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I'm a beginner trying to learn something about the Wightman axioms. I got the idea that from an abstract Hilbert space and an abstract Hamiltonian operator, I should be able to produce a trivial example of a "0+1-dimensional Wightman quantum field theory," but so far I haven't seen this spelled out.

Two bullet points on what I can see or guess, and one question:

  • Part of the Wightman axioms is a representation of the Poincare group on a Hilbert space. In 0+1 dimensions, the Poincare group is just time evolution. So our Hilbert space and $\exp(iHt)$ gives this part of the Wightman axioms.
  • Part of the Wightman axioms is a vacuum vector. This must be the ground state of $H$.

Here's my question:

  • Part of the Wightman axioms is a "field map." This is something like the data of operators $\phi(t)$ on the Hilbert space for each point $t$ in spacetime (= time, in $0+1$ dimensions). What are these operators supposed to be, in terms of the Hamiltonian? I got stuck trying to guess.
Qmechanic
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Upasker
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1 Answers1

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My view now is that the field map is additional information, not determined by the Hilbert space and Hamiltonian.

Wightman axioms imply the formula $\phi(t) = \exp(iHt) \phi(0)\exp(-iHt)$ holds, so the field map is determined by the operator $\phi(0)$. Or if there is more than one kind of field $\phi_1,\phi_2,\ldots$, then it is determined by the operators $\phi_1(0),\phi_2(0),\ldots$. No reason for these observables to be determined by the Hamiltonian $H$.

Example: If there is just one scalar field taking values in $\mathbf{R}$, then the Hilbert space is $L^2(\mathbf{R})$. The operator $\phi(0)$ is supposed to measure the strength of the field at time $0$ (and at the unique point of 0-dimensional space), i.e. it is the "position" operator on $L^2(\mathbf{R})$.

Corrections welcome!

Upasker
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