Heisenberg-Euler effective lagrangians

Transcription

1 Heisenberg-Euler effective lagrangians Appunti per il corso di Fisica eorica 7/ Fiorenzo Bastianelli We derive here effective lagrangians for the electromagnetic field induced by a loop of charged virtual particles of spin and /. Historically, Heisenberg and Euler derived in 936 an effective lagrangian that includes the virtual effects of electrons. Soon after Weisskopf extended that calculation to a charged scalar particle. For a particle of charge q and mass m the results, that include the classical lagrangian plus the leading effective quartic interaction in the electromagnetic field, are the following L = 4 F µν F µν + q4 6π m 4 88 F µν F µν + ] 36 F µν F νρ F ρσ F σµ L / = 4 F µν F µν q4 8π m 4 7 F µν F µν 7 ] 8 F µν F νρ F ρσ F σµ or, equivalently, L = E B + α 36 m 4 7 E B + 4 E B ] L / = E B + α 45 m 4 E B + 7 E B ] where α = q is the fine-structure constant when q is taken as the electron charge. 4π We use here a first quantized approach to derive these results. We consider first the case of a spin particle, it exemplifies the main points of the calculation. hen we extend it to the spin / particle, of more direct physical relevance. All charged particles induce similar nonlinearities, but since the outcome is proportional to m 4, it is the lightest charged particle that gives the dominant contribution, i.e. the electron. his nonlinear effect has not been detected experimentally yet, because of the smallness of the coupling. One may express the result in terms of the Lorentz invariants E B and E B. Using the components E E E 3 E E E 3 F µν = E B 3 B E B 3 B, F µν = E B 3 B E B 3 B E 3 B B E 3 B B one immediately finds F µν F µν = B E. As for the term F µν F νρ F ρσ F σµ one may first consider the case with only E, that annihilates the second invariant, and then the case E = B, that annihilates the first invariant, to find F µν F νρ F ρσ F σµ = E B + 4 E B.

2 Spin case We outline the calculation for a scalar particle of charge q and mass m, quoting the relevant formulas. he action to be used for the scalar particle is best described in hamiltonian form Sx, p, e; A] = p µ ẋ µ e πµ π µ + m 3 }{{} H where the minimal coupling to the electromagnetic field A µ has been introduced in the hamiltonian constraint H by using the covariant momentum π µ instead of p µ π µ = p µ qa µ x. 4 Eliminating p µ by its own algebraic equation of motion, one finds the configuration space action S c x, e; A] = e ẋ em + qa µ ẋ µ. 5 he calculation to be described is better performed using euclidean conventions, so that one performs a Wick rotation is c S E to find the euclidean action S E x, e; A] = e ẋ + em iqa µ ẋ µ. 6 aking the worldline to be a loop, i.e. a manifold with the topology of the circle S, one gets the worldline representation of the effective action ΓA] for the electromagnetic field as induced by the particle DxDe ΓA] = Z S A] = = VolGauge e S Ex,e;A] e m P Dx e Sqx;A] 7 where in the last line the path integral has been gauge fixed by setting eτ =, and the gauge fixed action without the mass term that is factored out is given by S q x µ ; A µ ] = 4 ẋµ ẋ µ iqa µ xẋ µ. 8 he overall normalization of 7, as well as a check on the integration measure over the proper time, can be fixed by comparing with similar formulas obtained directly from the quantum field theory of a charged Klein-Gordon field coupled to electromagnetism. he details are as follows. In QF the effective action ΓA] induced by a charged Klein-Gordon field φ with euclidean action Sφ, φ ; A] = d 4 x D µ φ D µ φ + m φ φ = d 4 x φ A + m φ is given by e ΓA] = DφDφ e Sφ,φ ;A] = Det A + m so that ΓA] = ln Det A + m = r ln A + m = r e A+m up to an irrelevant constant term. he proper time representation of the logarithm follows from the equality ln a b = e a e b.

3 he formula in 7 is a general representation of the effective action, but its evaluation in closed form is a difficult task. However, for external low-energy photons, one can consider the electromagnetic field F µν to be constant, and find an exact result. For such a constant configuration one may use the potential and the action takes a quadratic form S q x µ ; A µ ] = = A µ x = xν F νµ 9 4 ẋµ ẋ µ iq xµ F µν ẋ ν xµ δ µν τ iqf µν τ x ν. In the integration by part used to obtain the second expression no boundary term is produced, as one must use periodic boundary conditions i.e. there is no real boundary on the circle S. he path integral is gaussian and is computed exactly. he result is expressed in terms of functional determinants. In four spacetime dimensions it reads P Dx e Sqx;A] = d 4 x Det P 4π Det P τ iqf τ τ. o prove this formula, and its precise normalization, one may perform the path integral by defining it as an integration over the Fourier coefficients of the Fourier expansion of a generic periodic path x µ τ = n Z x µ n e πinτ. he zero modes x µ drop out of the action, and they can be identified with the space-time coordinates. One must integrate over them, as well as over the other Fourier coefficients. he latter produce a functional determinant over periodic functions without the zero modes. his is indicated by the prime in Det. For vanishing F µν it evaluates the path integral for a free particle, which has been computed earlier. Adapting notations one finds Dx e Sqx] = d 4 x Det P ] d 4 x τ = 4π. 3 P hen, for non vanishing and constant F µν one gets the expression in. he evaluation of the determinant is carried out as product of all eigenvalues. he determinants in can be simplified to Det P τ iqf τ = Det P τ + i qf. 4 τ Det P τ Let us consider the case of a single dimension with the matrix F replaced by a number. A set of eigenfunctions is given by the periodic functions Det P f n τ = e πinτ n Z 5 3

4 so that removing the zero mode one computes Det P τ + iw πin + iw Det = = + w P τ πin πn n Z,n n Z,n = w = sin w 4π n w. 6 n> Now, in four dimensions we have a matrix F, that one could diagonalize so to proceed by applying the result above. One finds the final result as a determinant of a 4 4 matrix, expressed in terms of its eigenvalues. hus one may just rewrite it as Det P τ + i qf sin Det = det 7 P τ where Det denotes a functional determinant and det the determinant of a finite dimensional matrix. Inserting this into, and then into 7, one finds ] ΓA] = d 4 e m sin x 4π det. 8 he effective lagrangian is then given by L = e m 4π det sin. 9 As it stands it contains ultraviolet divergences that must be renormalized away. hese divergences are seen as coming from the integration limit. o remove them let us consider the expansion of the determinant piece det sinx x = exp ln det sinx x = exp sinx ] tr ln x = exp = exp tr x 6 x4 8 + ] sinx ] ln det x = + trx + 88 trx + 36 trx4 +. Considering x =, one checks that only the first two terms give divergent contributions to the proper time integral in 9. One may subtract them with counterterms, and express the renormalized lagrangian, with the classical Maxwell lagrangian added, as L ren = 4 F µν F µν 6π e m ] det sin + q F µν F µν. his is the full, non perturbative result, for a constant electromagnetic field in a euclidean spacetime, known as the Heisenberg-Euler effective lagrangian induced by a scalar particle. 4

5 Expanding the term in square brackets, and keeping only the first contribution, i.e. the term quartic in F µν, one finds L ren = 4 F µν F µν q4 6π m 4 88 F µν F µν + ] 36 F µν F νρ F ρσ F σµ + he calculation has been done in euclidean time, but at the end one must perform the inverse Wick rotation, which amounts to a global change of sign plus the use of the Minkowski metric in raising and lowering indices. his sign can be understood as due to L M = V versus L E = + V in a Wick rotation, and considering that and give just a constant contribution to V. hus, back in Minkowski space, one finds the result anticipated in for the scalar particle. Spin / case For a spin / particle one starts from the action in phase space Sx, p, ψ, ψ 5, e, χ; A] = p µ ẋ µ + i ψ ψ µ µ + i ψ5 ψ 5 iχ π µ ψ µ + mψ 5 }{{} e πµ π µ + iqf µν ψ µ ψ ν + m }{{} H where the minimal coupling to the electromagnetic field A µ has been introduced in the susy constraint Q by using the covariant momentum π µ = p µ qa µ x, while the hamiltonian constraint H is found by imposing the Poisson bracket susy algebra {Q, Q} = ih. Eliminating p µ by its own algebraic equation of motion, one obtains the configuration space action S c x, ψ, ψ 5, e, χ; A] = e ẋ µ iχψ µ + i ψ ψ µ µ + i ψ5 ψ 5 em imχψ 5 Q 3 +qa µ ẋ µ ieq F µνψ µ ψ ν. 4 Again, the calculation is better performed using euclidean conventions, so that we perform a Wick rotation is c S E to obtain the euclidean action S E x, ψ, ψ 5, e, χ; A] = e ẋ µ χψ µ + ψ ψ µ µ + ψ5 ψ 5 + em + imχψ 5 iqa µ ẋ µ + ieq F µνψ µ ψ ν. 5 aking the worldline to be a loop, i.e. a circle S, one gets the worldline representation of the effective action ΓA] for the electromagnetic field as induced by the spin / particle DxDψDψ 5 DeDχ ΓA] = Z S A] = e S E VolGauge = Dx Dψ e Sqx,ψ;A] 6 e m 5 P A

6 where in the last line the path integral has been gauge fixed. his is achieved by considering periodic boundary conditions P for bosons and antiperiodic boundary conditions A for fermions; by setting eτ = and χτ = as gauge fixing conditions for the local symmetries; recognizing that ψ 5 becomes free and can be dropped at most it contributes to the overall normalization that is anyhow fixed by comparison with QF. At the end one has a gauge fixed action without the mass term that is factored out S q x, ψ; A] = 4 ẋµ ẋ µ + ψ µ ψ µ iqa µ ẋ µ + i µν ψ µ ψ ν. 7 he overall normalization of 6, as well as a check on the integration measure over the proper time, can be fixed by comparing with similar formulas obtained from the quantum field theory of a charged Dirac field coupled to electromagnetism 3. Again, the formula in 6 is a useful representation of the effective action, but its evaluation in closed form is not known. For external low-energy photons, one takes the external electromagnetic field F µν constant, and finds an exact result. For such a constant configuration one may use the potential and the action takes a quadratic form. It splits into two parts S q x, ψ; A] = = = S b q A µ x = xν F νµ 8 4 ẋµ ẋ µ iq xµ F µν ẋ ν + ψ ψ ] µ µ + i µν ψ µ ψ ν xµ δ µν τ iqf µν τ x ν + ] ψµ δ µν τ + i µν ψ ν x; A] + S q f ψ; A]. 9 he path integral is gaussian. he bosonic part has already been treated previously. he fermionic part reads A Dψ e Sf q ψ;a] = Det A τ + i Det A τ Det A τ where the last determinant corresponds to the free path integral that computes the trace of the identity as the hamiltonian vanishes in the Hilbert space of the worldline fermions. his Hilbert space is four dimensional, as it gives the number of spinorial components of a Dirac spinor in four dimensions Det A τ = r = = he details are as follows. In QF the effective action induced by a charged Dirac field Ψ with action SΨ, Ψ; A] = d 4 x ΨD/ + mψ is given e ΓA] = DΨD Ψ e SΨ, Ψ;A] = DetD/ + m so that ΓA] = r lnd/ + m = r ln D/ + m = r ln D/ + m = up to constant irrelevant terms. 6 r +m e D/ 3

7 he remaining determinants are evaluated as product of all eigenvalues. A set of eigenfunctions is given by the antiperiodic functions so that if one would have just a number Det A τ + iω = Det A πin + + iw τ πin + = n Z n Z = f n τ = e πin+ τ n Z 3 n= + w w = cos π n + w πn Now, in four dimensions we have a matrix F, that one could diagonalize so to use the result above. One finds the final result written as the determinant of a 4 4 matrix expressed in terms of its eigenvalues, which one rewrites it back as Det A τ + i Det A τ = det cos. 34 Inserting everything into 6 one finds ΓA] = d 4 x e m 4π det ] tan. 35 he effective lagrangian is thus given by L = e m 4π det tan. 36 As it stands it contains ultraviolet divergences that must be renormalized away. hese divergences are seen as emerging from the integration limit. o remove them let us consider the expansion of the determinant piece det tanx x = exp ln det tanx = exp tanx ] x ln det x = exp tanx ] tr ln x = exp x tr ] 9 x4 + = 6 trx + 7 trx 7 8 trx Considering x =, one checks that only the first two terms give divergent contributions to the proper time integral in 36, so that one subtracts them with counterterms, and expresses the renormalized lagrangian, with the classical Maxwell lagrangian added for comparison, as L / ren = 4 F µν F µν + 8π e m ] det tan q 6 F µν F µν. 38 7

8 his is the full, non perturbative result, for a constant electromagnetic field in euclidean time, which is known as the Heisenberg-Euler effective lagrangian. Expanding the term in square brackets, and keeping only the first contribution, which is quartic in F µν, one finds L / ren = 4 F µν F µν + q4 8π m 4 7 F µν F µν 7 ] 8 F µν F νρ F ρσ F σµ + 39 he calculation has been done in euclidean time. One may perform the inverse Wick rotation, which amounts to a global change of sign and the use of the Minkowski metric in raising and lowering indices. Back in Minkowski space, one finds the result anticipated in for the spin / particle 8