LIMITATIONS OF GAUGE INVARIANCE

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PREAMBLE • Physical processes involving electromagnetic interactions depend entirely on the electric and magnetic fields that are present. • These fields can be represented as derivatives of scalar and vector potential functions that are often mathematically convenient, but are actually only auxiliary quantities. (Each such set is called a gauge.) • The potential functions are not unique. There are many sets of these functions corresponding to the same fields and thus to the same physical processes. (They are connected by gauge transformations.) • The choice of which of the possible sets of potentials to use for a given problem is a matter of mathematical or conceptual convenience. • Exactly the same considerations about the choice of gauge apply to both classical and quantum phenomena. Not one of the above statements is entirely correct. 2
Furthermore… There is another set of statements that is accepted almost universally in the Atomic, Molecular, and Optical (AMO) physics community, and to some extent outside that community: • A gauge transformation is a unitary transformation. • Quantum transition amplitudes are manifestly gauge invariant. Less universal, but also widespread: • The most basic approach is to formulate all problems using a scalar potential. (Within the dipole approximation, this is called the length gauge.) If a different gauge is desired, this should be obtained through a gauge transformation from the length gauge. All of the bulleted statements above are false. 3
Page 1: LIMITATIONS OF GAUGE INVARIANCE and
Page 5 and 6: ELECTROMAGNETIC POTENTIALS Electric
Page 7 and 8: GAUGE TRANSFORMATION If is any sca
Page 9 and 10: THERE ARE HINTS OF BASIC PROBLEMS F
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 •
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LIMITATIONS OF GAUGE INVARIANCE

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