by Mostafa Husseinsupervised by Professor George Siopsis

PHYS-611: Quantum Field Theory IDepartment of Physics & Astronomy

The University of Tennessee - KnoxvilleDecember 2012

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World line Representation and

Derivation of the Amplitude for Pair Creation in Vacuum in an External Electric Field

using the Proper-Time Method

World line

2

Peter Nolan, “The Fundamentals

of The Theory of Relativity” http://www.farmingdale.edu/

faculty/peter-nolan/pdf/UPCh34.pdf

World line

2

Peter Nolan, “The Fundamentals

of The Theory of Relativity” http://www.farmingdale.edu/

faculty/peter-nolan/pdf/UPCh34.pdf

3

Stückelberg, E.C.G. HelveticaPhysica Acta, 14, 588-594 (1941).

On the world line of a charged particle, a particle is forwardly evolving, dx0 / dτ( ) > 0

τ is the proper time

can be interpreted as a particle with opposite charge backwardly evolving,

i.e. its anti-particle.dx0 / dτ( ) < 0

World line representation - Stückelberg

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Electric field between t1 and t2

World line representation - Stückelberg

Stückelberg, E.C.G. HelveticaPhysica Acta, 14, 588-594 (1941).

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Feynman, R.P. Physical Review, 74(8), 939-946 (1948).

World line representation - Feynman

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Derivation - Vacuum* Vacuum in an electric field results in the creation of virtual electrons and positrons ⇒vacuum polarization.

* The probability amplitude of the vacuum state to remain in the ground state when there is:

no electric field A=0 is

but with an electric field A≠0 is and the probability for pair creation will equal to

0+ 0−A≠0 2

≠1

0,t =+∞ 0,t = −∞ A=0 2≡ 0+ 0−

A=0 2=1

Dittrich et al. 1985, 2000, 2001

1− 0+ 0−A 2

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Derivation - Effective Action

Toms 2007 & Dittrich et al. 1985, 2000, 2001

* Let where is the effective action of the one-loop correction.

0+ 0− = eiW(1)[A] W (1)[A]

L =L(0) +L(1) ; L(0) = −

14FµνF

µν

where is the one-loop correction L(1)

* Using Schwinger action principle W (1)[A]= d 4∫ xL(1)(x)

* For fermions, using the Dirac field, with δW (1)[A]δAµ (x)

= 0 jµ (x) 0 A= −e lim

ʹ′x→xγ µ 0 T ψ( ʹ′x )ψ(x)( ) 0 = ie trγ γ

µG(x, x | A)⎡⎣ ⎤⎦

x = xµ (τ )

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Derivation - Schwinger Proper time Method* Our Green function should satisfy the Dirac Field Equationin coordinate space:

Schwinger 1951 & Dittrich et al. 1985, 2000, 2001

−iγ µDµ +m( )G(x, ʹ′x | A)=δ(x − ʹ′x ) Dµ = ∂µ+ ieAµ

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Derivation - Schwinger Proper time Method* Our Green function should satisfy the Dirac Field Equationin coordinate space:

Schwinger 1951 & Dittrich et al. 1985, 2000, 2001

−iγ µDµ +m( )G(x, ʹ′x | A)=δ(x − ʹ′x ) Dµ = ∂µ+ ieAµin momentum space:

−iγ µ Dµ +m( )G[A]=1

Dµ = pµ + ieAµ

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Derivation - Schwinger Proper time Method* Our Green function should satisfy the Dirac Field Equationin coordinate space:

Schwinger 1951 & Dittrich et al. 1985, 2000, 2001

−iγ µDµ +m( )G(x, ʹ′x | A)=δ(x − ʹ′x ) Dµ = ∂µ+ ieAµin momentum space:

−iγ µ Dµ +m( )G[A]=1

Dµ = pµ + ieAµ

Using Schwinger proper time method: X−1 = i d0

∞

∫ τ e−iτ X

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Derivation - Schwinger Proper time Method* Our Green function should satisfy the Dirac Field Equationin coordinate space:

Schwinger 1951 & Dittrich et al. 1985, 2000, 2001

−iγ µDµ +m( )G(x, ʹ′x | A)=δ(x − ʹ′x ) Dµ = ∂µ+ ieAµin momentum space:

−iγ µ Dµ +m( )G[A]=1

Dµ = pµ + ieAµ

G[A]= 1m− iγ µ Dµ

=m+ iγ µ Dµ( )

m− iγ µ Dµ( ) m+ iγ µ Dµ( )=

m+ iγ µ Dµ( )m2 + γ µ Dµ( )2

G[A]= m+ iγ µ Dµ( )i d0

∞

∫ τ e−iτ m2+ γµ Dµ( )2⎡

⎣⎢⎤⎦⎥

Using Schwinger proper time method: X−1 = i d0

∞

∫ τ e−iτ X

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Derivation - Ansatz L(1)

W (1)[A]= d 4∫ xL(1)(x)= d 4∫ x −

12

dττ0

∞

∫ e−iτm2trx,γ e

−iτ γµ Dµ( )2⎡

⎣⎢⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

Schwinger 1951 & Dittrich et al. 1985, 2000, 2001

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Derivation - Ansatz L(1)

W (1)[A]= d 4∫ xL(1)(x)= d 4∫ x −

12

dττ0

∞

∫ e−iτm2trx,γ e

−iτ γµ Dµ( )2⎡

⎣⎢⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

Indeed:δW (1)[A]δAµ (x)

= ie trγ γµG(x, x | A)⎡⎣ ⎤⎦

Schwinger 1951 & Dittrich et al. 1985, 2000, 2001

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Derivation - Ansatz L(1)

W (1)[A]= d 4∫ xL(1)(x)= d 4∫ x −

12

dττ0

∞

∫ e−iτm2trx,γ e

−iτ γµ Dµ( )2⎡

⎣⎢⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

Indeed:δW (1)[A]δAµ (x)

= ie trγ γµG(x, x | A)⎡⎣ ⎤⎦

Therefore L(1)(x)= − 1

2dττ0

∞

∫ e−iτm2trγ x e−iτ γ

µ Dµ( )2 x⎡

⎣⎢⎤

⎦⎥

Schwinger 1951 & Dittrich et al. 1985, 2000, 2001

9

Derivation - Ansatz L(1)

W (1)[A]= d 4∫ xL(1)(x)= d 4∫ x −

12

dττ0

∞

∫ e−iτm2trx,γ e

−iτ γµ Dµ( )2⎡

⎣⎢⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

Indeed:δW (1)[A]δAµ (x)

= ie trγ γµG(x, x | A)⎡⎣ ⎤⎦

Therefore L(1)(x)= − 1

2dττ0

∞

∫ e−iτm2trγ x e−iτ γ

µ Dµ( )2 x⎡

⎣⎢⎤

⎦⎥

Transformation amplitude

K( ʹ′x , ʹ′ʹ′x ;τ | A)≡ ʹ′x e−iτ γ

µ Dµ( )2 ʹ′ʹ′x = ʹ′x ,τ ʹ′ʹ′x ,0 = Dx(0)= ʹ′ʹ′x

x(τ )= ʹ′x

∫ x(τ )eiS x(τ )[ ].

Schwinger 1951 & Dittrich et al. 1985, 2000, 2001

10Itzykson et al. 1980

One dimensional Electric FieldIn the case of pure one dimensional time dependent electric field along the z axis choosing our gauge to be

A3 = E3x0 and A0 = A1 = A2 = 0

10Itzykson et al. 1980

One dimensional Electric FieldIn the case of pure one dimensional time dependent electric field along the z axis choosing our gauge to be

A3 = E3x0 and A0 = A1 = A2 = 0

Solving L(1)(x)= − 1

2dττ0

∞

∫ e−iτm2trγ x e−iτ γ

µ Dµ( )2 x⎡

⎣⎢⎤

⎦⎥

10Itzykson et al. 1980

One dimensional Electric FieldIn the case of pure one dimensional time dependent electric field along the z axis choosing our gauge to be

A3 = E3x0 and A0 = A1 = A2 = 0

Solving L(1)(x)= − 1

2dττ0

∞

∫ e−iτm2trγ x e−iτ γ

µ Dµ( )2 x⎡

⎣⎢⎤

⎦⎥

We obtain L(1)(x)= − 1

8π 2dττ 20

∞

∫ e−iτm2eE3 coth τ eE3( )−1⎡⎣ ⎤⎦

10Itzykson et al. 1980

One dimensional Electric FieldIn the case of pure one dimensional time dependent electric field along the z axis choosing our gauge to be

A3 = E3x0 and A0 = A1 = A2 = 0

Solving L(1)(x)= − 1

2dττ0

∞

∫ e−iτm2trγ x e−iτ γ

µ Dµ( )2 x⎡

⎣⎢⎤

⎦⎥

We obtain L(1)(x)= − 1

8π 2dττ 20

∞

∫ e−iτm2eE3 coth τ eE3( )−1⎡⎣ ⎤⎦

and

1− 0+ 0−A 2

=1− exp −i8π 2 d 4∫ x dτ

τ 20

∞

∫ e−iτm2eE3 coth τ eE3( )−1⎡⎣ ⎤⎦

⎧⎨⎩

⎫⎬⎭

2

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Endnotes✴World line representation can interpret pair production. Causality?

“the eventual creation and annihilation of pairs that may occur now and then, is no creation and annihilation, but only a change of directions of moving particles, from past to future, or from future to past” - Nambu Prog. Theor. Phys. 5, 82 (1950).

✴Derived the probability amplitude, considering up to one-loop correction of the Lagrangian, using Schwinger proper time method.

✴Proper time method useful in String Theory.

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Endnotes✴World line representation can interpret pair production. Causality?

“the eventual creation and annihilation of pairs that may occur now and then, is no creation and annihilation, but only a change of directions of moving particles, from past to future, or from future to past” - Nambu Prog. Theor. Phys. 5, 82 (1950).

✴Derived the probability amplitude, considering up to one-loop correction of the Lagrangian, using Schwinger proper time method.

✴Proper time method useful in String Theory.

Thank You