The Quantum State as a Function of The Quantum Field

[bhobba]

In answering another question, I came across a nice paper by Weinberg:

https://www.arxiv-vanity.com/papers/hep-th/9702027/

One thing that struck me was the following comment:

'In its mature form, the idea of quantum field theory is that quantum fields are the basic ingredients of the universe, and particles are just bundles of energy and momentum of the fields. In a relativistic theory, the wave function is a function of these fields, not a function of particle coordinates. Quantum field theory led to a more unified view of nature than the old dualistic interpretation of both fields and particles.'

I always considered the wave function as just the state in the position basis and the Quantum Fields as operators on the state (i.e. the Fock space). This view of it as a function of the fields is new to me.

Could someone explain this to me?

If true, that means the fundamental 'reality' (as far as we can tell today - Weinberg, of course, believes any theory will look like a QFT at large enough distances) are the Quantum Fields - the state is simply a tool used in ordinary QM. It's a view that, over the years, I am coming to suspect could be true.

Thanks
Bill

 

[PeterDonis]

I always considered the wave function as just the state in the position basis

Or the momentum basis, or any other basis of configuration space. (And then you have to add internal degrees of freedom like spin.) But that's a non-relativistic view. See below.

and the Quantum Fields as operators on the state (i.e. the Fock space).

This can't be right because QFT operators are operators at specific spacetime points. They don't operate on any "state" by your definition, because a "state" by your definition is not an object at a specific spacetime point.

In other words, in QFT (or at least relativistic QFT), you have to discard everything you understood about wave functions and states in non-relativistic QM, and rebuild everything from the ground up starting with quantum fields as operators at specific spacetime points. In this picture there aren't any "particle coordinates" to begin with, so of course there can't be any wave functions of particle coordinates.

 

[PeterDonis]

In a relativistic theory, the wave function is a function of these fields, not a function of particle coordinates.

Note that Weinberg actually says "functional", not "function". That's because the fields are operators, not coordinates.

 

[bhobba]

In this picture there aren't any "particle coordinates" to begin with, so of course there can't be any wave functions of particle coordinates.

Absolutely.

I will do another post about some interesting proposals for the emergence of space-time itself.

Although it involves QM (specifically string theory), the Relativity forum might be a better place for it.

Thanks
Bill

 

[PeterDonis]

Absolutely.

I will do another post about some interesting proposals for the emergence of space-time itself.

Although it involves QM (specifically string theory), the Relativity forum might be a better place for it.

Thanks
Bill

BTSM would be better, I think.

 

[bhobba]

BTSM would be better, I think.

Will move.

Thanks
Bill

 

[PeterDonis]

Will move.

This thread?

https://www.physicsforums.com/threads/the-emergence-of-space-time-in-string-theory.1058649/

 

[bhobba]

This thread?

https://www.physicsforums.com/threads/the-emergence-of-space-time-in-string-theory.1058649/

Yes.

Thanks
Bill

 

[gentzen]

Note that Weinberg actually says "functional", not "function". That's because the fields are operators, not coordinates.

I read Weinberg a bit past that point, and came to the conclusion that you interpret too much into him using the word „functional“. He just means a function of the field configurations (of 3d-fields) like everybody else.

 

[vanhees71]

In answering another question, I came across a nice paper by Weinberg:

https://www.arxiv-vanity.com/papers/hep-th/9702027/

One thing that struck me was the following comment:

'In its mature form, the idea of quantum field theory is that quantum fields are the basic ingredients of the universe, and particles are just bundles of energy and momentum of the fields. In a relativistic theory, the wave function is a function of these fields, not a function of particle coordinates. Quantum field theory led to a more unified view of nature than the old dualistic interpretation of both fields and particles.'

I'd have said it's a functional of these fields, but that's a detail.

I always considered the wave function as just the state in the position basis and the Quantum Fields as operators on the state (i.e. the Fock space). This view of it as a function of the fields is new to me.

There is no position basis in relativistic QFT (or at least you can define one only for massive particles or particles with spin $\leq 1/2$ since there's a position observable only for these cases of fields).

Fock states also can be defined only for (asymptotically) free fields.

As in any QT states in QFT are described by a statistical operator, and as in any QT you can express them in terms of the fundamental operators of the theory, which are quantum fields in QFT. E.g., a thermal state of a many-body system can be described by a grand-canonical operator,
$$\hat{R}=\frac{1}{Z} \exp[- \beta (\hat{P} \cdot u-\mu \hat{Q})], \quad Z=\mathrm{Tr} \exp[- \beta (\hat{P} \cdot u-\mu \hat{Q})],$$
where $\hat{P}$ is the total-four-momentum operator and $\hat{Q}$ some conserved charge (you can also have more than one conserved charge with corresponding chemical potentials of course); $\hat{P}$ and $\hat{Q}$ are given as a functional of the field operators.

Could someone explain this to me?

If true, that means the fundamental 'reality' (as far as we can tell today - Weinberg, of course, believes any theory will look like a QFT at large enough distances) are the Quantum Fields - the state is simply a tool used in ordinary QM. It's a view that, over the years, I am coming to suspect could be true.

Thanks
Bill

Of course, one must not forget, that physics is a description of what can be objectively observed in Nature. There are no Hilbert spaces, no self-adjoint operators, probability distributions, etc. in the lab but real-world equipment, and as it looks today, indeed almost everything so far knowncan be described with local relativistic QFTs, or even the Standard Model of elementary particle physics, despite the fact that most physicists do not consider the Standard Model to be complete, because it has no explanation for the strength of CP violation to explain "in a natural way" the matter dominance in the universe and the apparent necessity for some "dark matter", which should consist of some yet not discovered particle(s) (e.g., axions).

The great unsolved problem is of course a description of the gravitational interaction and/or a generic quantum description of spacetime (or spacetime as an emergent phenomenon describable from a all-comprehensive quantum theory of whatever type or something completely different ;-)).

What Weinberg wants to stress is that for sure local relativistic QFTs are very likely at least how any relativistic theory should look in the sense of an effective theory, i.e., below some energy scale, above which we ignore further details. That's of course based on the Wilsonian interpretation of the renormalization group, which indeed explains a lot, including the emergence of classical behavior of macroscopic many-body systems.

 

[vanhees71]

Absolutely.

I will do another post about some interesting proposals for the emergence of space-time itself.

In relativistic QFT spacetime is not emergent but used as an input. To formulate a specific QT you need to define an observable algebra, and in relativistic QFT that's done by looking for local realizations of irreducible unitary representations of the proper orthochronous Poincare group, which is the symmetry group of special-relativistic spacetime, i.e., Minkowski spacetime.

In general you can not define position observables in such a theory (though it's possible for all massive realizations but for massless ones only for spin 0 and spin 1/2).

On an operational level spacetime localizations come into the game as position and time of "detection events". E.g., photons as massless spin-1 quanta do not have a position observable. Nevertheless what's well defined are the probabilities for photon detection at a given time with the detector placed at a given position. If the detection mechanism is based on the photoelectric effect it's immediately clear that the corresponding probability density is given by the energy density of the electromagnetic field (using the usual dipole approximation of atomic/condensed-matter physics). See, e.g., Garrison&Chiao, Quantum Optics, about the theoretical description of all kinds of photon detections.

For a very nice didactical explanation, how spacetime coordinates enter the game in relativistic QFT in a consistent way (and why it doesn't work as in non-relativistic QM by assuming position coordinates as observables and only time as a parameter), see Sidney Coleman's lecture. The here relevant introductory parts
are available on the arXiv:

https://arxiv.org/abs/1110.5013v5

Otherwise the corresponding book is

S. Coleman, Lectures of Sidney Coleman on Quantum Field
Theory, World Scientific Publishing Co. Pte. Ltd., Hackensack
(2018), https://doi.org/10.1142/9371

 

[vanhees71]

I read Weinberg a bit past that point, and came to the conclusion that you interpret too much into him using the word „functional“. He just means a function of the field configurations (of 3d-fields) like everybody else.

No, it's a functional, and Weinberg indeed says so (@bhobba please quote precisely!):

In a relativistic theory the wavefunction is a functional of these fields, not a function of particle coordinates. Quantum field theory hence led to amore unified view of nature than the old dualisticinterpretation in terms of both fields and particles.

For a description of QFT in terms of such functionals (which is a formulation usually not treated in the textbooks), see

B. Hatfield, Quantum Field Theory of Point Particles and
Strings, Addison-Wesley, Reading, Massachusetts, 10 edn.
(1992).

 

[weirdoguy]

He just means a function of the field configurations (of 3d-fields) like everybody else.

Well, every functional is a function, per definition (almost everything is, if you squint hard enough), but if someone is using this particular word (functional) it's usually for a reason.

 

[bhobba]

No, it's a functional, and Weinberg indeed says so (@bhobba please quote precisely!):

Agreed. My bad.

Thanks
Bill