Spontaneous Breaking Local Gauge Symmetry: Hamiltonian Formalism?

[weejee]

Is it possible to describe spontaneous breaking of the local gauge symmetry in the standard Hamiltonian formalism, without any manual redefinition of the gauge field( A(r) -> A(r)+▽Λ(r) )?

Detailed description of my question is given below.(Quite lengthy.. sorry.)

1. By the 'standard Hamiltonian description of spontaneous symmetry breaking' I mean the following.
- There is a Hamiltonian and an associated continuous symmetry, under which the given Hamiltonian is invariant. Yet, the ground states don't possesses the same symmetry as the Hamiltonian. They can be transformed to one another by the given symmetry operation.
- For example, in the case of ferromagnetism, the Heisenberg Hamiltonian is invariant under rotation. Yet, its ground states are oriented to certain directions(they aren't rotationally invariant) and one can transform a ground state to another by applying the rotation operator.

2. A problem when we do the same thing for the local gauge symmetry?
- In quantum mechanics, when the local gauge invariance is discussed in textbooks, they say the physics is invariant under the local gauge transformation exp(iΛ(r)/hbar), if the gauge field is transformed as A(r) -> A(r)+▽Λ(r).
- However, while the Hamiltonian is transformed by a well-defined operation exp(iΛ(r)/hbar), the gauge field is manually transformed as A(r)->A(r)+▽Λ(r). (no such quantum operation exists). Furthermore, the Hamiltonian isn't invariant under exp(iΛ(r)/hbar), although it looks invariant once you redefine the gauge field as A(r) -> A(r)+▽Λ(r).
- This is clearly different from what we normally do when discussing other symmetries as in the part 1 of this thread.

3. What if we quantize the gauge field?
- One may expect that if we quantize the gauge field and write the total Hamiltonian( the electron part + the gauge field part + coupling term), we can use the formalism described in the part 1.
- Yet, the situation is not much better, since the gauge field can't be quantized without specifying the gauge(let's say, the Coulomb gauge). Then, the gauge transformation isn't a quantum operation.
- We can still construct an unitary operator made of photon creation and annihilation operators so that it can transform A(r) to something else. Howver, it will never have any longitudinal degree of freedom, whereas the term ▽Λ(r) appearing in the gauge transformation is solely longitudinal when Fourier transformed to the k-space.

4. What do we need? Higgs?
- I heard that in the electroweak theory, a scalar field(the Higgs field) is introduced to describe the spontaneous breaking of the local gauge symmetry, and the gauge bosons(W and Z) acquire the longitudinal degree of freedom and the mass.
- Then, do we need a fictitious scalar field(such as Higgs?) to describe the logitudinal degree of freedom of photons and achieve the full Hamiltonian formalism for the local gauge transformation, or is it simply impossible?


Cf. The global gauge symmetry - a simpler question
- Since the global gauge transformation multiplies 'exp(iθ)' to each particle, it can be written as 'exp(iNθ)' in the 2nd quantization notation, where N is the number operator.
- The Hamiltonian describing a superconductor or a superfluid is invariant under 'exp(iNθ)' but its ground states, namely the BCS ground state for the superconductor or the coherent state for the superfluid, are not invariant under exp(iNθ). Actually, exp(iNθ) transforms a ground state to another one(with a different phase). Therefore a complete analogy with the ferromagnetism is established.

Heisenberg Hamiltonian <-> superconductor(superfluid) Hamiltonian
rotational symmetry <-> global gauge symmetry
ferromagnetic ground state <-> BCS ground state (coherent state)

 

[Avodyne]

Sorry, but your question is just too long and messy to respond to effectively. It's not even clear which formalism you are working in, though it appears to be electrons described by the Schrodinger equation in a background classical EM field.

That said, there is a fundamental difference in that gauge symmetries are not spontaneously broken in the same way that global symmetries are. All physical states of a gauge theory are invariant under gauge transformations (when the gauge field is quantized).

 

[weejee]

Thank you so much! I was wondering if there would be anyone willing to answer such a long and clumsy question.

I was missing exactly what you pointed out. I didn't even notice that gauge transformations in QFT or QM don't transform the phyical state.

Half of the blame for my ignorance of such a simple fact should be layed on 'modern quantum mechanics' by Sakurai. It that book, the treatment of the gauge symmetry involves transformations of quantum states, which in itself is clear, but could be quite misleading.

 

[Avodyne]

Half of the blame for my ignorance of such a simple fact should be layed on 'modern quantum mechanics' by Sakurai.

Well, it's not my favorite book! But the invariance of physical states under gauge transformations is typically only discussed in rather advanced treatments.