Quantum Electrodynamics and Planck-Scale

Quantum Electrodynamics and Planck-Scale

Quantum Electrodynamics and Planck-Scale Rainer Collier Institute of Theoretical Physics, Friedrich-Schiller-Universität Jena Max-Wien-Platz 1, 07743 Jena, Germany E-Mail: [email protected] Abstract. This article examines the consequences of the existence of an upper particle momentum limit 3 ppP= ≤ ∗ in quantum electrodynamics, where P∗ = cG is the Planck momentum. The method used is Fourier analysis as developed already by Fermi in his fundamental work on the quantum theory of radiation. After determination of the appropriate Hamiltonian, a Schrödinger equation and the associated commutation rules of the field operators are given. At the upper momentum limit mentioned above, the divergent terms occurring in the Hamiltonian (the self-energies of the electrons and the zero-point energy of the electromagnetic field) adopt finite values, which will be stated and compared with each other. 1 Introduction To this day, it is not clear in what way the Planck-scale, i.e. particle energies or momenta in 2 the order of E∗∗= Mc or P∗∗= Mc, respectively, can be consistently related to the physics −5 of elementary particles and that of the gravitational field (with M∗ = cG =2,18 ⋅ 10 g denoting the Planck mass, the Planck constant, c the velocity of light in vacuo and G Newton‘s gravitational constant). There exist, though, several suggestions how to solve this problem ad hoc, i.e. without decisive references to experimental experience. Among these approaches there are the string theory, the doubly special relativity theory (DSR), loop quantum gravitation, the introduction of non-commutative geometries, the use of specially deformed Lorentz algebras, as well as several generalized uncertainty principles (GUP) in which the Planck momentum occurs. For some detailed overview articles, see refs. [2], [3] and [4]. To study the consequences of the existence of an upper particle momentum limit for a definite physical object, we had looked at the thermodynamics of the photon gas under this aspect and have proposed corrected Planck radiation laws modified for high photon energies [5]. For the mean spectral energy ε ()p of the energy level ω at the temperature T , e.g., we obtained the radiation law 1 ε ()p ε()p= , ε()p= ck = ω . (1.1) ε()pp− µ ε () exp −−1 kTB E∗ For the free energy of the photon gas F= − PV (the thermal equation of state), there followed P∗ ⌠ 2 4p pε() pp ε ()− µ FTV(,)()= gsB kTV ln 1−− 1 exp − dp . (1.2) 3 h ε ()p E∗ kTB ⌡ 1− 0 E∗ We also calculated the modified caloric equation of state U= UTV(, ), the number of particles NTV(, ) and the entropy STV(, ). For P∗∗=() Ec →∞, all these corrected radiation laws merge into the well-known laws of photon gas thermodynamics. We also developed a statistical thermodynamics with an upper particle momentum limit for all ideal quantum gases [6]. Because of presumed upper momentum limits, other authors also have derived thermodynamic laws of the photon gas that deviate from Planck’s laws, one example being a Planck radiation law modified for high photon energies on the basis of the Lorentz-invariant DSR theory [7]. To study the effects of a maximum particle momentum p≤= P∗∗ Ec also in quantum electrodynamics, we shall follow Fermi’s fundamental work on the quantum theory of radiation [1], in which he applies the method of Fourier analysis to the electrodynamic potentials a and ϕ . 2 The classical field equations Let us start from the Lagrangian density of the Maxwell-Dirac field (Gaussian cgs system) =ψγµ −e −ψ −⋅11 µν �c() pmmmc A mc0 44p Fµν F , (2.1) µ with ψψ, denoting bispinors, γ Dirac spin matrices, Aµ the four-potential of the electromagnetic field, e the electric charge, m0 the electron rest mass, FAAµν= ν|| µ − µ ν the electromagnetic field tensor and piµµ= ∂ the momentum operator. If we select qAA = (,ψ µ ) as the independent field variables, we obtain, by means of the Lagrange equations (∂∂AAqqAA||νν ) −∂∂= ( )0, the coupled system of field equations γψµ −−e = ()pµµc A mc0 0 , (2.2) 2 4p A= j , j= ecψγµ ψ , (2.3) µµc µ µ 2 22 where =∂µ ∂ =∂ ∂()ct −∇ and the Lorenz condition µ A |µ = 0 (2.4) has been used. In the metric ηµν =diag (1, −−− 1, 1, 1) , the 3- and 4-quantities are related as follows, µ 1 µµ γ=(,), βγ piµ = (c ∂∇t ,), A = (,),ϕ a j= ( cjρ ,) , (2.5) 1 a= βγ ⋅,= =−c a −∇ϕ ,, =∇∧a jr =ρ . (2.6) The field equation system (2.2) – (2.4), then, has the following 3-dimensional form: ∂ψ 2 i = ca ⋅−() pe a + e ϕ + mc βψ , (2.7) ∂t c 0 14∂∂22p 1 −∆ = −∆ϕ = pρ 22 aj , 22 4 , (2.8) ct∂∂ c ct 1 ∂ϕ +∇⋅a =0 . (2.9) ct∂ 3 An approach to the quantum field theory 3.1 Transition to field operators The classical electromagnetic potentials Aµ become field operators now by being subjected to certain canonical commutation relations, analogously to quantum mechanics. In that context, Fermi developed the electrodynamic potentials a and ϕ in Fourier series and regards the solely time-dependent coefficients atss(), ϕ () t of these series as the proper „coordinates“ of the fields, a=∑ atss( )sin Γ , (3.1) s ϕϕ=∑ ss(t )cos Γ . (3.2) s The phase Γs has the general structure Γ=sskr ⋅ +ϑ s , (3.3) 3 wherein r is the position vector, ks the wave vector and 2p ωs ||kkss= = = (3.4) λs c the wavenumber, λs the wavelength, ωs the angular frequency and ϑs an arbitrary phase shift of the sth partial wave. Let us now project the vector as onto three mutually perpendicular unit vectors nnn123sss,,, th with n1s pointing in the propagation direction ks of the s partial wave, ks as=++ an11 ss an 2 s 2 s an 33 ss , n1s= . (3.5) ks Now, the field variables of the electromagnetic field are ϕss(),tatatat123 (), ss (), () . (3.6) From the differential equations (2.8) and (2.9) we can now determine die time dependencies of the variables (3.6) by integration over the 3-volume V . With ΓΓ =1 d=p2 ∫ coss cos s′′dV2 V ss , g V8 c , (3.7) follow the differential equations ϕ+= ωϕ2 1 ρ Γ s ss g ∫ cos sdV , (3.8) +=⋅Γω2 11 ρ as ss agc∫ rsin s dV . (3.9) These two differential equations are obviously equivalent to the Maxwell equations if the Lorenz condition (2.9) is also written in terms of the new coordinates (3.6). Considering the orthogonality properties of the base vectors nnn123sss,,, we obtain ωϕssa1 += s 0 , (3.10) or, with repeated time derivation and (3.8), ω− ωϕ2 +1 ρ Γ= ssa1 ss g ∫ cossdV 0 . (3.11) 3.2 Transition to point charges Following Fermi, we now pass on from a continuous charge distribution to an ensemble of point charges. Let the point charges ei occupy the places ri so that the integrals in (3.8) to (3.11) can be replaced as follows: 4 rrΓ→ Γ Γ→ Γ ∫∫cos s dV∑∑ eicos si , rsin s dV ei r isin si , (3.12) ii Γs =kr s ⋅ +ϑϑ s →Γ si = kr s ⋅ i + s . (3.13) The Maxwell equations (with Lorenz condition) of a system of point charges will then adopt the following form ϕ+= ωϕ2 1 Γ s s s g ∑eicos si , (3.14) i +=⋅ω 2 11 Γ as s a s gc∑ eri isin si , (3.15) i ω+ϕ = ωω−2ϕ +1 Γ= ssa11 s 0 bzw. ss ae ssg ∑ icos si 0 . (3.16) i 3.3 Der Hamiltonian operator of the Maxwell-Dirac system The Hamiltonian of our Maxwell-Dirac system is formulated as follows: ei 2 2 =∑∑cai ⋅ p i − a ssin Γ si + e i ϕ s cos Γ+ si mc0 βi isc (3.17) 1 2 222 2 222 V +bga +ω −+ p g ωϕ , g= . ∑ ( s ss) ( s ss) 2 28gcs p The form of the term after the double sum in 2 we take from the Schrödinger form of the Dirac equation in (2.8) by substituting there the Fourier expansions for ϕ and a from (3.1) and (3.2). These terms correspond to the free Dirac field and to the interaction between the Dirac and Maxwell fields. The residual term after the single sum corresponds to the free Maxwell field. The Maxwell equations with sources can be shown to be contained in the above Hamiltonian (3.17). For this purpose, we use the above 2 to write the Hamiltonian canonical equations for the canonically conjugated electromagnetic variables (ϕss, p) and ( abss, ): ∂∂22p s 2 ϕs = =− , ps =−=−geωϕs s∑ icos Γ si , (3.18) ∂pssg ϕi ∂∂b 2 ==+22s =−=−+ωa1 Γ as , bs gs a s c ∑ eci( i )sin si . (3.19) ∂bs ga∂ s i By elimination of p s from (3.18), the Maxwell equation (3.14) appears again. If we additionally consider ri=∂∂=2 pc iia , the Maxwell equation (3.15) also appears after elimination of bs from (3.19). 5 Therefore, the fundamental equation of Fermi’s quantum theory of an interaction between the electromagnetic field and the electron field reads ∂Ψ i =2 Ψ . (3.20) ∂t Therein, the wave function Ψ has the structure Ψ=Ψ(as ,ϕs si , rt ,; i ), with the quantity s i characterizing the dependence on the electron spin. The differential equation (3.20) have to supplemented by the rules of commutation between the canonical coordinates and the associated momenta, [xpk, l] = id kl , [ abks, ls′] = iddkl ss ′ , [ϕs, p s ′′] = −i dss .

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