Romanian Reports in Physics, Vol. 66, No. 2, P. 319–335, 2014 THE SPINOR FIELD THEORY OF THE PHOTON RUO PENG WANG Peking University, Physics Department, Beijing 100871, P.R. China E-mail: [email protected] Received October 28, 2013 Abstract. I introduce a spinor field theory for the photon. The three-dimensional vector electromagnetic field and the four-dimensional vector potential are components of this spinor photon field. A spinor equation for the photon field is derived from Maxwell’s equations, the relations between the electromagnetic field and the four-dimensional vector potential, and the Lorentz gauge condition. The covariant quantization of free photon field is done, and only transverse photons are obtained. The vacuum energy divergence does not occur in this theory. A covariant “positive frequency” condition is introduced for separating the photon field from its complex conjugate in the presence of the electric current and charge. Key words: photon, spinor. 1. INTRODUCTION The electromagnetic interaction is the best studied one among the four known fundamental interactions. In the frame of quantum theory, the photon is the quantum of the electromagnetic field. There are several ways to represent the electromagnetic field: by the three-dimensional electric field and magnetic field vectors, by the 44× electromagnetic tensor, or by the four-dimensional potential vector [1, 2]. The electric and magnetic fields are directly related to the energy density of the electromagnetic field. The four-vector potential is directly related to the Lagrangian density of interaction. But however, none of these fields can be regarded as the photon field, because the photon density can not be expressed as inner products of these fields with their adjoint fields. In this paper, I introduce the spinor photon field that satisfies a spinor equation similar to the Dirac equation for the electron. The three-dimensional vector electric field, the three-dimensional magnetic vector field and the four-dimensional vector potential are components of this spinor photon field. The spinor equation for the photon field is based on the Maxwell equations, the relations between the electromagnetic field and the four- dimensional vector potential, and the Lorentz gauge condition. The Lagrangian densities for the free photon field and for the photon field in interaction with the 320 Ruo Peng Wang 2 matter are established. Covariant quantization of photon field is carried out, and only transverse photons emerge from the quantization procedure. The vacuum state of the photon field is found to have null energy. The solution for the photon field in the presence of the electric current and charge is found, and a covariant “positive frequency” condition is introduced for separating the photon field from its complex conjugate. The Maxwell equations, the relations between the electromagnetic field and the four-dimensional vector potential, and the Lorentz gauge condition are rewritten as two eight-component spinor equations in Sec. 2. The spinor photon field is introduced in Sec. 3. The quantization of the photon field is treated in Sec. 4, and the photon field in the presence of electric current and charge is analysed in Sec. 5. 2. THE SPINOR EQUATION FOR THE ELECTROMAGNETIC FIELD Maxwell’s equations for the electric and magnetic fields E and H in the presence of a charge density ρ and a current density j can be written in the following form: ∂ ()ε000EHj=∇×() µ − µ , (1) ∂x0 ∂ ()µ=−∇×ε00HE(), (2) ∂x0 0 =−∇⋅( µ0 H) , (3) 0 = ∇⋅( ε00E) − ε ρ, (4) where ε 0 is the vacuum permittivity, µ 0 is the magnetic permeability of the vacuum, and x 0 =ct. Because the electric field, the magnetic field, the current density and the charge density are real quantities, they are completely described by their positive frequency components. An alternative way for writing the above equations is to introduce an eight components spinor electromagnetic field and an eight components spinor electric current density defined by ψ=em ()x T (5) ()εεε01Ex() 02 Ex () 03 Ex ()0 µ 0 Hx 1 () µ 0 H 2 () x µ 0 H 3 ()0 x and 3 The spinor field theory of the photon 321 T je =µ01() jxjxjx() 2 () 3 ()0000 jx 0 () , (6) where jx()=ρ( c (),() xj x) is the four-vector current density. The Maxwell equations (1-4) now can be written as ∂ ψ=−⋅∇ψ−em()x α e em ()xjx e (), (7) ∂x 0 where 00 0−σiI22 000 − 00−σiI 0 00 0 22 α=ee12,, α= 000iIσ 22 000 iIσ−2200 0 000 (8) 00iσ 2 0 000−σi 2 α=e3 −σi 2 00 0 000iσ 2 with 010−i σ=22andI = . (9) i 001 We have α⋅α+α⋅α=δem en en em2, mn mn ,1,2,3. = (10) The electromagnetic field can be described by the four-vector potential Ax()=φ( ()/, x cA () x) . The relation between the electric and magnetic fields and the four-vector potential, and the Lorentz gauge condition can be written as: ∂ 1 ()ε=−∇ε−ε0000AE()A , (11) ∂xc0 1 0 =∇× εAH − µ , (12) ()00c ∂ ()ε=−∇⋅ε00A () 0A . (13) ∂x 0 The relations (11–13) can be rewritten as 322 Ruo Peng Wang 4 ∂ 1 ψ=⋅∇ψ−ψaea()x α ()xx em (), (14) ∂xc0 where the spinor potential field ψ a (x ) is defined by ε ψ=()xAxAxAxAx0 () () () ()0000 ()T . (15) a 123 0 One may observe that the equation (14) also holds if we replace the Lorentz gauge condition with the following one: ∂ 1 ()ε=−∇⋅ε−ψ00A () 0A em 8, (16) ∂xc0 where ψ em8 is an arbitrary scalar constant. In this case, the spinor electromagnetic field takes the following form ψ=em ()x T (17) ( εεε01Ex() 02 Ex () 03 Ex ()0 µ 0 Hx 1 () µ 0 H 2 () x µψ 0 H 3 () xem 8 () x) . According to properties of the electric field E(x ) , magnetic field H(x ) , and four- vector potential Ax( ) under continuous space-time transformations, we have the following relation for ψ em (x ) and ψ a (x ) under a Lorentz transformation ψ=−⋅ψ′′em()x exp( φl) em (),x (18) and ψ=′′aea()x exp(φα ⋅−ψ( l)) (),x (19) where v vv φ =+−−ln 1 ln 1 . (20) vc c Under a rotation characterized by the rotation angle φ , expressed as an axial vector, we have ψ=′′em()x exp(ixφ ⋅ψs) em (), (21) and ψ=′′aa()x exp(ixφ ⋅ψs) (), (22) 5 The spinor field theory of the photon 323 with ΣΣ00 i sl==,, (23) 00ΣΣ −i and 0000 00ii 0 0− 00 00−ii 0 0 000 0 00 Σ=,,. Σ= Σ= (24) 1230ii 00− 000 0000 0000 0000 0000 The following commutation relation holds for s : 3 [,s nmsi ]=ε∑ nmpp s . (25) p=1 Equations (7) and (14) are invariant under continuous space-time transformations (See the Appendices). 3. THE PHOTON FIELD One can use ψ em ()x or ψ a ()x to represent the electromagnetic field, but it is not possible to express the photon density as an inner product of ψ em (x ) or ψ a ()x with its adjoint field. Therefore neither ψ em ()x , nor ψ a ()x can be regarded as the photon field. The concept of photon is closely related to monochromatic electromagnetic plane waves, so we consider a monochromatic plane wave kk ψψ∝−em(x ), a (xikx ) exp( ) (26) +k +k in the absence of electric current and charge. Let ψ em ()x and ψ a ()x be the k k positive frequency parts of ψ em (x ) and ψ a (x ) . We find that the inner product ++kk† ψψem()x em ()x is equal to the time average of energy density, which should equal to, in the case of a monochromatic wave, the product of the photon density and the photon energy ck 0 . On other hand, we have 1 ε⋅=µ⋅EE++kk**()x ()xxx HH + k () + k ()=ψ++kk† ()x ψ ()x (27) 002 em em and 324 Ruo Peng Wang 6 1 ixxixxEA+kk**()⋅=−⋅= + () AE ++ kk () () EE ++ kk * () xx ⋅ (), (28) ck 0 thus the photon density is equal to ++kk†† ++ kk −ψixxixxaemema() ψ () + ψ () ψ (). (29) We have ψ +k ()x −ψixxixx++kk††() ψ () + ψ ++ kk () ψ () = ψ + k †† () x ψ + k () x τ em , (30) aemema() ema 2 +k ψ a ()x where 0 iβ e I 4 0 τ=2 with β=e , (31) −βiIe 00 − 4 and I 4 is the 4× 4 unit matrix. Based on the relation (30), we define the photon field ψ f (x ) as ψ ()x′ 1 4 em ψ≡f ()xkikxx d exp()()′ −. (32) 4 ∫ ψ ()x′ ()2π k 0 >0 a One may observe that the condition k 0 > 0 is covariant for the free photon field, because it does not contain Fourier components with k 0 < k . According to Eqs. (7) and (14), the free photon field satisfies the following equation: ∂ i ixixψ=−⋅∇ψ−βψfwff()α ()− () x, (33) ∂xc0 with α e 000 α w =β=,,− (34) 00−α e I 8 where I 8 is the 88× unit matrix. The invariance of Eq. (33) under continuous space-time transformations is assured by the invariance of Eqs. (7) and (14). We have ψ=′′ff()x exp(φΛ ⋅ψ) (),x (35) with −l 0 Λ = (36) 0 αle − 7 The spinor field theory of the photon 325 for Lorentz transformations, and ψ=′′fff()x exp(ixφ ⋅ψs ) (), (37) under space rotations, where s 0 s f = . (38) 0 s Eq. (33) is invariant also under space inversion and time reversal. It is easy to verify that τψ00f (,x −x ) and τ 30ψ−f (,)x x satisfy the same spinor equation Eq.