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For instance, the Hamiltonian for the hydrogen atom that is studied in elementary quantum mechanics has this form. What happens when we make the general gauge change (5.122) Substituting cP by cP' given by (5.122) in the Hamiltonian (5.129), we find the following expression for the atomic Hamiltonian in the new gauge: (5.130) Comparing (5.128) and (5.130), we see that we must take (5.131) Unfortunately, this does not work, as the resulting t yields (5.132) which can be rewritten as

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Looking at (5.130), we see that the last two terms in (5.133) should not have been present. The presence of these two terms shows that HA is not gauge invariant. But this is not very surprising, as HA does not include the radiation field. Any system of charged particles subject to acceleration would also involve electromagnetic radiation. Let us then complement HA adding the Hamiltonian of the radiation field, HF, which was derived in 2~ and a,yet u~known c:.ouplin~ ten!l ih. The complete Hamiltonian'is given by H = HA + HI + HF. As H'p = HF, HI must be such that when transformed, it generates terms that cancel the last two terms on the right-hand side of (5.133). In other words, writing HI as a functional of the electromagnetic potentials, HI [A, cPl, it must be such that (5.134)

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As you can easily verify, the minimal-coupling interaction derived in Section 5.2 is the solution of this equation. Without the minimal-coupling interaction, the Hamiltonian is not gauge invariant. Historically, the form of the coupling between matter and radiation was known long before the discovery of the symmetry under gauge transformations of the quantummechanical system of charged particles interacting with the electromagnetic field [315]. This symmetry was discovered by Fock in 1926 [215]. But the idea of gauge invariance as a basic unifying principle predates quantum mechanics and can be traced back [413,414] to Weyl's [636] failed attempt to unify electromagnetism and gravitation in 1919 [315]. Much later, in 1928, Weyl wrote [637]: "Butl now believe that this gauge invariance does not tie together electricity and gravitation, but rather electricity and matter in the manner described above." The quantum nature of the electromagnetic field gives rise to many physical phenomena, some of which we mentioned earlier. In 6 we use the formalism we have presented here to study spontaneous emission. First, we examine spontaneous emission in free space and show how it is triggered by the zero-point fluctuations in the field. Then we describe how a cavity can alter this zero-point field and the consequences it has for spontaneous emission.

RECOMMENDED READING For a discussion of minimal coupling as a consequence of gauge invariance using a field description for matter as well as forthe field, see Sec. 3.5 of [116]. For a gauge-invariant formulation of perturbation theory in QED, see [366]. The minimal coupling can also be seen as a consequence of Lorentz invariance [406]. For a nonrelativistic particle, a rephrasing of the principle of restricted Galilean invariance also leads to the minimal coupling Hamiltonian [320, 321]. For a fully relativistic derivation of the interaction between electric and magnetic dipoles with an external electromagnetic field, see [20]. For a discussion of the Aharonov-Bohm phase in this context, see also [397,572]. See the book by Allen and Eberly [13] for a detailed discussion of the experimental requirements for the validity of the two-level atom approximation. Richard P. Feynman once derived the Lorentz force and two of Maxwell equations from the assumption that a nonrelativistic point particle's position and momentum satisfy the well-known quantum-mechanical commutation relation and that the particle obeys Newton's second law of motion [176]. This crazy derivation can be reinterpreted as a proof that the most general velocity-dependent force that can be described in a Lagrangian or Hamiltonian formalism is one that has the same general form of the Lorentz force [302]. This is related to the question of whether knowing the equations of motion of a physical sys-

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