussion about it)


Below, I provide a bit of context to understand where these very specific questions arise from.

In ordinary (non-relativistic) QM one starts from a separable Hilbert space $\mathcal{H}$, whose vectors $|\psi\rangle\in\mathcal{H}$ are interpreted as "states" of physical systems with a finite number of particles (hence with a finite number of Degrees of Freedom [DoF]). Then, the Schrödinger equation $\hat{H}|\psi\rangle = i\hbar\partial_t|\psi\rangle$ provides the law of evolution, and given an observable quantity $\hat{M}$ the expected value in the state $|\psi\rangle$ is calculated as:

$$\langle\hat{M}\rangle_\psi =\langle\psi|\hat{M}\psi\rangle$$

and similarly, the transition probabilities from one state to another.

But QFT describes systems with non-finite number of DoF, in fact a state would require defining what happens at each point in space-time, so that states cease to be properties of particle systems and become "states of space-time". Therefore, now the fields become operators on those states, in particular a field is an operator-valued distribution, so the value of the field at a point in space-time is given by:

$$\langle \mathbf{F}(\boldsymbol{x}) \rangle_{\Omega_1}= \langle \Omega_1 | \mathbf{F}(\boldsymbol{x})\Omega_1\rangle$$

In QFT, mean values and probabilities are calculated, usually between "collision states", in which space-time contains only a number of asymptotic particles that approach, interact and separate, allowing effective cross sections to be determined, which can be compared with experiments. Usually, such calculations for the case of QED can be approached by perturbative methods in terms of Feynman diagrams. However, there are three issues that are not clear to me:

  1. How could we describe the Hilbert space underlying the states of space-time (one of which would be the quantum vacuum of the corresponding theory)?
  2. Why does not the equivalent of the Schrödinger equation in this state space seem interesting? Or, equivalently, the Heisenberg picture, of the evolution of operators of this theory?
  3. Why is the approach of trying to find the self-states of a field theory not useful in the case of QFT? Or is the underlying mathematical problem too complicated?

Added: I reformulate the same question here, and I recived some very interesting answers. So it can be seen that the question made sense and was relevant.

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    $\begingroup$ The Hilbert space of a relativistic QFT must by definition carry a unitary representation of the universal cover of the Poincaré group - that is a constraint of relativistic symmetry. The issue is that finding such a Hilbert space and the corresponding representation in a complete interacting theory is a very intricate open problem. Nevertheless, in free theories the problem is tractable, and when discussing the scattering problem such construction can still be used asymptotically. For a very good description of this see Weinberg's QFT textbook, volume 1, chapters 2 and 3. $\endgroup$
    – Gold
    Aug 6 at 0:02
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    $\begingroup$ @Davius You often don't use a Hilbert Space in QFT, you use a Fock Space (for states with changing particle number). en.wikipedia.org/wiki/Fock_space $\endgroup$
    – hft
    Aug 6 at 0:04
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    $\begingroup$ "Why did we never ask..." Who is "we"? Why do you assume we didn't ask such questions? $\endgroup$
    – hft
    Aug 6 at 0:05
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    $\begingroup$ @hft A Fock space is a special case of a Hilbert space. While the precise structure of the Hilbert space of interacting field theories is an open question, we know that it is not a Fock space. So, in fact, it is more accurate to refer to the Hilbert space of a QFT rather than to its Fock space (which would only make sense for a free QFT). $\endgroup$ Aug 6 at 0:23
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    $\begingroup$ @hft Wikipedia is a great resource. But not as authoritative as a good textbook on infinite-dimensional vector spaces. The issue is similar to claiming that a tensor space is not a special case of vector space. $\endgroup$
    – GiorgioP
    Aug 6 at 6:02

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