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Understanding the states in Quantum Field Theory

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I am self-studying quantum field theory, and I've been struggling to understand the nature of the states that emerge in quantum field theories.

After thinking about it, what I think one has in the state is like having a wavefunction at each point in space (or in momentum space). Basically, we are substituting a variable with a wavefunction, and we had a variable at each point (a classical field). This picture agrees with the standard QM being a zero-dimensional field.

For example, I've been watching a lecture in which phonons were explained by quantizing a field. The classical fields were the separation from the equilibrium position of each atom and its momentum. Fourier-transforming these we get a superposition of harmonic oscillators. Now, we promote these to operators. Ok, now what I imagine is that these quantum-harmonic-like operators should act on a quantum-harmonic-like space. Also, following the analogy, the one particle state for a specific p would look spread out in the basis of the eigenstates of the operator that corresponds to the amplitude of this Fourier mode.

So, to summarize, am I right in saying that the states in the Fock states are not just plane waves, but plane waves with a fuzzy amplitude? Does this fuzziness have any implications?

This post imported from StackExchange Physics at 2014-10-11 09:57 (UTC), posted by SE-user guillefix
asked Oct 9, 2014 in Theoretical Physics by guillefix (25 points) [ no revision ]
retagged Oct 11, 2014
I may answer this proprly later, but for now this might be helpful: physics.stackexchange.com/q/122570

This post imported from StackExchange Physics at 2014-10-11 09:57 (UTC), posted by SE-user DanielSank

Are "states in the Fock states [...] plane waves with a fuzzy amplitude?" Unless you're careful here to be clear that "fuzzy" is not a classical fuzziness of uncertainty about what the amplitudes of the field at different points of space and time are, what you're asking here is "are states classical?". So, unless you're careful about fuzziness, the answer to your final question is no, because there are nontrivial commutation relations at time-like separation. This is a quantum field theory.

1 Answer

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No, one cannot say this. The pure states of a quantum field theory are not fuzzy - they are functionals of the fields at a fixed time (diagonalizing the corresponding field operators); see the article by Jackiw,  Analysis on infinite dimensional manifolds: Schrodinger representation for quantized fields (p.78-143 of the linked document).

In the special case of a free field theory, where number operators make sense, one can write the states in the representation diagonalizing the number operator and the momentum operator, which leads to the Fock representation. For interacting fields, a number operator does not exist, and the Fock representation is only approximately valid (as in renormalized perturbation theory).

answered Oct 12, 2014 by Arnold Neumaier (12,365 points) [ no revision ]

Wow, I feel like I need to relearn all of calculus to work with this (functional analysis). I didn't understand everything but it seems to me that a functional of the field is just a wavefunction over the space of all possible field configurations, and that is what I meant by fuzzy, that the value of the field isn't certain at any point. However, I would need to learn more to say more really.
 

Also, do you what would be the advantages of learning functional analysis to go through this vs. the more standard approach (which seems to be based more on commutation relations)?

Thanks for the help.

The wave functional of a field is no more uncertain than the wave function of a single particle. But it is misleading to refer to fuzziness when the word uncertain (common in quantum mechanics) fully describes the situation. In a mathematical context, the word fuzzy has quite a different meaning.

The functional picture is very useful when you want to understand things beyond perturbation theory. See, e.g., the discussion in http://www.physicsoverflow.org/22342/

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