16. GAUGE THEORY AND THE CREATION OF PHOTONS

Size: px

Start display at page:

Download "16. GAUGE THEORY AND THE CREATION OF PHOTONS"

Fay Simmons
5 years ago
Views:

1 6. GAUGE THEORY AD THE CREATIO OF PHOTOS In the previous chapter the existence of a gauge theory allowed the electromagnetic field to be described in an invariant manner. Although the existence of this invariance had been known for some time, its full significance was not recognized until the late 960's [Kane93]. In quantum theory gauge invariance takes on a new meaning which leads to a cleared understanding of the photon as the carrier of the electromagnetic force. Using Postulate 3 to restate that quantum observables depend on ψ, the structure of the invariance can now be given as, ψ ψ= e i ψ, (6.) where is a constant. This transformation is global since ψ ( xt, ) transforms the same everywhere in space time. A local transformation could be constructed in which the phase of ψ at each point in space time can be chosen arbitrarily such that, ( ) ( ) χ i ( xt, ) ( ) ψ xt, =ψ xt, = e ψ xt, (6.) An exception to this local gauge transformation occurs when interference effects are measured and the interference intensities are sensitive to phase differences [Saku85]. It turns out that the local gauge transformation in Eq. (6.) does not exist in that form. If the Schrödinger wave function for a particle satisfies the time dependent equation, ( i ea ) ψ= i eφ ψ, (6.3) m t where e is the charge magnitude of the electron. 6. AIHILATIO AD CREATIO OPERATORS In Dirac's theory, the interaction between the electromagnetic field and atomic sized matter became an interaction between a large number of photons and the matter. Any change in the energy of the atom due to the electromagnetic field could be described as the creation and annihilation of photons in the electromagnetic field. Using the Fourier decomposition techniques developed in the classical expansion of the vector potential in the previous section, the linear combinations of Q and P are now given by, Copyright 000, 00 6

2 Gauge Theory and the Creation of Photons cˆ k = ( ω Qk ip k ),, ħω,, (6.4) cˆ k = ( ωqk ip k )., ħω,, (6.5) The commutation relations are given as, i i c ˆk, cˆk = Qk, Pk Pk, Qk,, ħ,, ħ,, =δ, δ kk,, (6.6) cˆ, cˆ = cˆ, c ˆk, = 0 (6.7) These commutation rules are evaluated at equal times, giving, cˆ, cˆ cˆ(), t cˆk, () t. (6.8) A new operator is created to aid in the development of the radiation field expressions, The new operator gives, cˆ c ˆ. (6.9) k, cˆ, k, = cˆcˆk, cˆk, cˆk, cˆk, cˆk,, = cˆ, cˆk, cˆk, cˆk, cˆk,, cˆk,, =δ δ cˆ kk,, k, (6.0) Similarly, cˆ, k, = δkk,, δ, c ˆk,. (6.) The quantized vector potential can now be written as, ( ) ikr ( ) ikr ( ˆ ˆ k, ) Ar (,) t = c () t ε e c () t ε e (6.) k =, Although this expression for the vector potential is nearly identical to Eq. (5.43) its meaning is very different. The vector potential in Eq. (6.) is an operator that acts on the state vectors in the occupation number space of the electromagnetic field. This operator has parameters of space and time, but he operator is not a function of the space and time 6 Copyright 000, 00

3 Gauge Theory and Creation of Photons coordinates of the photons representing the propagating electromagnetic field [Saki67]. The determination of the quantum mechanical Hamiltonian requires the calculation of the total energy of the radiation field, ( ) = E ˆ B ˆ H rad dv, (6.3) in which Ê and Bˆ are expressed in terms of canonical variables and ˆP. ˆQ The resulting expression for the quantum mechanical Hamiltonian is now, H ˆ ( ˆ ˆ ) P Q, (6.4) rad = k, ω k, k =, which is exactly the same form as the classical Hamiltonian. 6. PHOTOS STATES The algebra of the operators described in the previous section will now be applied the physical situation where the number of photons with a given momentum and polarization is increased or decreased using the operators, ĉ k,andĉ. The wave vector k will be defined as the photon momentum divided by ħ and again represents the polarization state (spin state) of the photon. The operator now represents the number of photons in state { }. In order represent a system in which there are many photons in many states, the product of the various states is given by,,,,, = (6.5) k, ki, i k, ki, i This state vector corresponds to the physical situation in which there are, photons in state { k, }. The number of photons,, is called k the occupation number for state { }. The state in which no photons are present is 0. This state will serve as the building block in which a general number of photons can be place by, Copyright 000,

4 Gauge Theory and the Creation of Photons k, k, ( cˆ k, ) nk i, i,, = 0 ki, i ki, i, (6.6) where the creation operator ĉ k, is applied to the general state vector, cˆ,,,, =,,,,. ki, i k, ki, i ki, i k, ki, i (6.7) where c ˆk i, i creates a photon in state { k, } number of states other than { k, }, leaving the occupation unchanged and the operator ĉ is applied to the general state vector, cˆ,,,, =,,,,,(6.8) ki, i k, ki, i ki, i k, ki, i where ĉ k, annihilates a photon in state { k, } number of states other than { k, } leaving the occupation unchanged. The operator does k i, i not change the occupation number, since it is diagonal and simply gives the eigenvalue of the number of photons in state{ k, }. Using Bose Einstein statistics in which the number of photons in a particular state is unrestricted, a many particle system can be constructed. [] 6.3 PHOTOS AS RADIATED FIELD EXCITATIOS In this manner the radiation field oscillator analogy is equivalent to the wave equation field operators which annihilate and create a particle at a given point in the field. The determination of the eigenvalues of this Hamiltonian is given by energy levels of linear oscillators, which is simply the radiation Hamiltonian, Bose Einstein statistics describes the mean number of bosons (photons) in mode n as βħω/ βħω ρ = e e. The probability P( n) of having n photons in nm = nρ nm where ( ) n the mode as a function of n and ( ) ( ) (Bose distribution law) is given as, Pn ( ) =. Contrary to the Poisson distribution law for classical particle states, the Bose distribution law decreasing with increasing n [Cohe89]. 6 4 Copyright 000, 00

5 Gauge Theory and Creation of Photons = ( ) cˆk cˆk cˆ ˆ k ck, H rad,,,, k =, ( k, ) = ħω. k =, (6.9) where again = cˆ c ˆ is the operator that measures the occupation k, number of a state of the system and has The annihilation c ˆk, and creation as its eigenfunctions. ˆk i, i c operators can now be used to construct the plane wave solution of the potential equation. The plane wave, ε e (6.0) ω ( ) i( ωt kr ) A k, = 4 π, which appears in the radiation field vector potential operator, k, k =, Aˆ = ( cˆ A c ˆ A ), (6.) as coefficients of the photon annihilation operators may be treated as the wave function of photons having momenta k and polarization ε ( ) Total Hamiltonian The total Hamiltonian of charged matter's interaction with the quantized radiation field is, where H obj is the Hamiltonian of the electrons, where the time dependent interaction terms is, H = H obj H int H, (6.) rad H obj = pi V( ri), (6.3) i m e e Vt () = ( pi, Ar ( i,) t ) A (,) rt φ(,) r i e t, (6.4) i mc mc which is summed over all electrons in the matter, H rad is the radiation field Hamiltonian and H int is the Hamiltonian of the interaction between the radiated field and the matter, given by, Copyright 000,

6 Gauge Theory and the Creation of Photons H e e = jrar () () ρ() rar () dv. (6.5) c mc int The relation between these three Hamiltonian's is shown in Figure 5.0. H obj e H int H rad Vbrg H obj e Figure 5.0 The Total Hamiltonian for a charged particle interacting with the electromagnetic field. The Hamiltonians for the radiation field and charged particle are combined with the interaction Hamiltonian to form the Total Hamiltonian. This description of the interaction of the electromagnetic field with charged matter is based on representing the radiation field's vector potential as a gauge particle field, whose interaction is governed through the dynamics of Hamiltonian mechanics Photon Polarization Like the polarization of the classical electromagnetic field, the quantified field contains a polarization element. In this case the quanta of the field, the photon, possesses a spin polarization. The photon like any other particle can possess angular momentum. In traditional descriptions of spin there are two components, the spin angular momentum and the orbital angular momentum. In the case of the photon, the momentum representation of the wave function Ak ( ) satisfies the transversallity condition Ak ( ) k =0. As a result the wavefunction cannot be arbitrarily specified for every component of the vector Ak ( ) at the same time. The result is that the photon's angular momentum and the photon's spin cannot be independently distinguished. The definition of the spin as the angular momentum of a particle at rest is inappropriate for the photon since is has no rest frame. Therefore only the total angular momentum of the photon has any meaning [Land8]. The photon state is characterized by its polarization vector ε ( ). Like the polarization of the classical electromagnetic field any polarization ε 6 6 Copyright 000, 00

7 Gauge Theory and Creation of Photons can be constructed as a superposition of two vector ε ( and ε ( ), which are orthogonal with a third vector such that ( ) ( ) ε ε = 0, where the polarization is given by, where ( ) ε and e and ( ) ε ( ) ( ) ε= eε e ε (6.6) e are the probabilities that the photon has a polarization respectively. Since ε ( ) transforms like a vector, the general theory of angular momentum allows the polarization vector to be associated with a unit of angular momentum. In the vocabulary of quantum mechanics, the photon has one unit of spin angular momentum. To find the spin components the circular polarization vectors are given by, ( ± ) () () ε = ε ± ε (/ )( i ). (6.7) The transformation properties of the polarization vector can be seen by rotating the right hand polarization vector ε R about an angle θ, Therefore, ε ε =ε cosθε sin θ, ε ε = ε sinθε cos θ. (6.8) ε R ε R = εcosθεsinθ i ( εsinθεcos θ), θ i = ( ε iε) e, θ i =ε e. R (6.9) Under an infinitesimal rotation around the propagation vector k, by an amount δφ, the circular polarization vectors are changed by, ( ± ) () () δε = δφ ε iε ( / )( ), = iδφε ( ± ). (6.30) ± so that the circular polarization vectors ε ( ) are associated with the spin components m =±, where the quantization axis has been chosen in the propagation direction k. The photon spin is either parallel or antiparallel to the propagation direction. Copyright 000,

8 Gauge Theory and the Creation of Photons ± The description of the polarization state with ε ( ) is called the circular polarization representation in contrast to the linear polarization ± representation. The orthogonal representations for ε ( ) are given by, ( ± ) ( ± ) ( ± ) ( ) ε ε = ε ε =, ( ± ) ( ) ( ± ) ( ± ) ε ε = ε ε = 0, ( ± ) k ε = 0. (6.3) The quantum mechanical expansion of the vector potential A's canonical variables leads to the idea that the excitations of the radiation field can be regarded as particles of zero mass and spin the photon. The field variables are simply the Fourier sum of the individual wavefunctions, where the coefficients multiplying each of the individual wavefunctions represent the probability of the creation and destruction of a quantum of that particular wavelength at any given point in the field and is referred to as the second quantization of the field. In this model the quantum field is equivalent mathematically to an infinite collection of harmonic oscillators. [] Since photons can act together in one energy state since photons obey Bose-Einstein statistics macroscopic effects can be observed from large assemblies of coherent photons. [3] What really interests me is whether God could have made the world in a different way; that is, whether the necessity of the logical simplicity leaves any freedom at all. A. Einstein [Holt75] Although the excitations of the electromagnetic field behave in many ways like particles, carrying energy and momentum, this process has limitations whose description is beyond this monograph. A similar situation occurs in the theory of harmonic vibrations of crystalline solids [Zima69]. The coupled motions of the crystal's atoms can be isolated into independent normal modes. each of these modes can be treated as a quantized harmonic oscillator. The excitations of the normal modes carry energy and momentum and are called phonons. The term was introduced by Frenkel [Fren3] with the following remark It is not in the least intended to convey the impression that such phonons have a real existence. On the contrary, the possibility of their introduction rather servers to discredit the belief in the real existence of photons. 3 Bose-Einstein statistics describe particles which occupy the same state as described by Pauli's exclusion principle. Photons obey Bose-Einstein statistics, where electrons obey Fermi-Dirac statistics and are excluded from occupying the same state while orbiting the nucleus of an atom. 6 8 Copyright 000, 00

Lecture 6 Photons, electrons and other quanta. EECS Winter 2006 Nanophotonics and Nano-scale Fabrication P.C.Ku

Lecture 6 Photons, electrons and other quanta EECS 598-002 Winter 2006 Nanophotonics and Nano-scale Fabrication P.C.Ku From classical to quantum theory In the beginning of the 20 th century, experiments