The Schwinger effect

The Schwinger Effect

Bachelorproject Natuur- & Sterrenkunde (12 EC)

Begeleider: Prof. dr. E.P. (Erik) Verlinde Tweede beoordelaar: dr. B.W. (Ben) Freivogel

FACULTEIT NATUURWETENSCHAPPEN, WISKUNDE & INFORMATICA

Pieter Bas Visser (6076041)

10 april 2014

Contents

Populair Wetenschappelijke Samenvatting 3

Abstract 5 1 INTRODUCTION 5 2 METHODS 7 2.1 Quantum tunneling . . . 7 2.1.1 The potential . . . 7 2.1.2 WKB approximation . . . 8 2.2 Path Integral . . . 8 2.2.1 Classical Mechanics . . . 8 2.2.2 Quantum Mechanics . . . 9 2.2.3 The Integral . . . 9 2.3 Semi-classical Limit . . . 10 2.3.1 wick rotation . . . 10

2.3.2 Euclidian path integral . . . 10

2.4 Instantons . . . 11

2.4.1 Derivation . . . 11

2.4.2 Double well potential . . . 13

2.5 Summary . . . 14

3 THE SCHWINGER EFFECT 16 3.1 Paths of charged particles . . . 16

3.1.1 Relativistic particle in an electric field . . . 16

3.1.2 Parameterization . . . 17

3.1.3 The action . . . 19

3.1.4 Perpendicular movement . . . 20

3.2 Probability . . . 20

3.3 Summary . . . 21

4 TWO CONDUCTING PLATES 22 4.1 Numerical example . . . 22

4.2 Discharge rate of the two plates . . . 23

4.2.1 Approximation . . . 23

5 DISCUSSION 25

6 CONCLUSION 26

Acknowledgements 27

A Derivation Path Integral 28

Populair Wetenschappelijke Samenvatting

Een korte samenvatting van deze scriptie voor een publiek dat het vak natuurkunde niet op universitair niveau beheersd.

Voor velen is natuurkundige kennis beperkt tot de klassieke limiet. Daarmee wor-den alle natuurlijke verschijnselen bedoeld die op menselijke schaal plaats vinwor-den. Natuurkunde wordt echter een stuk ingewikkelder en wonderlijker als je naar hele kleine of hele grote afstanden gaat kijken. Het Schwinger Effect is een voorbeeld van een natuurkundig verschijnsel op hele kleine schaal.

Voor het Schwinger Effect moeten we kijken naar het vacu¨um, de enorme, lege ruimte in het universum. In het vacu¨um zit eigenlijk niets, geen enkel deeltje (van een quark tot een molecuul) bevindt zich in die lege ruimte. Toch heeft een bekend natuurkundige, Planck, begin 20e _{eeuw aangetoond dat de energie van een deeltje}

of een veld, zoals het vacu¨um, niet nul kan zijn. Er is namelijk een hele kleine, minimale waarde voor de energie.

Op het moment dat je dit weet kan je je voorstellen dat er in het vacu¨um dan op de een of andere manier potenti¨ele energie zit, die kan worden gebruikt om deeltjes te cre¨eren. Alleen cre¨eer je dan eigenlijk deeltjes uit het niets, en dat kan weer niet volgens de wet van behoud van energie. Dus is er het idee dat er soms wel deeltjes uit het vacu¨um worden gevormd, maar dat ze elkaar direct weer vernietigen (anni-hileren). Als de tijd die dit kost kort genoeg is, schendt dit niet de wet van behoud van energie. Je zou dit nog het best kunnen zien als virtuele (niet echt bestaande) deeltjes, en dat dit ontstaan en annihileren een zogenaamde fluctuatie (schommel-ing) in het vacu¨um is. Hierbij ontstaat een deeltjespaar, namelijk een deeltje en een zogenaamd antideeltje. Een antideeltje is exact hetzelfde deeltje, maar dan met een tegenovergestelde lading. Aangezien plus en min elkaar aantrekken komen ze direct weer bij elkaar om te verdwijnen (zie figuur 1).

Stel nu dat je in het vacu¨um een heel sterk elektrisch veld zou aanleggen tussen twee grote platen met tegengestelde lading. De negatieve plaat zal dan posi-tief geladen deeltjes aantrekken en de positieve plaat zal negaposi-tief geladen deeltjes aantrekken. Volgens de formule van Einstein E = mc2_{, hebben deeltjes met massa}

een bepaalde ’rustenergie’, namelijk mc2_{. Stel nu dat je het elektrisch veld heel}

groot maakt, zodat de kracht die het uitoefent heel groot wordt, ongeveer zo groot als 2mc2_{, dan kan het zo zijn dat er door het elektrisch veld een deeltjespaar uit het}

vacu¨um wordt gecreerd. Doordat de deeltjes in tegenovergestelde richting worden aangetrokken, zullen ze elkaar niet meer uitdoven, maar daadwerkelijk ontstaan. In dit geval wordt de wet van behoud van energie niet geschonden, omdat er energie van het elektrisch veld wordt gebruikt. Met deze constructie is het dus mogelijk energie om te zetten in materie (zie figuur 1).

Julian Schwinger was de natuurkundige die deze mogelijkheid heeft bedacht en uitgewerkt, hij kwam uiteindelijk tot een uitdrukking die de kans op deze deelt-jescreatie per volume van het elektrisch veld per seconde geeft. Dit is een in-gewikkelde afleiding, waarvoor onder andere rekening moet worden gehouden met alle manieren waarop de deeltjes kunnen ontstaan. In de kwantummechanica is er namelijk niet ´e´en manier voor deeltjes om een bepaalde weg af te leggen, maar is

dat een kansverdeling van allerlei verschillende paden.

In deze scriptie worden alle kwantummechanische aspecten die een rol spelen uitgelegd en wordt de gehele berekening gegeven. Tot slot wordt gerekend met de specifieke constructie van twee platen met tegengestelde lading.

t

x

Elektrisch veld

Figure 1: Hier zie je een grafische weergave van het ontstaan van een deeltje (rood) en een antideeltje (blauw). Links is de normale situatie weergegeven van het ontstaan en annihileren van twee deeltjes. Rechts is het ontstaan weergegeven van twee deeltjes in een elektrisch veld, waarbij het positief geladen deeltje wordt aangetrokken door het elektrisch veld en het negatief geladen deeltje wordt afgestoten. Het antideeltje heeft een negatieve energie en wordt daarom volgens afspraak weergegeven alsof het deeltje terug in de tijd beweegt. Dit is echter een virtuele weergave, in de realiteit is dit natuurlijk onmogelijk.

Abstract

In this bachelor project, the Schwinger Effect will be derived via the use of instan-tons; localized maxima of the action of a particle. The Schwinger effect gives the probability of electrion-positron pair production under the influence of a strong electric field. One important concept in this derivation will be the path integral, which accounts for all the quantummechanical paths a particle can take. For that reason the path integral is derived at first. After the expression of the Schwinger Effect is found, it will be numerically solved for an electric field between two large parallel charged plates. The paper concludes with the derivation of an expression for the charge as a function of time. Since the created particles will be absorbed by the source of the electric field, this will lead to the slow discharge of the plates.

1

INTRODUCTION

In this paper the Schwinger effect, first proposed by Julian Schwinger in 1951 [1], will be derived and applied on a specific example. In short, the Schwinger effect is the creation of a particle and anti-particle in vacuum under the influence of a very strong electric field. This is a pure theoretical effect, which presently can not be tproved empirically.

Nowadays it is widely accepted that the minimum energy (on microscales) is not zero and that the energy distrubution is not continuous, but quantized by integer steps of hω. The so called zero-point energy was first proposed by Planck in his “second theory” in 1912 [2]. He assumed all energies between (n − 1)hν and nhν were equally likely, which lead him to the following equation

U = hν ehν/kT _{− 1}+

1

2hν (1)

Clearly, as T → 0 the energy U becomes 1₂hν, which is the zero-point energy. The derivation of this formula was later simplified by Einstein, using A and B coefficients for absorption and spontaneous and stimulated emission [3].

Not only particles have a zero-point energy, fields do as well. A fluctuating zero-point field exists even if no external sources are present. The vacuum field, or vacuum state |vaci, is the ground state of the free field . This state, like all stationairy states of the field, is an eigenstate of the Hamiltonian, but not of the electric and magnetic field operators, which do not have definite values. They fluctuate about their mean values of zero [4].

To illustrate this vacuum state, one can think of photons having a zero state with zero momentum and thus zero energy. When photons are absorbed they jump into this zero state (the vacuum) and when emitted they make a transition from this zero state to a physically observable state. Since the creation of photons in this way has no known limit, there is an infinite number of photons in this zero-state [5]. Consequently the energy of the vacuum state is infinite,P

kλ 1 2~ωk.

One of the effects of this zero-point field in the vacuum is the Casimir Effect [6]. In 1948 Casimir and Polder proved theoretically that two conducting plates without charge will experience an attractive force in vacuum. This effect has been

experimentally verified [7], which implies that the zero-point energy is not merely a mathematical or virtual correction, but that the vacuum field is real, just like the magnetic and electric field are real.

Another way of looking at this zero-point energy is the possibility of creation and direct annihilation of a particle and anti-particle in vacuum. The spontaneous creation of an electron and its counterpart, the positron, can be considered as vacuum fluctuations, which account for its zero-point energy. Although this would seem to violate the principle of conservation of energy, by using Heisenbergs un-certainty relation ∆E∆t ≥ ~

2 one can imagine that a disturbance in the energy is

possible as long as the time interval in which it takes place is very short. Hence, after creation the particle and anti-particle immediately annihilate back into the vacuum.

These vacuum fluctuations are the basis for Hawking radiation, proposed by Stephen Hawking in 1974 [8]. Consider the creation of a particle and antiparticle in the vincinity of a black hole. The energy-time uncertainty dictates that they have to annihilate each other in a certain amount of time. However, if one of the particles is absorbed by the black hole before annihilation, the remaining particle is suddenly real with mass/energy, which is thus radiated into space by the black hole. This means a black hole will lose mass over time and will eventually disap-pear. In some literature these vacuum fluctuations are used as an example that in quantum mechanics energy conservation can be violated for a short amount of time [9], although others argue this is not a legitimate reading of the energy-time uncertainty principle [10].

Precisely this potential creation and annihilation of a particle and anti-particle gives rise to the Schwinger effect. Consider for example a very large electric field between two conducting plates, a certain distance d apart, the force in between these plates can become large enough to create two particles. Since the rest-energy of a particle is mc2, the work of the plates has to equal W = 2mc2, hence the force has to be F = 2mc_d2. In this paper the probability of this pair creation per unit volume per unit time is derived using the method of instantons of quantum tunneling. I follow the proposition by Kim & Page [11] that the instantons of quantum tunneling may be related with the pair production of bosons.

First, the methods used for the derivation of the Schwinger Effect are given. Next these methods will be used to derive the actual Schwinger Effect. The formula is then adapted to get the probability in a specific electric field. Namely the electric field in between two conducting plates. By calculating the exact volume of the space between the plates we can compute the probability of pair creation. Since the particles will each be attracted by the other plate, the plates will slowly discharge. Subsequently we can derive an expression for the charge of both plates as a function of time. This will give us more insight into the actual behaviour of the Schwinger Effect.

2

METHODS

The current derivation of the Schwinger effect makes use of a number of techniques in the semi-classical limit, or in the limit where ~ → 0. In this section I will introduce certain methods and aspects which will make the actual calculation easier to carry out. As mentioned earlier an important aspect of the Swinger effect is the particle-antiparticle creation. In order to create these particles out of the vacuum, a potential barrier has to be surpassed. In the classical limit this would be impossible, because the potential barrier transcends the energy of the particles. However in quantummechanics this creation is possible through quantum tunneling.

To find the Schwinger Effect we have to find the expression of the tunneling probability in a potential that is analogous to the pair-creation in vacuum. On top of that we have to consider all the possible ways in which the particles can be created. Or, to make the analogy complete, the different paths the particles can take to tunnel through the (chosen) potential barrier. To make this calculation slightly more comprehendable we will do this calculation in a wick rotated metric. The specific use and applications of these techniques will be explained in the following section.

2.1

Quantum tunneling

2.1.1 The potential

The analogy of pair creation as tunneling through a potential barrier is best rep-resented by a double well potential. In this case we consider a particle in one of the minima of the potential, having an energy of zero. In quantum mechanics it is possible for the particle to tunnel through the barrier and end up in the other minimum. In this case we consider a double well potential with two minima at −d and d, so that V (x) = (x2_{− d}2₎2_{. An example of this potential is shown in figure}

1.

2.1.2 WKB approximation

In order to solve the tunneling probability within the above double potential well one can use the WKB approximation to obtain the (semi-classical) wave equa-tions in the forbidden area [10]. First we make use of an ansatz for the wave equation ψ(x) = A(x)eφ(x). Putting this equation in the schrodinger equation and rearranging terms gives the expression

A00+ 2ıA0φ0+ ıAφ00− A(φ0)2 = −p

2

~2A (2)

By splitting this equation into a real and imaginary part we can make this problem easier. We are left with two equations

A00− A(φ0)2 = −p

2

~2 and 2A

0

φ0+ Aφ00 = 0 (3) The second equation leads to an expression for A, namely A = √C

φ0, where C is

a real constant. The first expression can be solved by assuming the amplitude of the wave (A) varies slowly, so that A00 is negligible. This leads to the equation

dφ

dx = ±

p

~. Now putting these expressions in our ansatz and remembering that

p(x) =p2m[E − V (x)] leads to the following wave equations ψ(x) = eı Rq2m ~2(E−V (x))dx (E > V ) (4) ψ(x) = e Rq2m ~2(V (x)−E)dx (E < V ) (5)

The latter expression is useful for this research, since this is the wave equation in the region where E < V . Hence, the equation for the particle in the tunneling region. Now comparing this to our current problem, we can imagine this to be the equation for the path of the particle. This is an important comparison that will be clear after the following derivation of the path integral.

2.2

Path Integral

Having found the wave equations in the forbidden area of the double potential well, we also have to consider the possible paths the particles could follow while tunneling through the potential barrier. Some paths are more likely than others and the weight of each path is calculated in a path integral.

2.2.1 Classical Mechanics

In classical mechanics the path of an object, going from a to b, is determined by the minimum of its action (S). The action of a particular object is given by the formula

S = Z tb

ta

Where L( ˙x, x, t) is the Lagrangian of the object. In classical mechanics paths differing from the actual path taken, neutralize each other. This implies δS = 0, which ultimately leaves the classical lagrangian equation of motion:

d dt  dL d ˙x  − dL dx = 0 (7) 2.2.2 Quantum Mechanics

In quantum mechanics however, all possible paths contribute to the amplitude of a particle going from a to b. To illustrate this I will use an example from Feynman and Hibbs [12]. Consider the double slit experiment, where the path of a particle from the source a to a certain point b on the screen behind the slits is a superposition of the two different amplitudes for each slit. So in this case a particle can take two paths, from a through slit 1 to b and from a through slit 2 to b. Now if two more screens, each with a certain amount of slits, are placed between the source and the screen, there are more paths the particle can take. Each path has its own amplitude and the total amplitude of the particle going from the source to for example hole 1 is a sum of all the different amplitudes.

Next, imagine drilling more holes in these screens, creating more and more alternative paths for the particle to follow. Until eventually no screens are left. Now each possible amplitude is determined by an integral over the height of each screen. The following step would be to place an infinite amount of screens between the source and the detector and drill an infinite amount of holes in each screen. So that a path is now just the height as a function of distance, which can be integrated in between the boundaries, leaving a total amplitude as a superposition of all possible paths.

In order to specify this example even further, the time at which the particle passes each point in space can be considered. So that a path (in two dimensions) will be specified by the functions x(t) and y(t). The total amplitude is obtained by integrating this amplitude over all the alternative paths.

This means that, although the probability for a particle to take a path near the minimum action is greater, paths far away still contribute. Therefore the total amplitude is the sum over the contributions φ[q(t)] of each path [12].

K(b, a) = X

over all paths f rom a to b

φ[q(t)] (8)

The notation φ[q(t)] means that φ depends on the function q(t) (i.e. the path of the particle). All these contributions have a phase depending on the action S.

φ[x(t)] = Ae[(ı/~)S[q(t)]], _{A ∈ C} (9) 2.2.3 The Integral

The expression in 8 will become an integral if the amount of paths goes to infinity. This leads to the formula for the path integral. A specific, mathematical derivation of this integral is given in Appendix A.

Z

Dq(t)e ı S[q(t)] ₍₁₀₎

The path integral, which was first mentioned by Dirac in the 1930s, became very useful in the 1970s in quantum field theory [13].

2.3

Semi-classical Limit

2.3.1 wick rotation

Now that we have an expression for the path integral, we will have to apply it to our double well potential. This is easier when using the euclidian path integral. The path integral derived in Appendix A can be turned into an Euclidian path integral by wick rotating the minkowski metric. The minkowski metric is given by

ds2 = xµxµ= (c2dt2− dx2− dy2− dz2)

Where xµ _{= (ct, x, y, z) is the four-vector notation, often used in relativistic}

physics. Since this metric uses space-time dimensions, some problems can become very difficult to solve. Solving these problems in an Euclidian space can simplify the calculation. Transforming this into an euclidian problem is done by using the wick rotation, which transforms the time-component t → ıτ . So that the metric is now

ds2 = (−c2dτ2− dx2 _{− dy}2_{− dz}2₎

This is equal to the four-dimensional inner product. So that our derivation is now an Euclidian problem. After the calculations are carried out, the solution can be transformed back to the minkowski metric, to find the solution in space-time coordinates. The transformation is a 90 degrees counterclockwise rotation in the complex plane. This change of metric will significantly simplify our calculations. 2.3.2 Euclidian path integral

The next step is to wick rotate the path integral from 10 to get the Euclidian path integral. The action S[q(t)] is the integral over the Lagrangian,R Ldt, where L = _2mp2 − V (q). Knowing the impulse can be written as mass times the derivative over q, the transformation t → ıτ leads to

m 2( dq dt) 2_{− V (q) → −}m 2( dq dτ) 2_{− V (q)}

Similairly the interval of integration changes dt → ıdτ , which means the expression in the exponent is not imaginary, so that the path integral is transformed to

Z

[D(q)]e−1~SE[q(τ )]

Due to the wick rotation, the potential changed sign and has been rotated 180 degrees. This means that, instead of tunneling through the potential barrier, the

path now goes from one top to the other. This will make the calculation much easier, but first we will try to find the path integral in the forbidden area. Therefore we put the energy E = 0, so that the particle (in the classical case) would have to be in one of the minima of the potential. This also means the expression within the action can be rewritten. Since E = 0 means m₂(dx_dτ)2 _{− V (x) = 0, so that}

dx dτ =

q

2V (x)

m . Rewriting the Lagrangian gives a new expression for the action

SE = Z mdx dτ r 2V (x) m dτ = Z p 2mV (x)dx (11) Putting this expression for the action in the pathintegral gives an equation similair to the wave equation in the semi-classical area in WKB approximation (equation 5). This equality means we can approximate the path integral as something semi-classical.

2.4

Instantons

Knowing this expression for the path integral we will now look at quantum tun-neling again. We are trying to calculate the tuntun-neling probability of a particle from its initial position to its final position. To find this probability we can use the path integral, since it accounts for all the possible paths the particle can take. In this case the path is given by the amount of changes from one top to the other. Evaluating the integral will give the general tunneling probability, independent of the form of the potential. Since we are working in the semi-classical limit it is possible to use the Euclidian path integral. This means the particles are in a way localized in time on certain positions. In a tunneling situation this positions would for example be xi and xf.

2.4.1 Derivation

Starting with the euclidian path integral.[12] hxf|eHT /~|xii = N

Z

D(x)e−SE/~ ₍₁₂₎

The action is an integral over the Lagrangian from time −T /2 to T /2. Secondly the term D(x) denotes integration over all the possible paths or functions x(t) that obey the boundary conditions x(−T /2) = xi and x(T /2) = xf.[14] We can

generalize this function, because there are multiple paths to be taken. So assuming x(t) is a solution which obeys these boundary conditions, we can formulate a general equation for x(t)

x(t) = x(t) +X

n

cnxn(t) (13)

Where the functions xn(t) are orthonormal and zero at the boundaries, so that all

solutions obey the boundary conditions. This leads to an expression for D(x). The cn’s give the weight of each variation on x(t), so D(x) (the sum of all the paths

obeying the boundary functions) is nothing more than the sum of these cn’s, apart from a constant. D(x) =X n 1 2π~cn (14)

In order to evaluate the path integral over the exponential we will be using the method of steepest descent.[15] We are working in the limit where ~ is small (the semi-classical limit). This means the stationary points of the action contribute the most to the integral, because the most likely paths contribute most to the action. One can imagine there is only a limited amount of paths that make up most of the total action and thus contribute to the equation. Assuming one such stationary point is given by x0, we can conclude the contributing values are in the vincinity

of this x0. Secondly the first derivative of the action has to be zero, since we are

looking at a maximum. Taylor expanding the action and keeping only the lower terms (which is allowed due to the sharp peak in the action of a contributing path), we are left with the following expression

e−S(x)/~ ≈ e−S(x0)/~−S00(x0)/~ ₍₁₅₎

Just like Sydney Coleman we can then choose the xn’s to be eigenfunctions of the

second variational derivative of S at x0, −d

2_xn

dt2 + V 00_(x

0)xn = λnxn. After replacing

the second derivative of the action in the exponential (eq. 15) with the right hand side of the previous equation, we are left with an integral over λnxn. Since the

first term in the exponential is a constant and the second term depends on xn, we

can write

e−S(x0)/~

Z

e−1~R λnxndxn (16)

This equation is solvable for each n. First we integrate over λnxn so that we

have a gaussian integral. We can solve this integral using the standard integral ?? of Appendix C. Which gives an expression for each λ (so that we get a sum over all λ’s, P

nλn). The next step is to put this expression into the original path

integral. Putting in the expression for D(x) as well and noting that the constants in both expressions are neutralized, this leaves the tunneling probability from xi

to xf. Where [1 + O(~)] accounts for the error margin in approximating the action

(we are working in the semi-classical limt) hxf|eHT /~|xii = N e−S(x0)/~

X

n

λ−1/2_{[1 + O(~)]}

= N e−S(x0)/~[det(−∂_t2+ V00(x))]−1/2_{[1 + O(~)]} (17) The second expression in 17 contains the determinant of the second derivative of the action. To see where this comes from, consider a general function ∆ with eigenvalue λ and eigenfunction φ. Next, define φλ(t) to be a solution of this

general function. This means it obeys the boundary conditions, φλ(−T /2) = 0

the eigenfunction has to obey the boundary condition φλn(T /2) = 0. Similairly

det(∆ − λn) = 0 and now putting in the second derivative of the action for ∆ gives

the determinant in 17.

2.4.2 Double well potential

Figure 3: The inverted double well potential −(x2_{− d}2₎2 _{with d = 2.}

The previous expression for the tunneling probability is a general one. We now turn to a specific case, the double well potential (x2 − d2₎2_{, so that ±d are its}

minima. However, due to our wick rotation earlier, the potential is rotated 180 degrees. So we are now looking at two maxima (see figure 3). First of all we put E = 0, so that our particle is located on top of one of the hills of the potential. Now there are two possibilities for the particle, it can either stay fixed on top of the hill, or it can move to the top of the other hill. In the first case the transition probability is given by

h−d|e−HT /~| − di = hd|e−HT /~|di (18) The second case is more interesting. The particle start on top of one hill (at either d or −d) at time −T /2 and ends up on top of the other hill at time T /2. This transition probability is given by

hd|e−HT /~| − di = h−d|e−HT /~|di (19)

Eventually we will take T to infinity. Hence, to find a solution to this transition probability we will focus on the solution in the limit T → ∞. So that the particle is located at one of the tops at T minus or plus infinity.

We have set E = 0, so that the particle starts out at one of the tops of the inverted potential. This means m₂ dx_dτ2− V (x) = 0, taking natural units m = 1 the expression becomes dx_dτ =p2V (x). Integrating both sides gives

t = t0+

Z x

0

Here t0 is the constant for which x = 0, or the time at which the particle is

located on the top of the hill. The graph for this expression is in the form of an hyperbolic tangent with maxima −d and d.[14] This solution is called an instanton with center t0. An anti-instanton moves in the opposite direction and is found by

replacing t for −t. For large T however the solution does not just consist of one instanton or anti-instanton, but can consist of multiple (anti-)instantons, which can be thought of as going from d to −d and back a number of times. All of these instantons are centered at certain points in time, tn. So that for n (anti-instantons)

the centers of the positions are given by

T /2 > t1 > t2... > tn> −T /2 (21)

Integrating over al these intervals gives the weight of each (anti-)instanton, with Tn_{/n! as a solution.}

Now in order to calculate the complete transition probability we have to eval-uate all the components of equation 17. The total action for n widely seperated objects (since T is very large, the distance between a certain ti and ti+1 is also

relatively large) is n times the action of one such an object, S0.

The next step is to look at the determinant from equation 17. We know it has eigenvalues λn. Following the proof in appendix B we obtain the determinant for

a single well potential w

π~

1₂

e−ωT /2. However we are now working with a double well potential. So the intervals of the instantons and anti-instantons influence this formula. Since these intervals are small (tn << T ), we can correct the formula

with a factor Kn_.

[det(−∂_t2+ V00(x))]−1/2= ω π~

1/2

e−ωT /2Kn (22)

Now we can put all these values into the expression for the transition probability (equation 17). If the particle stays fixed on one of the hills (for example at −d), only the even paths count, so the expression becomes a sum over all even n. If the particle ends up on top of the other hill, only the odd paths count and so the expression becomes a sum over all odd n. Since we are dealing with a sum of the formPxn

n!, these expression are respectively a hyperbolic cosine and a hypberbolic

sine. So that the general expression becomes h±d|e−HT /~| − di =w π~ 1/2 e−ωT /21 2[e Ke−S0/~T _{∓ e}−Ke−S0/~T_{] (23)}

This is the expression for the tunneling probability for a particle in a double well potential. In the next section we have to find out how this coincedes with the Schwinger effect.

2.5

Summary

To recapitulate, first we derived the path integral in order to account for all the possible paths taken by a particle in a potential. Afterwards we applied the wick rotation t → ıτ to change the metric to a four-dimensial euclidian space. This rotation was also applied to the path integral, changing the action and thus

the expression within the integral. This integral was similair to the wave equa-tions of the WKB approximation, which meant the path integral could be seen as semi-classical. This simplified the calculation of the tunneling probability, which followed from the method of steepest descent and applying the expression to a (wick-rotated) double well potential. Ultimately leading to the expression for the tunneling probability in a double well potential.

3

THE SCHWINGER EFFECT

Now that we have established all necessary techniques, we can start to derive the actual Schwinger Effect using the method of instantons.

3.1

Paths of charged particles

3.1.1 Relativistic particle in an electric field

In a strong electric field a positively charged particle will move in the direction of the electric field and a negatively charged particle will move in the opposite direc-tion of the electric field. So when a particle and antiparticle are in the same electric field they will move in opposite directions. This is the basis for the Schwinger Ef-fect, where a particle and antiparticle are created due to the strong electric field and will be moving away from each other afterwards. So that they will not anni-hilate each other. To find the exact displacement of each particle we will need to derive its equation of motion. Since the particle and anti-particle are equal, except for charge, the equations of motion are similar apart from the sign. Calculating the equation of motion of one particle will be enough. Since the particles move relativistically we have to start with the relativistic impulse.

p = F t = γmv (24)

Since γ depends on v as well, rewriting this equation leaves us with v = F t

m q

1 + _mcF t
2

(25) Integrating both sides will give the equation of motion of the particle

x(t) = mc 2 F   s 1 + F t mc 2 − 1   (26)

Obviously this is a hyperbola, so the movement of the particle is called hyperbolic movement. In case of the anti-particle a minus sign comes in front of the expression. Both these paths are shown in figure 4. The question is how these paths relate to the Schwinger Effect. Consider a particle in a strong electric field. It moves along the dotted line in figure 4. Suppose at a certain point this particle encounters a barrier of width d and tunnels through this barrier. Imagine this barrier to be wide enough, so that tunneling through it costs as much energy as creating two particles. In that case calculating the tunneling probability of this particle will be the same as calculating the particle creation in an electric field. The minimum distance d should be

Z

F dx = 2mc2 (27)

Since the electric force is given by ~Fe = q ~E, the strenght of the electric field should

t

x

Figure 4: The paths of the particle and anti-particle in a strong electric field. 3.1.2 Parameterization

Looking at the creation and direct annihilation of a particle and an anti-particle, we can imagine the anti-particle as travelling ’back in time’. So that we can approximate the pair-creation by a small spherical loop. In order to work with this concept, we first need to find the equations of motion of the two particles. Starting with the electric force and aligning our set-up so that only the field lines along the x-axis contribute, we know

dpx

dt = qEx (28)

However, we are looking at relativistic particles, so we want to look at the relativistic momentum in the eigentime. Since the relativistic momentum is given by p = γmv and γ can be written as γ = _dτdt and the velocity is of course the time derivative of position, we get px = m_dτdtdx_dt. Next we want to take the derivative of

the momentum to the eigentime, so that it equals the second derivative of position to the eigentime times the mass of the particle. If we now look at equation 28 again, we can also write the derivative of the momentum to the eigentime as

dpx dτ = dt dτ dpx dt = qEx dt dτ (29)

Putting the equations for dpx_dτ together and noting that x and t can be written in 4-vector notation, we come across the first equation of motion similair to the one found in the article of Dunne.[16]

d2_x 1 dτ2 = − qEx mc dx0 dτ (30)

In this derivation we looked at the electric force and its influence on the path of the particle. But we can also look at the situation from a different angle by looking at the energy of the particle, which is given by γmc2. If we differentiate both sides to the eigentime we will get

mc2dγ dτ =

dU

dτ (31)

Noticing that the eigentime derivative of γ is just the second eigentime-derivative of time t changes the left-hand side. The particle gets its energy from the work done by the electric field. So that dU_dτ = _dτdtdU_dt = _dτdtqExdx_dt. Filling in these

expres-sions into equation 31 and again turning to the 4-vector notation gives the second equation of motion for the charged particle in the electric field.

d2_x 0 dτ2 = − qEx mc dx1 dτ (32)

It is important to see where these two equations came from. One was derived using the force of the electric field and the momentum or velocity the particle gains due to that force. The second equation came from the energy that the particle obtains as it is created and accelerated along the field lines of the electric field. Together these equations represent the motion of the particle. They obviously represent two hyperbolic functions and thus hyperbolic movement. The boundary conditions at τ = 0 are t = 0 and x should equal the energy needed to create one particle in vacuum. Using equation 27 this last condition leads to x = mc_qEx2, so that

x₀ x1  = mc 2 qEx  ± sinhqExτ mc ∓ coshqExτ_mc  (33) Before we can use these expressions to find the tunneling effect, we have to turn back to Euclidian space. First writing the hyperbolic functions as a normal sine and cosine and then applying the familiar rotation τ → ıτ to neutralize the ı’s transforms these equations of motion to

x₀ x1  = mc 2 qEx  ± sinqExτ mc ± cosqExτ_mc  (34) Apparently in Euclidian space the motion of the particles is circular. This is our virtual approximation of the tunneling motion in the Schwinger effect. Again thinking of the anti-particle travelling back in time the creation of the two particles can be considered as a circular loop. Due to the strong electric field, the loop does not close and the creation is half a circular loop of which the movement is given by equation 34. For a visual representation of this effect, see figure 5.

x t

Figure 5: A visual representation of the creation of a particle and anti-particle due to a strong electric field.

3.1.3 The action

Now that we have found the equations of motion of a charged particle in an electric field we can start to evaluate the action. Since we are dealing with relativistic particles within an electric field, the Lagrangian is not simply T - V . On top of that the wick rotation has rotated our composition with 90 degrees, but the electric field is still aligned along the x-axis. So the proper representation of this situation would be a field perpendicular to the electric field. This is clearly a magnetic field

~

B along the z-axis. In order to find the action of the path of the particle, we need the lagrangian for a relativistic particle in a magnetic field. Since the potential of the magnetic field is ~A, where ~B = ∇ × ~A. And the action of a free relativistic particle is mc times the eigenlenght of the worldline of the particle (or mc2 times the eigentime along the worldine of the particle), the total action is the integral over the following lagrangian

S = mc2 Z τ1 τ0 dt γ + q I ~ A · ˙~xdt (35) The first integral gives the eigentime τ = τ1− τ0. In this case the eigentime is the

period of the equations of motion (the sine and cosine) of the particle. So the first part of the action is πm_qEx2c3.

Using the fact that ˙~xdt = dx and applying Stokes’ theorem, the second integral can be rewritten as a surface integral over the rotation of the vector field ~A. This

is just the magnetic field ~Bz, so that d~a is the surface of the circle of the path.

Using ~Bz = ~ Ex

c , this leaves us with

q Z ~ Bz· d~a = qπ Ex c R 2 (36) Using equation 34 to get the radius of the motion, we find for the second part of the action πm2c3

qEx . Putting the two expressions together gives the total action of the

movement. However, we have now calculated the action over a whole circle while we are looking at the path of a semicircle. So we need to put a factor 1₂ in front. Next we have to remember that the movement along the semicircle can go back and forth. Since we are only interested in the tunneling effects where particles are actually created, we need to consider movements an odd amount of times along the semicircle. So that the motion ends at the other end of the semicircle. Taking all this together the final expression for the action becomes

S = πm

2_c3

qEx

(2n + 1) (37)

3.1.4 Perpendicular movement

Before we use this expression for the action to find the Schwinger effect, we have to consider the possibility that the created particles will not only move parallel to the electric field lines. It is possible for the particles to gain velocity in a different direction upon creation. That is, the transverse momenta ki (i = y, z) might not

be zero as we have assumed so far. In order to include this possibility, we have to introduce the transverse momenta into the equation. The momenta perpendicular to the field lines of the electric field were encapsulated into the mass. Yet we now want to allow for the perpendicular momenta to be greater than zero. In that case we have to split up the total momentum, noting that momentum can be given by mc and that we have a factor c3 in our expression for the action, we can write

m2c3 → (m2 ec

2_{+ ~p}

⊥2)c = (m2_ec2+ (~ki)2)c where i = y, z (38)

3.2

Probability

The next step is to find the actual probability of particle creation. Looking at the calculation of the tunneling probability in the double well potential, we see that we also have to sum over all the locations of the instantons and anti-instantons. In order to wheigh their contributions. These locations are the centers of each instanton. Since the path of one instanton is a semicircle, we need to sum over the centers of each semicircle. For n instantons this summation becomes

∞

X

n=1

1

π2_{(2n + 1)} (39)

Putting these values together, we are very close to the result Schwinger obtained in 1951.[1] We only have to include an expression in front, _8π~qE, and note that we

have to take the integral over the transverse momenta. It is important to keep the dimension sound. We are trying to find the probability of pair creation per unit time per unit volume, so we are looking for dimension _[m]13_[s]. With the integral

over ki our probability becomes

P = qE ~ Z d2_k 8π3 ∞ X n=0 1 (2n + 1)exp  −π(m 2_c2_{+ (~k} i)2)c ~qE (2n + 1)  (40) This is in accordance with other literature. [17, 18] Dimensions are sound; the expression in the exponential is dimensionless as we would expect and our probability is indeed given per unit time per unit volume. Solving the integral over k2 _{gives a factor} qE

~c(2n+1) in front. This leads to the actual Schwinger effect,

the probability a pair-antipair is created in a strong electric field per unit time per unit volume P = (qE) 2 ~2c 1 8π3 ∞ X n=0 1 (2n + 1)2 exp  −πm 2_c3 ~qE (2n + 1)  (41)

3.3

Summary

We started this section with the tunneling probability for a particle in a double well potential with the use of instantons. We then considered the path of a charged, relativistic particle to find the minimum strenght for the electric field to create a particle pair, E = 2mc_qd2. Next we found the equations of motion of the particle in an electric field. These two equations were derived on the one hand from the force of the elctric field and the momentum the particle gains due to that force and on the other hand from the energy the particles obtain as they are created and accelerated along the field lines of the electric field.

With those equations we were able to establish the action and thus the path integral of the particle creation. Considering the transverse momenta lead us to the final expression of the Schwinger effect. A rather elegant solution to our problem that depends on only one variable, the strenght of the electric field. In order to make this effect more tangible I will now apply the Schwinger Effect to an actual physical situation.

4

TWO CONDUCTING PLATES

An electric field can be created in a number of situations as long as there is some charged source. Here we consider a relatively easy situation; an electric field be-tween two large, thin, parallel plates with opposite surface charge (±σ).Assuming the thickness of the plates is negligible in comparison to its surface and arrang-ing our set-up so that the surface of the plates is much larger than the distance between the two plates (A >> d), the electric field will be easily derived using Gauss’s law.

4.1

Numerical example

We align our set-up so that the surface of the plates is perpendicular to the x-axis and the plates are located at x = −d₂ and x = d₂. Using Gauss’s law and the symmetry of two oppositely charged plates gives us the electric field

~ E = Ex= σ 0 ˆ x (−d 2 < x < d 2) (42) ~ E = Ex= 0 (|x| > d 2) (43)

Since we know the value of 0, the strenght of the electric field only depends on

the surface charge σ = Q_A.

Our next step is to fill out the known values into equation 41. We have an expression for the electric field and we know the values of q, 0, ~, c and me.

Putting in these values in SI-units, leaves P = 3.515 × 1030_σ2 ∞ X n=0 1 2n + 1e −3.683×107 σ (2n+1) (44)

Recall that this is the probability of pair creation per unit volume per unit time. The volume of our construction is given by V = A × d and time is of course measured in seconds. In order to find a first numerical value of the surface charge of the plates, we could assume the surface area to be A = 100 m2_{, that the distance}

between the plates is d = 0.1 m and that we are looking at the configuration for 1 hour or t = 3600 s. Since terms with larger n are small compared to n = 0, we will only look at n = 0. Let’s say we want to have a probability of 1 percent, in that case

0.01 = 1.265 × 1035σ2e−3.683×107σ ⇒ σ = 332227 C m−2 (45)

In other words, in this approximation the charge over the plates would have to be Q = 3.32×107 C to have a 1 percent chance of creating a pair of particles within one hour. In table 1 different probabilities and charges for different time intervals are given. Increasing the surface of the two plates would increase the volume of our set-up, but simultaneously the surface charge is decreased quadratically, which leads to a smalle probability. So in the table the surface is kept constant at 100 m2 as is the distance between the two plates (d = 0.1 m).

Probability (P) Charge (in C) Time Surface (in m2₎ 0.01 3.32 × 107 C 1 hr 100 m2 0.01 3.07 × 107 _{C 1 yr 100 m}2 0.01 2.63 × 107 _{1 Gyr 100 m}2 1 × 10−10 2.86 × 107 C 1 hr 100 m2 1 × 10−10 2.67 × 107 _{C 1 yr 100 m}2 1 × 10−10 2.32 × 107 _{C 1 Gyr 100 m}2 2.94 × 10−1599457 1000 C 1 Gyr 100 m2

Table 1: Probabilities of pair creation for different charges in different time inter-vals

Changing the time interval from 1 hour to 1 year or even 1 billion years only lowers the required charge on the plates minimally. A normal charge for the plates of 1000 C during 1 billion years leads to a probability of pair creation of practically zero. This confirms the difficulty of empirically testing the Schwinger effect.

4.2

Discharge rate of the two plates

As soon as a particle pair is created within the electric field of the two conducting plates, the particles will be accelerated in opposite directions. The positive particle will be attracted by the negative plate and vice versa for the negative particle. Both plates will inevitably absorb the particles, causing a small discharge on the plates. By integrating this discharge rate, one can find an expression for the charge of both plates as a function of time. Naturally, after a certain amount of time the charge of the plates will have reached a treshold value for which the chance of particle creation is practically nihil.

When using this discharge rate, it is important to take into account that the probability of pair creation depends on the elecric field (i.e. the charge of the two plates), while the charge of the two plates at a certain time depends on the probability of pair creation at the same time.

4.2.1 Approximation

Here I will derive an approximation of the discharge rate dQ_dt, by only considering the term n = 0. Again this term dominates terms with larger n, so this is al-lowed. The rate is given by the amount of particles absorbed per second (i.e. the probability of pair creation per second times the charge of a particle), so that

dQ

dt = −q(P × V ) (46)

Here we have multiplied the Schwinger Effect with the volume of our set-up. In order to find a high probability for the Schwinger effect, the electric field has to be very large. We can approximate the Schwinger Effect to be P ≈

(qE)2 ~2c 1 8π3exp h −πm2_c3 ~qE i

, this leads to the following approximation for the discharge rate (where the volume is given by A × d, with A the surface of the plates and d the distance between the plates)

dQ dt = − q(qE)2 ~2c 1 8π3exp  −πm 2_c3 ~qE  × dA = (47)

Next we can fill in the expression for the electric field between the two plates and gather similair terms at either side, afterwards substituting πm2c30A

~qQ for x and

taking the integral on both sides (we start at some time t0, where Q0 = Q(t0))

gives − ~q πm2_c3_A 0 Z Q0=Q(t) Q0_=Q0 dxex = − q 3 ~2cA220 1 8π3dA Z t t0 dt (48)

Integrating both sides leads to the following expression

− ~q πm2_c3_A 0  exp  −πm 2_c3_A 0 ~qQ(t)  − exp  −πm 2_c3_A 0 ~qQ(t)  = − q 3 ~2cA20 1 8π3d(t − t0) (49) By rearranging terms and taking the logarithm on both sides, we can get an expression for the charge of the plates Q(t). It is not entirely accurate, because we have neglected the higher order terms for n, but it is very close.

Q(t) = πm 2_c3_A 0 q~(ln  eπm2c3A0~qQ0 +q2m2c2d ~38π20(t − t0)  ) (50)

As can be seen from this expression, at t0, the charge will be Q0 as expected. We

could pull the term Q0 out of the logarithm, to show this more directly. Also

setting t0 = 0 for clarity, leads to

Q(t) = Q0 1 + ln  1 + q2m2c2d ~38π20e −πm2c3A0 ~qQ0 t  (51)

This shows that it is impossible to keep the probability of creation constant in this specific configuration. Obviously the particles will always be attracted by the charged source causing the electric field, so this discharge of the source will happen in most configurations. Unless the source is somehow protected from the absorption of the created particles.

As a consequence in general no set-up will be fully stable when the electric field is large enough to cause the Schwinger Effect.

5

DISCUSSION

The method of instantons used in this paper to derive the Schwinger Effect has been proved effective. The results agree with the expression found by Schwinger himself [1]. The factor qE

8π~ has not been fully derived but stems from a similar

integral as the one over the transverse momenta.

The numerical example is of course an approximation, but is fairly accurate because the n = 0 term dominates the expression. This gave some insight in the charge needed for an electric field to stimulate the pair creation. As expected these values were very high and at a more regular charge of 1000 C, the probability of pair creation was near zero. Secondly, the set-up of two large charged plates in vacuum is not very realistic. It would be hard to create or simulate such a situation and keep it stable for a long enough time to make the probability significant.

The last expression, of the charge as a function of time, gives more insight into the discharging of both plates and is quite accurate as well, despite the use of just the n = 0 term. It touches on Hawking radiation, where a black hole absorbs particles created in vacuum due to its gravitational strenghts. This also leads to the slow decay or evaporation of the black hole. However, similar to the Schwinger Effect, this decay takes a huge amount of time, because of the small charge of the created particles.

All in all, the results correspond to theory and confirm the difficulty of exper-imentally testing the Schwinger Effect.

6

CONCLUSION

The structure of this paper has been an introduction of the methods associated with the Schwinger effect. After explanation of these concepts the actual derivation of the Schwinger effect. Finally the resulting expression was used to give some numerical examples and an equation for the charge of the source of the electric field as a function of time.

The main concepts used in this derivation were the path integral and the method of instantons. The path integral is a very elegant technique for quantum-mechanical situations, because it actually considers the infinite amount of paths and the fact that certain paths are more likely and thus have higher contributions in the integral. Apart from that, the instanton method has made the calcula-tion simpler by looking at the problem in euclidian space and approximating the particles as ’instantons’, critical points of the action of the situation.

After treating these methods we started looking at a specific situation, the pair-creation in a strong electric field. Determining the behaviour of relativistic particles in electric fields and the euclidian action we found the expression for the Schwinger Effect. This expression corresponded with the one found in other literature and the expression Schwinger himself had derived. This confirms the fact that the use of instantons is a sound method for deriving the Schwinger Effect.

Furthermore we considered the evolution in time of a situation that satisfied the conditions of the Schwinger Effect. Since created particles will inevitably discharge the source of the electric field, this touches the idea of Hawking Radiation of a black hole, where a particle is absorbed by the gravitational strenght of the black hole. The final expression for the charge as a function of time is a small approximation, because again only the term n = 0 is considered. Nevertheless this is by far the most contributing factor, so our result is quite accurate.

Despite the fact that Julian Schwinger’s paper has been published long ago, it has been one of the most influential papers in physics, there are 2441 citations in SPIRES, which only dates back to 1975. It has not been tested experimentally yet, but an experimental set-up with the use of graphene has been proposed and one using optical lasers [19, 20]. Empirical prove of the effect would be a great support for different fields in particle physics and quantum field theory.

Seeing the derivation of the Schwinger Effect is a long calculation, that makes use of a lot of different aspects, this paper was mostly intended to give an extensive calculation. By extensively discussing the methods needed to derive the expression and by giving a specific example, I hope it can serve as a clarification for physics bachelor students.

Acknowledgements

I would like to thank my supervisor Prof. dr. Erik Verlinde for the realization of this paper. His helpful remarks and comments on this subject, both in our meetings and through our e-mail correspondence, have made the difficult aspects more clear to me.

Appendix

A

Derivation Path Integral

Following the steps of an assignment used in a third years workshop in theoretical physics at the University of Amsterdam it is possible to derive the path integral.

In quantum mechanics we are interested in the solutions of the Schodinger equation

ı∂

∂t|ψ(t)i = ˆH|ψ(t)i

Since |ψ(t)i are eigenvalues of the hamiltonian, the expression for these eigenvalues is

|ψ(t)i = e−~ı ˆ

Ht_|ψ(0)i

For every time t we have ψ(q, t) = hq|ψ(t)i, so that ψ(q, t) = hq|e−~ıHtˆ |ψ(0)i

Now using the completeness of the states |qi we get ψ(q, t) =

Z

dq0hq|e−ı~Htˆ |q0iψ(q0, 0)

We want to demonstrate that the transition amplitude on the right side of this equation can be rewritten as a path integral over paths q(t) from q0 = q(0) to q = q(T ).

hq|e−~ıHtˆ |q0i =

Z

Dq(t)e~ıS[q(t)] (A.1)

This derivation makes use of a seperation of the path into N steps with time-intervals of ∆t, where

N ∆t = T Eventually we will send N → ∞ and thus ∆t → 0.

The left side of A.1 can be rewritten. Noting that the formula in the exponential can be written as −ı

~HN ∆t we get

hqN|e−N ı

~H∆tˆ |q₀i

Furthermore we can again use the completeness of the states |qi, for example multiplying with R dq |qN −1ihqN −1| gives

Z dqN −1hqN|e− ı ~ ˆ H∆t_|q N −1ihqN −1|e−(N −1) ı ~ ˆ H∆t_|q 0i

N −1 X i=1 Z dqihqN|e− ı ~H∆tˆ |q_{N −1}ihq_{N −1}|e− ı ~H∆tˆ |q_{N −2}i . . . hq₁|e− ı ~H∆tˆ |q₀i (A.2)

Next we make use of the completeness of the momentum-states |pi, implementing this in a particular factor gives

hqi−1|e− ı ~H∆tˆ |q_ii = Z dp1hqi−1|piihpi|e− ı ~H∆tˆ |q_ii

Now, knowing that ˆH = _2mpˆ + V (q, t), we can write hpi| ˆH|qii as

hpi| ˆ p 2m + V (q, t)|qii = hpi| ˆ pi 2m|qiihpi|V (q, t)|qii

And since hp|ˆp|qi = php|qi and hp|V (q, t)|qi = V (q, t)hp|qi we are thus left with hpi| ˆH|qii = H(pi, qi)hpi|qii

By Taylor expanding the exponential in the bra-ket, discarding powers n  1 (since ∆t → 0) and applying the above equation we get

hpi|e− ı ~H∆tˆ |q_ii = hp_i|1 − ı ~ ˆ H∆t|qii = hpi|qii(1 − ı ~H(pi, qi)∆t) = e −ı ~H(pi,qi)∆thp_i|q_ii

From the given rules of the bra-ket notation, we have hp|qi = e−ıpq and thus hqi−1|piihpi|qii = eı piqi−1e−ı piqi = eı pi(qi−qi−1)

Using these equalites and putting them back into the integral we are left with

N X i=1 Z dqidpiepi(qi−qi−1)− ı ~H(pi,qi)∆t (A.3)

In the limit N → ∞ the factor qi − qi−1 becomes infinitely small, in other words

dq. Clearly ∆t(= ti− ti−1) becomes dt. The factors within the hamiltonian, qi and

pi, are in the sum from 1 to N the total values of q and p. Finally the integrands

dqi and dpi reduce by definition to Dq and Dp, so that

Z

Dq(t)Dp(t)e~ıR (p ˙q−H(p,q)) dt (A.4)

Notice that the expression in the exponent is equal to the first order lagrangian L(p, q, ˙q) = p ˙q − H(p, q)

If we assume the hamiltonian is of the form H(p, q) = _2mp2 + V (q), the integral can be written as Z Dq(t)Dp(t)e~ıR (p ˙q)dte− ı ~ R p2 2mdte− ı ~R V (q)dt (A.5)

Now taking the integral over p gives a standard integral (expression C.1) Z Dp(t)e~ıR pdqe− ı ~ R p2 2mdt

Solving this integral and using the fact that _2m1 (dq_dt)2_{− V (q) = L(q, ˙q) leads to the}

final expression of the path integral. Z

Dq(t)e~ıR L(q, ˙q)dt (A.6)

This path integral accounts for all quantum mechanical paths a particle can take from A to B and is thus instrumental in deriving the Schwinger Effect.

B

The Determinant (following the proof of

Cole-man [14])

The general expression for the transition probability in a potential well consists of a determinant over the second derivative of the action.

det[−∂t+ V00(x)] (B.1)

Where λn are the eigenvalues of the second derivative of the action. These

eigen-values correspond with certain eigenfunctions that obey the boundary conditions. So at T/2 this eigenfunction is zero, φλ(T /2) = 0. Similairly the determinant of

the second derivative of the action minus the corresponding eigenvalue is zero. So we can write

det[−∂t+ V00(x) − λ] = φλ(T /2) (B.2)

Next we can define a quantity N by

det[−∂t+ V00(x)]

φ0(T /2) = π~N

2 _(B.3)

Rewriting this equation gives

N det[−∂t+ V00(x)]−1/2 = [π~φ0(T /2)]−1/2 (B.4)

If we set the derivative of the potential to be V00(x) = ω2. We can find an expression for the eigenfunction φ0(t).

φ0 = ω−1sinh ω(t + T /2) (B.5)

After putting this expression into equation C.4 we are left with N det[−∂t+ V00(x)]−1/2 =

 ω π~

1/2

[sinh(ωT )]−1/2 (B.6) Or, since e−ωT goes to zero for large T:

N det[−∂t+ V00(x)]−1/2 =

 ω π~

1/2

C

Standard Integrals

References

[1] J. Schwinger - Phys. Rev. 82 (5), 664-679 (1951). [2] M. Planck - Ann. d. Phys. 37, 642-656 (1912). [3] A. Einstein - Phys. Zs. 18, 121-128 (1917).

[4] P. W. Milonni - The Quantum Vacuum, An Introduction to Quantum Elec-trodynamics. San Diego: Academic Press, 1994, p. 525.

[5] P. A. M. Dirac - Proc. Roy. Soc. Lond. A114, 243-265 (1927).

[6] H. B. G. Casimir - Kon. Ned. Akad. Wetensch. Proc. 51, 793 (1948). [7] M. J. Sparnaay - Phys. 24, 751-764 (1958).

[8] S. W. Hawking - Nature 248, 30-31 (1974).

[9] Martin, B. R., Shaw, G., Particle Physics 3rd ed. Chichester: John Wiley & Sons Ltd. 2008. p. 442.

[10] Griffiths, D. J., Introduction to Quantum Mechanics 2nd ed. New Jersey: Pearson. 2005. p. 428.

[11] S. P. Kim, D. N. Page - Phys. Rev. D 65, 65-73 (2002).

[12] Feynman, R. P., Hibbs, AR., Qantum Mechanics and Path Integrals. New York: McGraw-Hill, Inc. 1965. p. 365.

[13] Rivers, R. J., Path integral methods in quantum field theory. Cambridge: University Press, 1987. p. 339.

[14] Coleman, S., Aspects of Symmetry. Cambridge: University Press, 1985. p. 402.

[15] A. L. Koshkarov - Theor. Math. Phys. 102, 153-157 (1995). [16] G. V. Dunne - Eur. Phys. J. 55, 327-340 (2009).

[17] T. Friedman & H. Verlinde - Phys. Rev. D 71, 064018 1-15 (2005).

[18] R. Brout, R. Parentani & Ph. Spindel - Nucl. Phys B 353, 209-236 (1992). [19] D. Allor, T. D. Cohen & D. A. McGady - Phys. Rev. D 78, 096009 1-5 (2008). [20] D.B. Blaschke et al. - Eur. Phys. J. D 55, 341-358 (2009).