# Energy and properties in quantum fields

I'm trying to understand where energy is located in a quantum field world.

For force fields, there are conserved properties that (in my mind's eye) are always stuck in that field. These arise from symmetries in the fields, by Noether's Theorem. As such, I imagine that properties such as charge can move around in the field but never leave or appear.

Noether's Theorem also explains conservation of energy as related to a symmetry, the invariance of the laws of nature at any point in time. It seems fair, then, to say that energy is a property of the field that makes up spacetime itself.

When a charge is carried by a particle, that particle can have different masses with equal charge. I am not sure how I should picture the difference in the fields.

A muon, for instance, has both mass and charge. My question is then, which fields contain the muon's properties?

• Is it all in the muon field? And does this mean that both charges and energy can transfer between fields?

• Or is a muon a sympathetic coupling between the muon field, EM field (charge) and spacetime curvature (energy)?

• Haven't you dealt with quantum mechanics before? Energy is an operator, the actual value of the energy of a physical configuration is encoded in the quantum state. – Prof. Legolasov Oct 25 '16 at 7:29
• @SolenodonParadoxus No I haven't. I did well in an introductory undergrad course but did not continue in that field. My curiosity remains, however, and I'm glad to have increased my understanding through this thread. I have to admit your statement is semantically confusing, are you implying that energy as we know it classically does not have a physical presence at all in quantum mechanics? I have a lingering memory of the Hamiltonian and applying operators on the wavefunction. – Ketil Tunheim Oct 25 '16 at 15:06
• That's not my words. I was saying that energy as we know is classically is encoded in the quantum state, not in the operator. There is a single quantum operator for energy (the Hamiltonian), contrast to the multitude of values that energy of the system can take. These are encoded in quantum states. – Prof. Legolasov Oct 28 '16 at 6:05

Noether's theorem is a statement about classical field theory. The "conserved quantity* being just a function of the fields is a classical field theoretical statement. And it is only in classical field theory that you should think of the fields encoding the state of the world.

In quantum field theory, the fields are no longer assingments of numbers to spacetime points, they are assignments of operators to spacetime points (or regions, really), and those operators act on the state of space. This state of space is distinct from the space of possible configurations of the fields, that is, in quantum field theory, the fields do not encode the state of the world. This is crucial, and why statements like "the energy is in the field" make sense classically (because the field really is what we encode a state with a certain energy in), but are non-sensical quantum mechanically. It's not so much that it is a false statement, it's not even wrong, it just doesn't even begin to be a meaningful statement in quantum field theory that could be true or false.

The quantum analogon to Noether's theorem are the so-called Ward-Takahashi identities, basically stating that up to certain singular terms, the classical conservation equations of Noether's theorem hold within expectation values. That is, if you take the classically conserved expression that is now an operator on all states and not a function of a state itself and take its expectation value with respect to a particular state (to make it into a simple number that's a function of a particular state again), then you'll find it still fulfills the conservation equation (up to those singular terms I don't want to discuss here).

So, in the end, the energy, or whatever else, is "in the state". But a quantum state is an elusive object not amenable to our intuition. In ordinary QM, you might represent it as a wavefunction - the state itself is a function of all of space. In QFT, the dynamical variables are not just spatial position, but field themselves dependent on spacetime. So the analogon of the wavefunction is a so-called wave-functional, a complex-valued function of the fields. Just like the position in QM was not the state, and the energy is not "in the position", so are the fields not the same as the state, and the energy is not "in the fields". It's in the quantum state, of which I am afraid you will not get any concrete or intuitive description.

As for where the properties of the muon come from - they indeed come from the muon field, because to each field, we associate a range of particle states that it can more or less "create". But without the electromagnetic field there is no notion of charge, and without the weak field no notion of hypercharge, so the properties of the particle states created by a field also depend on the other fields present in our theory. Except for mass, to some extent: "Spacetime curvature" does not exist in standard QFT, QFT is special relativistic, not general relativistic. Mass is simply a property of each field/particle itself, although it might get modified through interactions.

Get rid of the idea that quantum fields are these things out there that have physical properties. Quantum fields are a theorist's attempt to encode into the equations of physics that the world is local i.e that what happens here can be talked about and understood without knowing what happens in alpha centauri. This is ,roughly speaking, the "cluster decomposition principle". Careful examination of Quantum Field Theory shows that there are such things as "field redefinitions" i.e we have a piece of physics we need to understand but two physicists do not actually need to use the same quantum field in order compare calculations with experiments.

A lot of confusion about this issue has been brought about by physicists saying things like,"we think of quantum fields as the fundamental thing and particles as merely excitations". This is wrong. There is nothing fundamental about quantum fields and so it is mistaken to look for where anything physical is in them.

• This is interesting, could you elaborate or link to any resources? If I understand correctly, you are implying that the choice of fields is arbitrary, and that assigning different laws to different fields is only a tool for calculation? I am not sure how to fit the Higgs field into this. – Ketil Tunheim Oct 23 '16 at 22:01
• The assignment of fields is not arbitrary there are well defined things called quantum fields that obey specific equations and that one can use to do calculations but again there are specific ways of changing these quantum fields so that the physics stays the same but the quantum fields are different. The only reference I know with a discussion of this is in Steven Weinberg's book on "Theory of Quantum Fields" the last section of Chapter 7. Here is a link of a discussion of a specific example physics.stackexchange.com/questions/45262/… – Amara Oct 23 '16 at 23:15
• With regards to the Higgs field, again forget about the Higgs field, it is the Higgs particle that you ought to care about. The particle is the invariant thing not the field. – Amara Oct 23 '16 at 23:19