Talk:Canonical commutation relation

message: "The non-relativistic Hamiltonian for a particle of mass m and charge q in a classical electromagnetic field is 
:H=\frac{1}{2m} \left(p-\frac{qA}{c}\right)^2 +q\phi
where A is the three-vector potential and \phi is the scalar potential. This form of the Hamiltonian, as well as the Schroedinger equation H\psi=i\hbar \partial\psi/\partial t, the Maxwell equations and the Lorentz force law are invariant under the gauge transformation 
:A\to A^\prime=A+\nabla \Lambda
:\phi\to \phi^\prime-\frac{1}{c} \frac{\partial \Lambda}{\partial t}
:\psi\to\psi^\prime=U\psi
:H\to H^\prime= U HU^\dagger 
where 
:U=\exp \left( \frac{iq\Lambda}{\hbar c}\right)
and \Lambda=\Lambda(x,t) is the gauge function."
It is not true that this H is invariant under a gauge transformation. Substituting for the transformed potentials with simple algebra H'  becomes
:H^\prime= U HU^\dagger-\frac{q}{c} \frac{\partial \Lambda}{\partial t}
not the expression given in the article. However the Schroedinger equation, the Maxwell equations and the Lorentz force law are invariant under the gauge transformation. Xxanthippe 04:02, 11 June 2007 (UTC)
replies: text: The Hamiltonian of semi-classical electrodynamics, given above, is, by elementary algebra (consider the potential term), not gauge invariant. The field equation that it is associated with, the Schrödinger equation is gauge invariant because of the time derivative of the wavefunction that occurs in it. On the other hand, the Hamiltonian and the field equations of quantum electrodynamics are all gauge invariant. Puzzled? The reason is that semi-classical electrodynamics is an incomplete theory because it does not take account of the internal dynamics of the (electromagnetic) field as it describes the field by externally specified potentials. Quantum electrodynamics is a complete theory in this respect but it suffers from disadvantages like giving infinite results. However, it can be shown that the probability amplitudes for transitions of the matter field in semi-classical electrodynamics are gauge invariant. So semi-classical electrodynamics is a pretty good theory for most of chemistry and much of physics, so much so that some people don't realise its shortcomings. Xxanthippe (talk) 00:34, 20 June 2010 (UTC)
thread_title: Gauge invariant?

message: "It is relatively easy to see that two operators satisfying the canonical commutation relations cannot both be bounded." 
- Can anyone provide a reference for this claim?  —Preceding unsigned comment added by 142.151.156.48 (talk) 00:46, 5 March 2009 (UTC)
replies: text: Self-evident: Trace.  Cuzkatzimhut (talk) 01:15, 4 March 2013 (UTC)
replies: text: Would you like to expand for the benefit of those with less intelligence than yourself? Xxanthippe (talk) 01:22, 4 March 2013 (UTC)
replies: text: But under no circumstances in the article: less is more. As the aside invitation to take the trace of the commutation relation suggests, the right hand side is of order N for N-dimensional vector spaces and infinite for the Hilbert space in question. The left hand side, by the cyclicity of the trace is 0 for finite dimensional spaces, and ambiguous for Hilbert space, that is as N grows infinite, the zero and the emergent delta functions conspire to yield an infinite expression equal to the right hand side. To sum up, for finite N, the commutation relation cannot hold, as seen by thus tracing, but it tends to the one given in the article provided either x or p are unbounded, in our case both. My favorite reference is  where that limit emerges magnificently, but  it would be overkill to foist it on the reader right here. The reader is steered to the Stone-von Neumann theorem, and might go to standard QM texts if interested. I think just the plain statement and the exhortation to take the trace suffice to let the reader in the know recall the fact, and the one who is not to shrug it off.... One could always cite the canonical derivation of Ch VIII of Reed and Simon, , based on this commutator, but I fear this is overkill for an elementary introduction−−maybe the cite could belong in the wiki on boundedness.  Cuzkatzimhut (talk) 12:59, 4 March 2013 (UTC)
thread_title:  Boundedness? 

message: Para 2 starts out: "In contrast..."
In contrast to what?  How is the reader supposed to guess? P0M (talk) 01:40, 22 June 2009 (UTC)
replies: text: ? I can't guess what your question asks. "By contast" to the two conjugate variables of the preceding paragraph, of course, which fail to commute. What else? Where is the lack of clarity hiding? Cuzkatzimhut (talk) 00:56, 3 December 2012 (UTC)
thread_title:  Poor writing