Commuting observables for Fermion fields?

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Discussion Overview

The discussion centers on the concept of commuting observables in the context of fermion fields, particularly free Dirac fields, within quantum field theory (QFT). Participants explore the challenges of defining a complete set of commuting observables for fermions, contrasting it with the more straightforward cases for bosons.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes that in nonrelativistic quantum mechanics (QM), a complete set of commuting observables is used to define the Hilbert space, providing examples from QM and QFT for boson fields.
  • Another participant questions what a complete set of commuting observables would look like for a free Dirac field, suggesting quadratic expressions in the fermion fields as a potential direction.
  • A subsequent reply acknowledges the idea of using quadratic expressions and inquires about the possibility of forming eight commuting Hermitian operators based on the real and imaginary parts of the four components of the field at each space point, asking about their spectra.
  • Another participant points out that the problem is ill-defined since the fields are distribution-valued, which leads to a lack of spectra, but suggests that discretizing space could allow for a finite collection of qubits that can be diagonalized.
  • A later post raises the question of whether the operators would have eigenvalues of zero and one, framing it as a yes/no question regarding the presence of a particle at a point, and asks about the number of such operators.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the definition of a complete set of commuting observables for fermion fields, with multiple competing views on how to approach the problem. The discussion remains unresolved, with no consensus on the specifics of the operators or their spectra.

Contextual Notes

Participants note limitations related to the distribution-valued nature of fields and the implications for defining spectra, as well as the potential need for discretization to facilitate analysis.

maline
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In nonrelativistic QM, we usually describe the Hilbert space by choosing a complete set of commuting observables, so that the set of states that are eigenstates of all the observables can be used as a basis. For instance, the "wavefunction" is the state as expressed in terms of "states" with definite positions for all particles (not normalizable, but that can be dealt with), which works because the positions (3 for each particle) are a complete set of commuting observables. Another classic example is the n,l,m,s basis for the states of the hydrogen atom.

For free boson fields, this concept seems to carry over well into QFT. For instance, in the case of a complex scalar field, the real and imaginary parts of the field at each point (on a spacelike hypersurface) are Hermitian and commute, so we can think of the well-defined field configurations as "basis states", and then a general state will be a functional giving a quantum amplitude to each configuration.

This "simple" setup obviously does not work for fermion fields. There, we are given anticommutation relations, which don't seem very helpful for defining basis states. Even the real and imaginary parts of the field at the same point do not commute.

What would a complete set of commuting observables look like for a fermion field, say a free Dirac field, expressed in terms of the field operators?
 
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maline said:
What would a complete set of commuting observables look like for a fermion field, say a free Dirac field, expressed in terms of the field operators?
quadratic expressions in the fermion fields
 
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A. Neumaier said:
quadratic expressions in the fermion fields
Yes, I guessed that that could be a good direction. Do you have a detailed idea for a complete set? Can the products be arranged so as to form, say, eight commuting Hermitian operators (for the real & imaginary parts of the four components) at each space point? What would be the spectra of these operators?
 
maline said:
Yes, I guessed that that could be a good direction. Do you have a detailed idea for a complete set? Can the products be arranged so as to form, say, eight commuting Hermitian operators (for the real & imaginary parts of the four components) at each space point? What would be the spectra of these operators?
Strictly speaking, this is an ill-defined problem as the fields are distribution-valued only and hence have no spectra.

But if you discretize space you get a finite collection of qubits, and you can diagonalize each one separately. Then take a continuum limit.
 
Okay, so we're talking about operators with eigenvalues zero and one? A yes/no answer to "is there a particle at this point"?
Would there be four or eight such?
 

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