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davidge
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What equation in QM show us that symmetry in quantum-mechanical phase implies charge conservation?
davidge said:What equation in QM show us that symmetry in quantum-mechanical phase implies charge conservation?
Thanks. This is more difficult to understand than I thought.PeterDonis said:The symmetry that implies charge conservation is usually referred to as electromagnetic gauge invariance. See, for example, here:
Thanks for posting this in such detail. It is not covered in any of my textbooks.vanhees71 said:A conservation law is always due to global (not local!) gauge symmetries. Since a local symmetry implies a corresponding global symmetry, you also have a conservation law (in this case of a charge-like quantity)
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Thus now indeed the Lagrangian is invariant under local gauge transformations. Of course, the current is still conserved, because the Lagrangian is still invariant under global gauge transformations.
Ok. But where this term comes from?vanhees71 said:there are only the ##\delta \eta## in the entire posting.
So, at the beggining of your post, ##\Psi## is a function of ##x, \eta##, but as you are holding ##x## constant, ##\delta \Psi = \delta \eta \ \tau (x, \Psi)##?vanhees71 said:It's the "infinitesimal" group parameter (in our example the phase).
Oh yea, I see. Thanks.vanhees71 said:I don't understand the confusion. Of course ##\psi## is a function of ##x## only. It denotes a general set of fields, and ##\eta## are a set of parameters of the Lie group. Take ##\psi## an ##n##-dimensional field, ##t^a## a set of generators of the Lie group. Then the Lie group acts on the fields via
$$\psi \rightarrow \psi'=\exp(-\mathrm{i} \eta_{a} t^a) \psi.$$
For an infinitesimal transformation you have
$$\psi \rightarrow \psi+\delta \psi, \quad \delta \psi=\delta \eta_{a} t^{a} \psi$$
and thus
$$\tau^{a} =t^a \psi.$$
For the Abelian case discussed as an example you just have a one-dimensional Lie group ##\mathrm{U}(1)## and ##t^a=1##.
vanhees71 said:A conservation law is always due to global (not local!) gauge symmetries.
QM phase symmetry, also known as quantum phase symmetry, is a fundamental principle in physics that states that the laws of nature should remain unchanged under a transformation of the phase of the wave function. This means that the physical properties of a system should remain the same regardless of the overall phase of the system.
Charge conservation is a direct consequence of QM phase symmetry. This principle ensures that the total electric charge of a system remains constant, even as the system undergoes transformations. This is because any change in the overall phase of the wave function would alter the charge distribution, violating the principle of charge conservation.
QM phase symmetry is important because it is a fundamental principle that helps to explain the behavior of particles at the quantum level. It is a key component of quantum mechanics, which is essential for understanding the behavior of particles and systems at the atomic and subatomic levels. Without QM phase symmetry, our understanding of the quantum world would be incomplete.
While QM phase symmetry is a fundamental principle, it can be broken under certain conditions. For example, in systems with strong electromagnetic fields or at very high energies, the phase of the wave function can become distorted, causing QM phase symmetry to be violated. However, in most everyday situations, QM phase symmetry holds true.
QM phase symmetry not only ensures the conservation of electric charge, but it also leads to the conservation of other physical quantities, such as energy and momentum. This is because the laws of nature should remain invariant under transformations of the wave function's phase, which includes transformations that involve these physical quantities. Therefore, QM phase symmetry has far-reaching implications for our understanding of the fundamental laws of physics.