A Small Insight on Quantum Field Theory

I’ve been thinking about quantum field theory for a while now, trying to
understand it.. Here is a small insight I recently made.

One way of thinking about the configuration space is through the field
perspective. There are observables at every point in space, they obey the
Klein-Gordon equation with nonlinear perturbations. Taking the Fourier
transform, you get at every momentum a set of harmonic oscillators, and the
interactions are couplings between them. Quantizing this, each oscillator gets
an integer spectrum which corresponds to how many particles of a given type
there is in a given momentum. So now we get the particle perspective in momentum

Another perspective, which is usually used in less rigorous descriptions of QFT,
is the particle presentation in position space. For example, this is implicit
in the “cloud of virtual particles” intuition. It seems to work, but nobody uses
it in calculation. Presumably it can described precisely by taking the Fourier
transform of the momentum space particles to get position space particles. Like
the field presentation, it seems to be manifestly local.

My small insight is this: there is a direct way to describe this perspective in
terms of the field perspective. It is this: a field at a given point over time
behaves somewhat like a harmonic oscillator. Take the basis derived from the
fixed energy states of these harmonic oscillators to get position-space particle

Some consequences:

  • A massless field doesn’t behave like a harmonic oscillator at a fixed position because there is no spring constant. This explains a claim I heard that position
    space doesn’t work for massless particles. However, extrapolating the creation
    operator as the spring consant goes to zero, it looks like something similar can
    be made to work when thinking of particle count as an index for a
    Taylor-expansion-like decomposition of the wavefunction into polynomials.
  • This position space perspective is not Lorenz-invariant. This partially
    explains the claim I heard that position space also doesn’t work for massive
    particles in the relativistic theory. It also explains why nobody uses this
    perspective seriously.

1 thought on “A Small Insight on Quantum Field Theory

  1. In fact people do often do Feynman diagrams in position space; in this approach you label the vertices of the diagram with positions, unlike the momentum space picture where you label the edges of the diagram with momenta. It is related to the momentum picture by a Fourier transform, and it is manifestly Lorentz invariant if you work in the Heisenberg picture. People use it less often in flat spacetime (in part because the propagator for a massive particle is a complicated Bessel-type function), but it is often more useful in situations where the particles are propagating in a background field that doesn’t have translation symmetry (e.g. in curved spacetime).

    However, the position space picture does not have the locality properties one might naively attribute to it. For example, if I start with the vacuum (in Minkowski) and then act with a local operator at t = 0, in some spatial region R, then I get a 1 particle state with the position inside of R, but this state does NOT have the property that at t = 0 the local operators in the complement of R take on their vacuum expectation values.

    Also, I don’t think you can get this picture in the way you suggest, by quantizing the particle at a given spatial position (varying time), because it is actually very far from being true that “a field at a given point over time behaves somewhat like a harmonic oscillator”. How the field at a given point evolves with time depends on what is going on at nearby points, and its evolution need not be anything like a sinusoid, except in the sense in which any function can be Fourier-decomposed into sine waves.

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