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The gauge principle says that the Dirac Lagrangian density L
is invariant
under gauge transformation. Here the gauge function Φ(x)
is an arbitrary function
of the four space-time coordinate, denoted by x.
The transformation of the 4-potential is
and the transformation of the Dirac function is
(see [1], p. 78).
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ψ(x) →
exp(iΦ(x))ψ(x)
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(2)
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Here the following problem arises: The electromagnetic Lagrangian density
ℒ is invariant under gauge transformation. Therefore, also the
action
is invariant under gauge transformation. Now the phase is
the action divide by ℏ. It follows that the phase
of a Dirac particle should not vary under a gauge transformation.
Problem:
The de Broglie principle says that the phase depends on dynamical
properties of the electron. Now,
if the action is invariant under a gauge transformation then
why the phase of the Dirac function (2) and its relevant dynamical properties
vary under the same gauge
transformation?
It turns out that this is the root of many contradictions. A further
discussion can be found
here.
References:
[1] M. E. Peskin and D. V. Schroeder An Introduction to Quantum Field
Theory (Addison-Wesley, Reading Mass., 1995).
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