LIMITATIONS OF GAUGE INVARIANCE

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NEWTONIAN EQUATION <strong>OF</strong> MOTION The trajectory of motion of a particle of charge q and mass m subjected to an electromagnetic field is given by the solution of the Newtonian equation of motion The trajectory depends only on the force. The force depends only on the fields E and B. The fields are unchanged by a gauge transformation. Therefore the solution of the problem is gauge invariant. The analysis above has long been accepted as conclusive evidence for gauge invariance as a fundamental and unambiguous principle of electrodynamics. 8
THERE ARE HINTS <strong>OF</strong> BASIC PROBLEMS For example: • The equations of quantum mechanics are all in terms of potentials and not fields. Many attempts have been made to write the Schrödinger equation directly in terms of E and B, but none have succeeded. All such attempts have yielded nonlocal formulations. • In textbooks on classical mechanics it is common to find demonstrations that alternative formulations (Hamiltonian, Lagrangian, …, expressed in terms of potentials) can be shown to infer Newton’s equations, but not the converse. • In strong-field physics, two gauges are in common use, and they can be shown to contradict each other in fundamental ways. 9
Page 1 and 2: LIMITATIONS OF GAUGE INVARIANCE and
Page 3 and 4: Furthermore… There is another set
Page 5 and 6: ELECTROMAGNETIC POTENTIALS Electric
Page 7: GAUGE TRANSFORMATION If is any sca
Page 11 and 12: These equations can be combined to
Page 13 and 14: GAUGE EQUIVALENT - BUT DIFFERENT Th
Page 15 and 16: NEWTONIAN MECHANICS DIFFERS FROM OT
Page 17 and 18: CAUTIONS ABOUT THE “PHYSICAL GAUG
Page 19 and 20: MIXED FIELDS The physical gauge for
Page 21 and 22: ATOMIC UNITS 1; m 1; e 1, or q 1 e
Page 23 and 24: V(r) LASER PHYSICS IN THE LENGTH GA
Page 25 and 26: Intensity (a.u.) Intensity (W/cm 2
Page 27 and 28: QUALITATIVE IMPLICATIONS OF THE VEL
Page 29 and 30: DOMAINS FOR AN ELECTRON IN A PLANE-
Page 31 and 32: VG-LG COMPARISON It has just been s
Page 33 and 34: CONTRADICTIONS Much success has bee
Page 35 and 36: PONDEROMOTIVE ENERGY A charged part
Page 41 and 42: ANOTHER FUNDAMENTAL PROBLEM The pon
Page 43 and 44: RIGOROUS DERIVATION OF AN S MATRIX
Page 45 and 46: By hypothesis, the laser pulse is f
Page 47 and 48: The end result of this alternative
Page 49 and 50: TRANSITION MATRIX ELEMENTS ARE NOT
Page 51 and 52: The SFA includes A 2 (t); the HFA h
Page 53 and 54: REVISIT ITEMS FROM THE PREAMBLE •

LIMITATIONS OF GAUGE INVARIANCE

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