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Particle pair production in strong EM field, Imaginary temperature and field emission
Boris Levchenko SINP MSU, Moscow



Tunneling processes (Nature, adv. technologies) Electron field emission from metals: quantum theory by Fowler and Nordheim





Results by Sauter and Schwinger for electric field in vacuum
Condensed matter .vs. Vacuum & EM field



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Critical electric field in QED

(F. Sauter, 1931)

A similar phenomenon already was studied in quantum mechanics in 1928 !

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Theory of cold field emission from metals (Fowler-Nordheim ,1928)
Image potential + external field F

Penetration coefficient (WKB)

Electron emission current

B = 6.8x10 7 3 / 2V / = 5.4x10 8V / cm = 4eV
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Results by Sauter and Schwinger for strong field in vacuum

Figures from the paper by W. Heisenberg, H. Euler (1936)

WKB method

QED: The probability of vacuum decay in a uniform and static electric field with e+e- pairs creation (Schwinger, 1951)

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Wick rotation: some examples
Imagine that we are calculating a certain correlation function in the euclidean formulation of the gauge theory. This means averaging over all possible fields A with the weight equal to:



Minkowski space-time



Euclid space by

t = i

exp(iS(A))





Statistical Mechanics



Quantum Mechanics

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Instantons:
Minkowski space-time -> Euclid space by t -> i BPST-instanton, self-dual equation (Belavin, Polyakov, Schwartz, Tyupkin) U(1) (`t Hooft)

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Field emission at T0 vs. Particle production at H0
FE from a metal at T=0

J

FN

( F ) = A F exp (

2

Fcr F
Ecr E

)

Scalar or e+e- pair production in electric field, H=0 (at WKB level, the Dirac and Fock-Klein-Gordon equations are identical)

2 w( E ) = 2 E exp (

)

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Field emission at T0 vs. Particle production at H0
FE from a metal at T 0 (Murphy, Good, 1956)

Pair of scalar particle production from vacuum, H 0 (Ritus,1969),

We are observing a very intriguing features !
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Field emission atT0 vs. Particle production at H0
If recall that

With the replacement (a sort of Wick rotation)

One get from FE Eq, the equation for a pair production from vacuum

J ( F , T ) w( E, H )
Magnetic field in vacuum plays a role of the imaginary temperature
How is unusual our conclusion ?
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Ising Model in D=2

The first result obtained for H 0 is the famous circle theorem of Lee and Yang (1952): the zeroes of the partition function of an Ising model on a finite size lattice in D dimensions lie on the unit circle |z| = 1:

The imaginary magnetic field
Lee and Yang also found that for the free energy of the Ising model can also be exactly solved for Ev = Eh.

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Weierstrass theorem

When V is finite the grand partition function Qgr (z , T; V) is an entire function of z which is positive on the positive z axis. We may use the Weierstrass factorization theorem to express Q gr in terms of its zeros and thus

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QFT pays a reward ?
Production a pair of spinor particles from vacuum (Nikishov, 1969)

Inverse rotation (anzac)

Do we get the emission current from a metal with account of the electron spin ?

?
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No that simple. Experimental data on FE compared to a theory (FE+thermionic em. + image ch. eff.)

x sin x

x t an x

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Conclusions



FE from a metal and particle production from vacuum in strong EM field are compared Both phenomena are related via a type of Wick rotation



H ic 0 k BT



Magnetic field in vacuum plays a role of the (imaginary) temperature for scalar particles

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Backups
Scattering Off the Vacuum a study of the scattering of an electron beam off the vacuum. The issue is that the "vacuum" pipe of a particle accelerator is not a true vacuum, but contains room-temperature blackbody radiation as well as residual gas molecules. In the process called "inverse" Compton scattering by astrophysicists, a beam electron can amplify the energy of a blackbody photon by a factor 4І. A room-temperature photon an energy 1/40 eV can attain energies of order 1 GeV when scattered by a 50 GeV electron beam. The experiment consisted of the 50-GeV LEP beam at CERN, a single lead glass block, and "no" target.

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Scheme of probing strong-field QED in the collisions of a 50-GeV electron beam with a focused laser beam.

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Multiphoton pair creation by light can arise from a two-step process in which a high-energy photon from the previous reaction interacts with the laser beam

This process is the strong-field variant of Breit-Wheeler pair creation.

(a) Laser-on and -off spectra of positrons from reaction (7). (b) Subtracted spectrum. The solid line is a model calculation.
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governs the importance of multiple photons in the initial state
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governs the importance of "spontaneous" pair creation

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