Structure of Matter in Quantum Field Theory

[DarMM]

Yes.

These are not different sets of fields. They are different descriptions in terms of fields. I look at it the same as a change of inertial frames in SR: you're describing the same underlying thing, just in different coordinates. Similarly, symmetry breaking doesn't change the underlying thing, but it does change which description of it is most useful.

Fascinating. And what would you think of the fact that fields don't have well defined values at a point, i.e. $\phi(x)$ is undefined? That the fundamental things are the smeared fields, $\phi(f) = \int_{\mathcal{M}}{\phi(x)f(x)d^{4}x}$?

 

[PeterDonis]

what would you think of the fact that fields don't have well defined values at a point, i.e. $\phi(x)$ is undefined? That the fundamental things are the smeared fields, $\phi(f) = \int_{\mathcal{M}}{\phi(x)f(x)d^{4}x}$?

Well, since you and I are not point particles, I don't see the problem.

Seriously, since we never make measurements of anything at an exact point, I don't see the problem. "Smeared fields" seems like a better description of what we actually measure anyway.

 

[DarMM]

Well, since you and I are not point particles, I don't see the problem.

Seriously, since we never make measurements of anything at an exact point, I don't see the problem. "Smeared fields" seems like a better description of what we actually measure anyway.

Yeah I agree. This would tend to suggest that the fundamental object is really the local observable algebra and that any choice of fields is a particular basis/decomposition of it. That genuinely is invariant and carries the representations of symmetries we expect and is detached from the observer dependent notion of a Hilbert space.

Of course one could ask is this really what I'm made of, or if it simply defines a set of admissible and complementary classical descriptions of what I'm made of (i.e. the old Copenhagen "QM doesn't tell you anything about reality" that you also see in the Consistent histories view). However that's going into interpretations so I won't bother.

Regardless the fundamental thing is the sheaf of local observables.

 

[PeterDonis]

This would tend to suggest that the fundamental object is really the local observable algebra and that any choice of fields is a particular basis/decomposition of it.

I would agree.

 

[A. Neumaier]

And what's that if not particles?

We are made of matter described by states of a quantum field theory. Just as in classical mechanics we were supposed to be made of particles described by phase space coordinates.

 

[Peter Morgan]

And what would you think of the fact that fields don't have well defined values at a point, i.e. $\phi(x)$ is undefined? That the fundamental things are the smeared fields, $\int_{\mathcal{M}}{\phi(x)f(x)d^{4}x}$?

Those $\phi$'s really need an indication that they're operators, indexed by a set of test functions, $\hat\phi_f$, which, taken independently of any other observables we can take to represent a random variable, with the vacuum state providing a probability density. We can't measure $\hat\phi_f$ at a point, for $f$ a delta function, insofar as the variance is infinite (or, better, undefined, as you say) even for the free field, but, thinking very loosely, for any finite region an infinite sum of infinite variance random fields can perfectly well be finite and finite variance.
I find it helpful to use signal analysis language, with the test functions performing a function very close to that performed by "window functions" (signal analysis), Chris Fewster calls them "sampling functions" (and perhaps others do, but I haven't seen it from others). In signal analysis terms, we can think of $\hat\phi_g|0\rangle$ as a multiplicative modulation of the vacuum state, so that when a test function is used in this way it's appropriate to call a test function a "modulation function" (so signal analysis again).
In signal analysis terms, there's no a priori reason to think that the (pre-)inner product $\langle 0|\hat\phi_f^\dagger\hat\phi_g|0\rangle$ has to be a linear functional of $f$ and $g$ at all length scales and at all amplitudes, provided it's complex-linear in $\langle 0|\hat\phi_f^\dagger$ and $\hat\phi_g|0\rangle$, indeed our usual experience of nonlinearity in signal analysis suggests that we should expect it not to be, and yet the Wightman axioms insists it must be (for no physically justified principle), and the Haag-Kastler axioms, to approximately the same effect, insist on Additivity. Loosening this axiom results in a plethora of (what I find) interesting nonlinear models, as a result of which we can naturally construct multi-point operators (by polarization) that can be used to represent bound states (the account I've given here is obviously much too fast: a development that is as good as I could manage a few years ago can be found in arXiv:1507.08299, still very early days yet). FWIW, I see a connection between this account and the discussion in this thread of bound systems, with apparently no resolution, whereas for me this kind of approach offers at the least some possibilities — which, moreover, are moderately principled and empirically grounded in signal analysis concerns.