Derivative expansion for the one-loop effective Lagrangian in QED

Gusynin, V. P.; Shovkovy, Igor

doi:10.1139/p96-044

Cited by 80 publications

(113 citation statements)

References 23 publications

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“…Heisenberg and Euler derived the Lagrangian of an effective description of this interaction for constant fields [5], which was later rederived by Schwinger [6]. Derivative expansions of this effective interaction [7][8][9] and numerical worldline calculations [10] imply that "constant" is to be taken with respect to the Compton time h/mc 2 for electron mass m. This suggests a good approximation of the effect for time-dependent fields with a much longer period than the Compton time is to simply insert them in place of the constant fields in the Heisenberg-Euler Lagrangian. In particular, the polarised vacuum supports the phenomenon of self-interaction when two electromagnetic waves couple via virtual electron-positron pairs and the principle of superposition no longer holds.…”

Section: Introductionmentioning

confidence: 99%

Vacuum high-harmonic generation in the shock regime

Böhl¹,

King²,

Rühl³

2015

Phys. Rev. A

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Electrodynamics becomes nonlinear and permits the self-interaction of fields when the quantised nature of vacuum states is taken into account. The effect on a plane probe pulse propagating through a stronger constant crossed background is calculated using numerical simulation and by analytically solving the corresponding wave equation. The electromagnetic shock resulting from vacuum high harmonic generation is investigated and a nonlinear shock parameter identified.

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Section: Introductionmentioning

confidence: 99%

Vacuum high-harmonic generation in the shock regime

Böhl¹,

King²,

Rühl³

2015

Phys. Rev. A

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“…In fact, the degeneracy is given by the (innite) total ux, which in a nite area A is BA=2. Thus the proper time propagator (5) (6). The sum over Dirac energies in (11) may also be performed explicitly because the spectrum is discrete: …”

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confidence: 99%

An All-Orders Derivative Expansion

Dunne

1997

Int. J. Mod. Phys. A

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We e v aluate the exact QED 2+1 eective action for fermions in the presence of a family of static but spatially inhomogeneous magnetic eld proles. This exact result yields an all-orders derivative expansion of the eective action, and indicates that the derivative expansion is an asymptotic, rather than a convergent, expansion.The eective action is a fundamental tool for the study of quantum eld theory. Using the proper-time technique, Schwinger 1 showed that the QED eective action can be computed exactly for either a constant or a plane wave electromagnetic eld. This result was later adapted to QED 2+1 with constant elds by Redlich 2 . For general electromagnetic elds the eective action cannot be computed exactly, so one must resort to some sort of perturbative expansion. A common approach is the derivative expansion 3;4 in which the eld is assumed to vary only very`slowly'. A useful computational procedure for the derivative expansion 5;6 is provided by the supersymmetric path integral representation of the proper-time eective action 7 . However, even rst-order derivative expansion calculations of the eective action are rather cumbersome, and it is very unclear just how convergent (or otherwise) the expansion is. Thus, the physical signicance of a rst-order correction term must be examined with care. In this paper we seek a better nonperturbative understanding of the derivative expansion by considering a new exactly solvable model which has a spatially varying magnetic eld. The exact integral representation for the eective action may then be expanded in an asymptotic series, and we nd that the rst few terms agree with (independent) derivative expansion computations.Specically, w e show that the QED 2+1 eective action for massive fermions can be computed exactly 8 in the presence of time-independent but spatially varying

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“…This is what we set out to find, and we see that it arose through the divergence of the derivative expansion. I stress that this exponential correction is not accessible from low orders of the derivative expansion, such as those studied in [8][9][10]. But the situation is even more interesting than this result (17) suggests.…”

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confidence: 86%

“…For a general inhomogeneous background field we need some sort of approximation to determine the effective action. One such approximation is the derivative expansion [7][8][9][10], in which one assumes that F µν is "slowly varying", so that one can make the following formal expansion…”

mentioning

confidence: 99%

“…First, the derivative expansion is not actually a series. This is because as one goes to higher orders, there is a rapid proliferation of new terms that do not have a counterpart at previous orders [8][9][10]. Second, it is extremely difficult to calculate any of these terms at very high order, so we could not expect anyway to observe any asymptotic behaviour of the expansion coefficients.…”

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confidence: 99%

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QED effective actions in inhomogeneous backgrounds: Summing the derivative expansion

Dunne¹

2001

AIP Conference Proceedings

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Abstract. The QED effective action encodes nonlinear interactions due to quantum vacuum polarization effects. While much is known for the special case of electrons in a constant electromagnetic field (the Euler-Heisenberg case), much less is known for inhomogeneous backgrounds. Such backgrounds are more relevant to experimental situations. One way to treat inhomogeneous backgrounds is the "derivative expansion", in which one formally expands around the soluble constant-field case. In this talk I use some recent exactly soluble inhomogeneous backgrounds to perform precision tests on the derivative expansion, to learn in what sense it converges or diverges. A closely related question is to find the exponential correction to Schwinger's pair-production formula for a constant electric field, when the electric background is inhomogeneous.This talk is concerned with the one-loop QED effective action [1,2]:Here D µ = ∂ µ + ieA µ is the covariant derivative, and so S[A] is a functional of the classical background field A µ (x). The effective action is the generating functional for one-fermion-loop Green's functions, which describe the nonlinear QED effects to this order. Ideally, one would like to know S[A] for any field A µ (x), but this is not feasible. However, for the special case where the field strength tensor, F µν = ∂ µ A ν − ∂ ν A µ , is uniform, the effective action can be evaluated in closed form [3][4][5]:Here the first term is the familiar classical Maxwell action, while the next term gives the leading quantum correction, which is quartic in the field strengths. The fine structure constant α = e 2 4π in these units. All higher terms in the expansion (2) are known explicitly [3][4][5]. It can also be shown that when the background field is a constant electric field of strength E, the effective action has an exponentially small imaginary part

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Derivative expansion for the one-loop effective Lagrangian in QED

Cited by 80 publications

References 23 publications

Vacuum high-harmonic generation in the shock regime

Vacuum high-harmonic generation in the shock regime

An All-Orders Derivative Expansion

QED effective actions in inhomogeneous backgrounds: Summing the derivative expansion

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