This is related to the Osterwalder-Schrader posting, but is much, much more elementary. I’ll write up some basic facts about the quantum harmonic oscillator and explain what bothers me about the relation to Osterwalder-Schrader.
Every quantum mechanics course covers the quantum harmonic oscillator, generally in the Schrödinger picture, with states functions of space and time. The Hamiltonian is a second order pde, one finds its eigenfunctions and eigenvalues. For a version of this that I wrote, see chapter 22 here.
Free quantum field theories are just infinite collections of such harmonic oscillators, but in QFT one wants to use the Heisenberg picture (the Schrödinger picture would be very awkward). For a single quantum harmonic oscillator in the Heisenberg picture, one has two operators $Q(t),P(t)$, with Hamiltonian
$$H=\frac{1}{2}\left(P^2 +\omega^2 Q^2\right)$$
(here I’m rescaling so that $m=\hbar=1$). The Heisenberg equations of motion are
$$
\frac{d}{dt}Q=i[H,Q]=P,\ \ \frac{d}{dt}P=i[H,P]=-\omega^2Q
$$
Subsituting the first in the second, one gets the second-order equation of motion
$$\left(\frac{d^2}{dt^2}+\omega^2\right)Q=0$$
Such equations can most easily be solved by complexifying (allowing not just real, but complex linear combinations of solutions). Using complex linear combinations of operators, one can write
$$a=\sqrt{\frac{\omega}{2}}Q+i\sqrt {\frac{1}{2\omega}}P,\ \ a^\dagger=\sqrt{\frac{\omega}{2}}Q-i\sqrt {\frac{1}{2\omega}}P$$
which turns the first order equations into
$$\frac{d}{dt}a=-i\omega a,\ \ \frac{d}{dt}a^\dagger=i\omega a^\dagger$$
with solutions
$$a(t)=a(0)e^{-i\omega t},\ \ a^\dagger(t)=a^\dagger(0)e^{i\omega t}$$
The Heisenberg commutation relations are the time-independent
$$[a,a^\dagger]=1$$
and the Hamiltonian is
$$H=\frac{\omega}{2}(aa^\dagger +a^\dagger a)$$
One can then easily show that the state space has a basis
$$\ket{0},\ket{1},\ket{2},\ldots$$
with
$$H\ket{n}=\omega (n+\frac{1}{2})\ket{n}$$
This in some sense is the simplest possible quantum system and easily extends to a quantum field theory describing arbitrary numbers of non-relativistic particles of mass $m$. Just put together an infinite collection of such oscillators, with operators $a_{\mathbf p},a^\dagger_{\mathbf p}$, parametrized by the possible momenta $\mathbf p$, with
$$\omega=\omega_{\mathbf p}=\frac{|\mathbf p|^2}{2m}$$
If one wants to describe fermions, just change commutation relations to anti-commutation relations. This system is exactly the starting point of many-body physics methods for dealing with condensed matter systems.
The usual field operators are the Fourier transforms of these operators parametrized by momenta to operators parametrized by space:
$$\widehat{\psi}(t,\mathbf x)=\frac{1}{(2\pi)^{\frac{3}{2}}}\int_{\mathbf R^3} e^{i\mathbf p \cdot \mathbf x}a_{\mathbf p}(t) d^3\mathbf x$$
This is a wonderfully simple story, but it bothers me that it doesn’t seem to fit at all the Euclidean QFT philosophy of starting with an imaginatry time theory, then using OS reconstruction to get the physical theory.
The simplest case of the Osterwalder-Schrader theory would describe a harmonic oscillator in a more complicated way, using not the first-order equations of motion but the second order equation. Still complexifying, $a$ satisfies the second-order equation
$$\left(\frac{d^2}{dt^2}+\omega^2\right)a=\left(\frac{d}{dt}+i\omega\right)\left(\frac{d}{dt}-i\omega\right)a=0$$
This has twice as many solutions as our earlier version, with the new solutions complex conjugates of the old ones. Physically the problem with them is that they have negative energy.
One can deal with the new solutions by defining a separate state space and separate operators $b,b^\dagger$, solving the negative energy problem by interchanging the role of annihilation and creation operators. Now, besides states of quanta, one also has “anti-quanta”, which one can metaphorically describe as “quanta traveling backwards in time.”
This is a theory of a quantum complex harmonic oscillator, with two adjoint operators
$$a(0)e^{-i\omega t} +b^\dagger (0) e^{i\omega t}\ \ \text{and}\ \ a^\dagger(0)e^{i\omega t} +b(0) e^{-i\omega t}$$
To get back to the usual state space with just quanta, one can identify quanta and anti-quanta, i.e. $a=b, a^\dagger=b^\dagger$. Then there is just one kind of operator, the self-adjoint
$$a(0)e^{-i\omega t} +a^\dagger (0) e^{i\omega t}$$
This last theory is a relativistic real scalar field in 0+1 dimensions. It has a sensible imaginary time version and the OS reconstruction theorem applies. For more about the details of how this works, see for examples section VII.4 of this paper.
A simple question that’s bothering me is that I haven’t run across a discussion of OS reconstruction that applies to the case of the complex harmonic oscillator. If someone is aware of such a thing, please let me know about it.
For the case of the simplest possible description of the harmonic oscillator as given in the beginning of this posting, I’ve always been bothered not just by the fact that something like Osterwalder-Schrader doesn’t seem to apply, but even more by the fact that it’s hard to come up with a consistent path integral formalism that would describe it, even in imaginary time.
During one period in my life I spent a great deal of time thinking about this. There’s a whole subject of “coherent state path integrals” (although they’re not really integrals), with a large literature. A good discussion of the subject is chapter 6 (“path integrals and holomorphic formalism”) of Jean Zinn-Justin’s Path Integrals in Quantum Mechanics (for a more public domain version see here).
Besides the harmonic oscillator case (quantization of $\mathbf C$) case, even simpler should be the spin degree of freedom (quantization of the Riemann-sphere). I ended up convinced that the only way to make sense of such a path integral would be with a supersymmetric path integral, of the sort been related to the index theorem. For an early write up of some of this, see here.
My current point of view is that what one wants is not a purely Euclidean path integral, but a formalism holomorphic in the time variable, so in the realm of complex analysis rather than real analysis. Still stuck on some of the details of this, hope to soon have the energy to get back to that and get something written up.
In case it’s not clear, the ultimate motivation of this is to come up with a better way of understanding some of the things that are confusing about the Standard Model, in particular the treatment of chiral spinor fields. I’ll try to write soon the promised blog entry about the other Osterwalder-Schrader paper, the one dealing with Euclidean Fermi fields.
