Perturbation Field Theory Methods for Calculating Expectation Values

Perturbation ﬁeld theory methods for calculating expectation values

QºK.AK. YÔg@ hAÖÞ African Institute for Mathematical Sciences (AIMS)

Supervised by: Dr. W. A. Horowitz University of Cape Town, South Africa

19 May 2016 Submitted in partial fulﬁllment of a structured masters degree at AIMS South Africa Abstract

We use a new technique to tackle the problem of studying strong interactions, which is the holographic principle. We find an equivalent description for the quantization of the Klien-Gorden scalar field and Green’s function in (n + 1) dimensions. Furthermore, we study the cross section formula in (n + 1) dimensions. Finally, we explore the Gell-Mann-Low theorem and Keldysh formalism as two different methods for calculating the expectation value.

Declaration

I, the undersigned, hereby declare that the work contained in this research project is my original work, and that any work done by others or by myself previously has been acknowledged and referenced accordingly.

Samah Ahmed Mohamed AHMED, 19 May 2016

i Contents

Abstract i

1 Introduction 1

2 The Klien-Gordon field3 2.1 The Lagrangian field theory...... 3 2.2 Quantization of the free scalar field...... 4 2.3 The Klein-Gordon field in space-time...... 5 2.4 The Interacting Field Theory...... 6

3 Interacting Fields 12 3.1 The cross section and the S-Matrix...... 12 3.2 The Gell-Mann-Low theorem...... 17 3.3 The Keldysh formalism...... 19

4 Conclusion 22

References 24

ii 1. Introduction

Unlike what is expected from an energetic collision at Brookhaven’s Relativistic Heavy Ion Collider (RHIC), experimenters found a strongly coupled liquid with almost no viscosity, instead of finding a very weakly coupled gas of quarks and gluons(Jacak and Steinberg, 2010). That liquid they found is a new phase of matter formed when the quark-gluon plasma is strongly coupled under high enough temperatures, so quarks and gluons are no longer bound to hadrons. Such a liquid has a unique property η that the ratio of its shear viscosity to its entropy density, s , is nearly zero (Jacak and Steinberg, 2010); making it a nearly perfect fluid with conditions that haven’t been available since the very early age of the universe. The study of Quark-Gluon Plasmas (QGP) is based on an understanding of the interactions between the quarks and the gluons. The main theory that is devoted to studying these interactions is Quantum Chro- modynamics (QCD). QCD describes strong interactions with two properties: The first is confinement, that states that quarks can still interact when they are separated by a distance, and it is demonstrated in lattice QCD. Lattice QCD states the ratio between the temperature and the energy density, /T 4, controls the degrees of freedom of a thermal system. Figure 1.1 shows how this ratio changes with 12 temperature to jump near a critical degree Tc about 170 MeV(2×10 K). This jump indicates the existence of two different phases: a law-temperature phase of hadrons and a high-temperature QGP phase(Jacak and Steinberg, 2010). The second property is the asymptotic freedom that predicts the existence of the strongly coupled QGP, which can’t be described by perturbative QCD. Therefore, a non-perturbative QCD is needed to study the QGP(Johnson and Steinberg, 2010).

Figure 1.1: The dependence of temprature to the ratio /T 4, as calculated from QCD by lattice- approximation techniques(Jacak and Steinberg, 2010)

A conjecture has been formulated by the string theory community to solve such a problem. This conjecture comes from the fact that string theory is able to describe the strong interactions in terms of some properties of black holes in a higher dimensional space-time(Johnson and Steinberg, 2010). Also, a new technique has been presented to tackle this problem which is the holographic principle. This principle states that any theory of quantum gravity in a (n + 1) dimensional space-time has an

1 Page 2 equivalent description in terms of a theory that can roughly be thought of as living on the spacetime’s n dimensional boundary (Johnson and Steinberg, 2010). The best understood example of holography is the so-called AdS/CFT (anti–de Sitter/conformalfield theory) correspon-dence. The analogue of the Minkowski spacetime is the negative cosmological constant Anti-de Sitter ADS, that its n + 1 dimensional spactimes have a volume that has a radial slice of a n dimensional Minkowski spacetimes. The holographic duality works for fields because they have a well defined expectation values at asymptotic infinity(?). In this present work we will discuss some theories to compute the expectation value of a propagato. As such we will start with with a review for the Klien-Gordon field in (n+1) dimensions in chapter 2. In chapter 3 we then drive the S-matrix formula in (n + 1) dimensions. We shall introduce the Gell-Mann- Law theorem and the Keldysh formalism that uses the Schwinger-Keldysh closed time contour. 2. The Klien-Gordon field

In this chapter we review the classical ﬁeld theory that uses a quadratic Lagrangian with an action that is invariant under the Lorentz transformation. This will help us to build a classical, and linear, equation of motion by using the Euler-Lagrangian equations. We also introduce the Klein Gordon ﬁeld and discuss second quantization. We then compute the correlation functions (Greens functions) from the interaction theory.

2.1 The Lagrangian ﬁeld theory

The Lagrangian formalism is a fundamental approach in classical mechanics in which the action, S, can be expressed as the time integral of the Lagrangian, L. Also, the Lagrangian L can be written as the spatial integral of the Lagrangian density, L (Peskin and Schroeder, 1995):

Z Z S = Ldt = d(n+1)xL. (2.1.1) where n + 1 is the number of space-time dimensions. The Lagrangians encountered in the field theory depend only on fields and their derivatives L = L(φ, ∂µφ). We study fields that are locally evaluated at a certain space-time point x. We apply Hamilton’s principle, that the classical fields configuration extremizes the action on Equation (2.1.1). by varying the action S with respect to the fields (φ) and their derivatives (∂µφ)

δS = 0, Z (n+1) ∂L ∂L = d x δφ + δ(∂µφ) . (2.1.2) ∂φ ∂(∂µφ) We know that we can’t change the ﬁelds and their derivatives independently, hence we state:

δ(∂µφ) = ∂µ(δφ), (2.1.3) and by integrating the second term of equation (2.1.2) by parts, one ﬁnds:

Z (n+1) ∂L ∂L δS = d − ∂µ δφ (2.1.4) ∂φ ∂(∂µφ)

We require the variation of the ﬁeld to vanish δφ = 0, at the temporal beginning and end of a (n + 1)- dimensional space-time region that is, the surface term is zero. Since the integration is required to be zero and hence the variation of the action depends on some term multiplied by ∂φ, this the quantity that multiplies δφ must vanish for all the values of δφ. Thus we write

∂L ∂L ∂µ − = 0. (2.1.5) ∂(∂µ)φ ∂φ

3 Section 2.2. Quantization of the free scalar ﬁeld Page 4

This is known as the Euler-Lagrange equation of motion. To make an easier transition to quantum mechanics we will use Hamiltonian formulation. The conjugate momentum for a continuous system is deﬁned as ∂L π(x) ≡ , (2.1.6) ∂φ˙(x) where φ˙(~x) is the derivative of the ﬁeld. Thus the Hamiltonian can be written as

Z Z (n) (n) H = d x π(~x)φ˙(~x) − L[φ, ∂µ] ≡ d xH. (2.1.7)

We consider the case of a single scalar ﬁeld whose Lagrangian is given by: 1 1 1 L = φ˙2 − (5φ)2 − m2φ2, 2 2 2 1 1 = (∂ φ)2 − m2φ2. (2.1.8) 2 µ 2

After applying the Euler-Lagrangian equation on (2.1.8). we obtain the equation of motion as:

µ 2 (∂µ∂ + m )φ = 0, (2.1.9)

µ ∂2 2 where ∂µ∂ is ∂t2 − 5 . This equation is the well known Klein-Gordon equation of a free scalar ﬁled. The momentum density and the energy density of the scalar ﬁeld is given by

π(~x) = φ˙(~x), (2.1.10) and Z 1 1 1 H = dnx φ˙2(~x) + (5φ)2 + m2 . (2.1.11) 2 2 2

We studied the case of free scalar ﬁeld and obtained its equation of motions. Next, we will quantize this scalar ﬁeld theory.

2.2 Quantization of the free scalar ﬁeld

Canonical quantization is an algorithm for turning classical systems into quantum systems. In quantum mechanics our observables are Hamiltonian operators. Since the quantum ﬁeld theory is a combination of special relativity and quantum mechanics, we need to promote our observables φ and π to operators, and impose a suitable set of commutation relations:

[φ(x), π(y)] = iδn(x − y).

[φ(x), φ(y)] = [π(x), π(y)] = 0. (2.2.1) Here we are using the delta Dirac function instead of the Kronecker delta because we encounter ﬁelds φ(~x,t) that introduce ~x with a continuous index. Section 2.3. The Klein-Gordon ﬁeld in space-time Page 5

In quantum mechanics the energy fluctuates, where as in special relativity the matter and energy are equivalent. Therefore, quantum field theory requires a new language to deal with systems in which fluctuating energy can turn into mass (Zee, 2010). Such a condition is satisfied in free particle systems; systems where no interaction between particles can occur. This is why we considered the simple harmonic oscillator as our system of study. Now we decompose the Klein-Gordon scalar field φ(~x,t) in terms of ladder operators a and a† in the Fourier space as follow:

Z n d 1 ip.x † −ip.x φ(x) = n p ape + a e , (2.2.2) (2π) 2wp and similarly the conjugate momentum will be

Z dn rw π(x) = (−i) p a eip.x − a†e−ip.x . (2.2.3) (2π)n 2 p

From the above two equations we ﬁnd a commutation relation between the annihilation operator a and the creation operator a† 0 [a , a† ] = (2π)nδn(p − p ), (2.2.4) p p0

† † [ap, ap] = [ap, ap] = 0.

The Hamiltonian in terms of the ladder operators is:

Z d(n+1)p 1 H = w a†a + [a , a†] . (2.2.5) (2π)n p p p 2 p p

† Using this expression for the Hamiltonian in terms of ap and ap, it is easy to evaluate the commutators

† † [H, ap] = wpap and [H, ap] = −wpap. (2.2.6) ap|0i = 0 where |0i is the ground state, or the vacuum and has E = 0. The other eigen states can be found by operating on |0i with the creation operators. Now we specify that ap destroys a particle of † 0 momentum p and ap creates a particle of momentum p whose state is given by

q † |pi = 2wpap|0i. (2.2.7)

2.3 The Klein-Gordon ﬁeld in space-time

In the previous section we introduced the quantization of the scalar ﬁeld in the Schrödinger picture which considers the time-independent quantities. We now turn to the Heisenberg picture and consider the time-dependent quantities. The operators φ and π will be

φ(x) = φ(x, t) = eiHtφ(x)e−iHt, (2.3.1)

π(x) = π(x, t). Section 2.4. The Interacting Field Theory Page 6

The time dependent picture of equation (2.2.2) and equation (2.2.3) will be

Z n d p 1 −ip.x † ip.x φ(t) = n p ape + ape , (2.3.2) (2π) 2Ep ∂ and π(t) = φ(t). ∂t

2.4 The Interacting Field Theory

In this section we aim to ﬁnd a description that is closer to the real world, a description that in- cludes interactions and scattering between particles. Thus we need to add new non-linear terms to the Hamiltonian. The interaction Hamiltonian will be of the form

Z Z n n Hint = d xHint[φ(x)] = − d xLint[φ(x)]. (2.4.1)

We note that the Hamiltonian we’re interested in is a function of the ﬁelds only, not their derivatives, and the example we discuss is called "phi-fourth" theory:

1 1 λ L = (∂ φ)2 − m2φ2 − φ4, (2.4.2) 2 µ 2 4! λ is a dimensionless coupling constant. After applying the Euler-Lagrange equation on equation(2.4.2) the equation of motion for φ4 theory is −λ (∂2 + m2)φ = φ3. (2.4.3) 3!

2.4.1 Perturbation Expansion of Correlation Functions. In this section we discuss the perturbation theory for interacting ﬁelds, in order to ﬁnd a formalism that will allow us to express the perturbation series as space time processes. Let us start with calculating the two-point correlation function,

hΩ|T φ(x)φ(y)|Ωi, (2.4.4) in the φ4 theory. The notation|Ωi represent the ground state of the interacting theory which is diﬀerent from the ground state of the free theory |0i, and T is the time ordering symbol. Deﬁnition Time ordering is a function that places the latest operators to the left. For example:

(φ(x)φ(y) when x0 > y0 T [φ(x)φ(y)] = (2.4.5) φ(y)φ(x) when y0 > x0.

There is another type of ordering which is normal ordering. Normal ordering can be deﬁned as placing † all the annihilation operators ap’s to the right of the creation operators ap’s. For example:

† † : apakaq := akapaq. (2.4.6) Section 2.4. The Interacting Field Theory Page 7

2.4.2 The Correlation Function. The correlation function can be interpreted physically as the ampli- tude for the propagation of a particle or excitation between y and x (Peskin and Schroeder, 1995);

n n 0 Z d pd p 0.y D(x − y) ≡ h0|T φ(x)φ(y)|0i = h0|a a†0|0ie−ip.x+ip , (2π)n p p Z n d p 1 −ip.(x−y) = n e . (2π) ωp

2.4.3 Green’s Function. Green’s functions of the Klien-Gorden equation satisfy:

(∂2 + m2)G(x − y) = −iδn(x − y). (2.4.7)

Using the Fourier transform to write G(x − y) in (n+1) dimensions

Z d(n+1)p G(x − y) = e−ip.(x−y)Gˆ(p), (2.4.8) (2π)(n + 1 Gˆ(p) is deﬁned as (Peskin and Schroeder, 1995):

(−p2 + m2)Gˆ(p) = −i. (2.4.9)

Then rewriting equation (2.4.8)

Z d(n+1)p i G(x − y) = e−ip.(x−y). (2.4.10) (2π)(n+1) p2 − m2

0 This formula has an indefinite answer when on mass shell condition p = Ep. To get a definite value we need to evaluate the p0 integral first by adding an infinitesimal value i to the dominator slightly of the real axis, as shown in Figure 2.1. We close the integration contour by adding a semicircle in the positive half of the complex plane when x0 − y0 < 0, shown in Figure 2.1. So the Feynman propagator is:

Z dnp ie−ip.(x−y) h0|T φ(x)φ(y)|0i = D (x − y) = , (2.4.11) free F (2π)n p2 − m2 + i and its Green’s function can be deﬁned as

(D(x − y) for x0 > y0, DF (x − y) = (2.4.12) D(y − x) for x0 < y0, where T is the time ordering function deﬁned in equation (2.4.5). For the second case when x0 < y0, instead of adding we subtract i, and we close the the integration contour by adding a semicircle in the negative half of the complex plane, as shown in Figure 2.2. We then get another Green’s function called the anti Feynman propagator DF˜’

Z d(n+1)p ie−ip.(x−y) D ˜ = h0|T˜ φ(x)φ(y)|0i = , (2.4.13) F (2π)(n+1) p2 − m2 − i Section 2.4. The Interacting Field Theory Page 8

Figure 2.1: Feynman Propagator,the poles shifted in diﬀerent directions.

Figure 2.2: Anti-Feynman Propagator, the poles in opposite directions. , where T˜ is the anti time ordering function:

(φ(y)φ(x) for x0 > y0 T˜ φ(x)φ(y) = (2.4.14) φ(x)φ(y) for y0 > x0 and its Green function can be deﬁned as

(−D(y − x) x0 > y0 D ˜(x − y) = (2.4.15) F −D(x − y) y0 > x0.

Shifting the poles in the same direction will give us another two Green’s Functions, the ﬁrst one is the Section 2.4. The Interacting Field Theory Page 9 retarded Green’s function

Z d(n+1) ie−i.p(x−y) D (x − y) = , (2.4.16) R (n+1) 0 0 (2π) (p − wp + i)(p − wp + i) where poles are shifted in lower imaginary plane as shown in ﬁgure 2.3. The second one is the advanced

Figure 2.3: Retarded propagator, the poles are shifted in the same direction.

Green’s Function DA(x − y):

Z d(n+1) ie−i.p(x−y) D (x − y) = , (2.4.17) A (n+1) 0 0 (2π) (p − wp − i)(p − wp − i) where the poles are the poles shifted in the upper imaginary plane as shown in Figure 2.4.

Figure 2.4: Advanced propagator, the poles are shifted in the same direction Section 2.4. The Interacting Field Theory Page 10

We now want to see how this expression for the correlation function will change in the interacting theory. 4 First we introduce the Hamiltonian of the φ theory as the sum of the free H0 Hamiltonian and the interaction Hamiltonian Hint: H = H0 + Hint. (2.4.18) We note here that there are three ways of representing the time evolution of the system. First the Schrödinger picture, which represents the operator as time independent and the state as time dependent. Second, the Heisenberg picture, which represents the operators as evolving under the full Hamiltonian H and the states that are time independent and ﬁnally the interaction picture which represents both the operators and the states as a time dependent. We re-write equation (2.3.1)for t 6= t0 in the Heisenberg picture as follows: iH(t−t0) −iH(t−t0) φ(t) = e φ(t0, x)e , (2.4.19) when λ= 0 H becomes H0 so the previous expression will be

iH0(t−t0) −iH0(t−t0) φ(t)|λ=0 = e φ(t0, x)e ≡ φ1(t, x). (2.4.20)

Now we express the full Heisenberg ﬁeld φ in terms of φ1

φ(t) = eiH(t−t0)e−iH0(t−t0)φ1(t,x)eiH0(t−t0)e−iH(t−t0) (2.4.21)

† ≡ U (t, t0)φ1(t, x)U(t, t0), (2.4.22) while the interaction picture time evolution operator U is deﬁned by

iH0(t−t0) −iH(t−t0) U(t, t0) = e e . (2.4.23)

Diﬀerentiating this equation with respect to time will give us a Schrödinger-like equation which has the unique solution U(t, t0), with the initial condition U(t0, t0) = 1 ∂ i U(t, t ) = eiH(t−t0)(H − H )e−iH(t−t0), ∂t 0 0 iH0(t−t0) −iH0(t−t0) iH0(t−t0) −iH(t−t0) = e (Hint)e e e ,

= HI (t)U(t, t0), where Z λ H (t) = eiH0(t−t0)(H )e−iH0(t−t0) = dnx φ4. (2.4.24) I int 4! I The Solution that satisﬁes the initial condition is the following series in λ:

Z t Z t Z t1 2 U(t, t0) = 1 + (−i) dt1HI (t1) + (−i) dt1 dt2HI (t1)HI (t2) t0 t0 t0 Z t Z t1 Z t2 3 3 +(−i) dt1 dt2 dt HI (t1)HI (t2)HI (t3) + ··· . (2.4.25) t0 t0 t0

The factors of HI in (2.4.25) can be written in terms of the time ordering operator. For example, 2 consider the case of HI as an example (Peskin and Schroeder, 1995)

Z t Z t1 1 Z t Z t dt1 dt2HI (t1)HI (t2) = dt1 dt2THI (t1)HI (t2). (2.4.26) t0 t0 2 t0 t0 Section 2.4. The Interacting Field Theory Page 11

As shown in ﬁgure 2.5 below (Peskin and Schroeder, 1995) the integrand THI (t1)HI (t2) is symmetric about the plane t1 = t2 hence the double integral on the right hand side counts everything twice. We apply this identity for the higher orders:

Z dt1 Z t1 Z n−1 1 Z t dt2 ··· dtnHI (t1) ··· HI (tn) = dt1 ··· dtnTHI (t1) ··· HI (tn). (2.4.27) t0 t0 t0 n! t0

Figure 2.5: The ﬁgure illustrate the symmetric about the plane t1 = t2 .

Summing up the all the terms in the Dyson series we ﬁnd :

∞ Z t (−i)2 Z t U(t, t ) = X U (t, t ) = 1 + (−i) dt H (t ) + dt dt TH (t )H (t ) + ··· , 0 n 0 1 1 1 2! 1 2 1 1 1 2 n=0 t0 t0

Z t 0 0 ≡ T [exp[−i dt H1(t )] . (2.4.28) t0 3. Interacting Fields

In this chapter we will continue exploring the correlation functions. First we will derive the cross section formula. Then we will introduce theorems like the Gell-Mann-Low theorem to calculate the expectation values in the interaction picture of ﬁelds in the non-interacting vacuum. Eventually, we will introduce the Keldysh formalism for calculating the the expectation values of operators.

3.1 The cross section and the S-Matrix

Our aim in this section is to find ways to compute some quantities that can be measured like the cross section. First we review the definitions of these quantities before we prove the formula. The cross section In any experiment of colliding two beams of particles with well-defined momenta, the cross section is an intrinsic quantity to the colliding particles that expresses the likelihood of a final state(Peskin and Schroeder, 1995). We thus see the probability of finding the final state |fi given the initial state |ii is,

P (i → f) ∝ |hf|ii|2, (3.1.1) where |ii is the initial set of wave-packets and |fi is the ﬁnal set of wave-packet. A wave packet representing some desired state |φi can be expressed as:

To prove the above statement we need to ﬁnd |φi and hφ| in order to ﬁnd hφ|φi as follows:

Z dnk 1 √ n φ(k)|ki, (3.1.4) (2π) 2Ek Z dnk0 1 φ∗(k0)hk0|. (3.1.5) (2π)n q 0 2Ek

Then hφ|φi will be:

Z dnk Z dnk0 1 hφ|φi = φ(k)φ(k0)∗hk0|ki, (3.1.6) (2π)n (2π)n q 0 2Ek2Ek

12 Section 3.1. The cross section and the S-Matrix Page 13

Putting this result back into the integral (3.1.6) we obtain

Z n Z n 0 d k d k 1 0 ∗ n ( 0 hφ|φi = φ(k)φ(k ) 2Ek(2π) δ k − k ), (2π)n (2π)n q 0 2Ek2Ek Z dnk = |φ(k)|2 = 1. (2π)n

- The probability that we wish to compute:

2 P = |hφ1φ2 · · · |φAφBi| . (3.1.7)

Note that the wave packets are localized in space, so we constructed two wave packets in the far past hφAφB| independently from another n wave packets that we constructed in the far future hφ1φ2 ··· φn|. So we write the initial state as

Z dnk Z dnk φ (k )φ (k )e−ib·kB A B A A√ B B |φAφBiin = n n |kA, kBiin labeleq1 (3.1.8) (2π) (2π) 2EA2EB where b is the impact parameter which is a perpendicular distance between the path of a projectile and the center of a potential ﬁeld that an object creates when the projectile is approaching. See Figure 3.1 The S matrix can be expressed in terms of the T matrix as

S = 1 + iT, (3.1.9) where the S matrix is a unitary operator with the following properties (Carr, 2009):

S(t, t) = 1, S(t0, t) = S†(t, t0), S(t, t0)S(t0, t00) = S(t, t00),

0 and ﬁnally S(t, t ) can be written as U(t, t0) in chapter 2:

Z t 0 0 0 S(t, t ) = T exp[−i dt HI (t )] . t0 1 in equation (3.1.9) represents the forward part where there is no interaction between particles and T represents the interaction part. Section 3.1. The cross section and the S-Matrix Page 14

The ﬁnal (out) state can be written as:

Z dnp φ∗(p ) hφ φ · · · | = Y f f hp p · · · |. (3.1.10) out 1 2 (2π)n p2E out 1 2 f f

The matrix elements of S should reﬂect 4-momentum conservation, that is why S or T always has a n+1 P factor of δ (kA + kB − pf ). This factor represents the conservation of momentum, inﬁnity when the momentum is conserved and zero when it is not conserved. n+1 P Extracting the factor δ (kA + kB − pf ) we introduce M as the invariant matrix element by:

n n+1 X hp1p2 · · · |iT |kAkBi ≡ (2π) δ (kA + kB − pf ) · iM(kA, kB → pf ). (3.1.11)

Thus the probability for the initial state |φAφBiin is:

dnp 1 P (AB → 12 ··· n) = Y | hp ··· p |φ φ i |2 (3.1.12) (2π)n p2E out 1 n A B in f f

The cross section of a number of scattering events shown in Figure 3.1 is:

n Figure 3.1: incident wave-packets φB on a object A with impact parameterb(Peskin and Schroeder, 1995) .

N N Z σ = = = dnbp(b). (3.1.13) nBNA nB.1 We derive a formula for σ by combining equations (??),(3.1.12) and (3.1.13). First combine (3.1.12) and (3.1.13). Z dnp 1 σ = dnb Y | hp ··· p |φ φ i |2. (3.1.14) (2π)2 2E out 1 n A B in f f Section 3.1. The cross section and the S-Matrix Page 15

instead of squaring the term we multiply by the conjugate, after substituting outhp1 ··· pn| and |φAφBiin from equation (??)

Z dnp 1 Z dnk φ (k) Z dnk φ (k) Z dnk¯ φ∗ (k¯) n Y A √A B √B A A σ = d b n n n n q (2π) 2Ef (2π) 2EA (2π) 2EB (2π) f 2E¯A n¯ ∗ ¯ Z d kB φ (k) ¯ √B i·b(kB −kB ) ∗ n e outhp1 ··· pn|φAφBiin outhp1 ··· pn|φAφBiin (3.1.15) (2π) 2EB

dnp 1 Z Z dnk φ (k ) Z dnk¯ φk¯ Y n Y i √i i i √ i = n d b n n (2π) 2E (2π) (2π) ¯ i f f i=A,B 2Ei 2E ¯ ib.(kB )−kB ) ¯ ×e outhpf |kiiin outhpf |kiiin (3.1.16)

dpn 1 Z dnk φ (k ) Z dnk φ(k¯ ) Y Y i √i i B √ i dσ = n n n (2π) 2E (2π) (2π) ¯B f f i=A,B 2Ei 2E (n) ¯ 2 n (n) X n (n) ¯ X ×δ (kB − kB)|Mki → pf | (2π) δ (ki − pf ).(2π) δ (ki − pf ). (3.1.17)

n n (n) ⊥ ¯⊥ Note that the integral d b is giving a factor of (2π) δ (kB − kB ). Writing the last two terms in equation (3.1.17) in terms of M we will obtain more delta functions. Using the delta functions n (n) ⊥ ¯⊥ (n) n P P (2π) δ (kB − kB ) ∗ (2π) δ ( ki − pf ) to evaluate the six conjugate integrals, only those over ¯z ¯z kA and kB will remain: Z ¯Z ¯z ¯Z ¯Z X z ¯ ¯ X dkAdkBδ(kA + kB − pf )δ(EA + EB − Ef ). (3.1.18)

¯z ¯z P z R ¯z ¯z ¯z P z Evaluating the ﬁrst integral over kB by putting kA + kB = pf so dkB(kA + kB − pf ) = 1, we then ﬁnd Z q q ¯z ¯2 2 X z ¯ 2 X dσ = dkAδ( k A + mA + pf − kA + mB − Ef ).

R P f(xi) This integral is in the form f(x)δ(g(x))dx, which can be evaluated as 0 . In our integration |g (xi)| q q P z ¯z ¯z 2 P z ¯z 2 2 P ¯z f(xi) = 1 and xi = pf −kA and taking ( k A + mA + ( pf − kA) + mB − Ef ) to be g(kA), so

q q 0z ¯z2 2 X 2 X ¯z ¯z2 2 X g(kA ) = ( kA + mA + p zf − 2 pf kA + kA + mB − Ef , −1 X X 1 X 0 0z ¯z 2 2 ¯ 2 z − 2 ¯z g (kA ) = (kA + mA) .kA + ( pf − 2 pf kA) .(− pf + kA). This can be written as ¯ P 2 ¯z 0 0z kA pf + kA g (kA ) = q − q , ¯z2 2 (P pz − 2 P p k¯z ) + k¯z2 + m2 kA + mA f f A A B ¯z ¯ 0 ¯z kA kB 1 |g (kA)| = − = . EA E¯B |VA − VB| Section 3.1. The cross section and the S-Matrix Page 16

¯z ¯z P z ¯ ¯ P where kA + kB = pf , EA + EB = Ef and |VA − VB|, is the relative velocity of the beams as viewed from the laboratory frame (Peskin and Schroeder, 1995).

Recalling the normlization factor (3.1.3) we evaluate the integral over kA and kB. Hence the ﬁnal form of the relationship between the S-matrix elements and the cross section is (Peskin and Schroeder, 1995): 1 dnp 1 dσ = Y 2E 2E |V − V | (2π)n 2E A B A B f f 2 n X ×|M(pA, pB → pf )| (2π) (pA + pB − pf ). (3.1.20)

We simplify the expression for dσ in (3.1.20) by partially evaluating the phase-space integrals in the center of mass frame by considering a ﬁnal state of two particles with momenta p1 and p2, so the integral over the ﬁnal state momenta is: Z Z dp dp dΩ d Y = 1 2 (2π)δ(E − E − E ), (3.1.21) (2π)n2E 2E cm 1 2 2 1 2

q 2 2 q 2 2 where E1 = p1 + m1, E2 = p1 + m2 and Ecm is the total initial energy. This integral delta function as a function can be evaluated by following the same procedure we followed to evaluate the integral (3.1.18); consider:

g(b) = (Ecm − E1 − E2), (3.1.22) q q 2 2 2 2 = (Ecm − p1 + m1 − p1 + m2). (3.1.23) Diﬀerentiating the last equation gives g(p):

0 2 2 − 1 2 2 − 1 g (p) = −(p1 + m1) 2 p1 − (p1 + m2) 2 p1, (3.1.24) p p |g0(p)| = 1 + 1 , (3.1.25) q 2 2 q 2 2 p1 + m1 p1 + m2 1 = . (3.1.26) | p1 + p1 | E1 E2 Q Putting this result back to the integral of d 2 we obtain: Z Z p2 p p −1 d Y = dΩ 1 1 + 1 , (3.1.27) 16π2E E E E 2 1 2 1 2 1 |p1| = dΩ 2 . (3.1.28) 16π Ecm Substituting this simplification into equation (3.1.20) we find a final form for the cross section: dσ |M|2 = 2 2 . (3.1.29) dΩ cm 64π Ecm Section 3.2. The Gell-Mann-Low theorem Page 17

3.2 The Gell-Mann-Low theorem

As an extension to the interaction theory we discuss the Gell-Mann-Low theorem as a method that allows us to relate the ground state of the interacting theory to the corresponding non interacting ground state. To prove the theory we start by writing the ground state (the vacuum) of H in |Ωi in terms of the ground state |0i of H0 that evolves through time as follows:

X e−iHt|0i = e−iEnT |nihn|0i. (3.2.1) n Note here that we multiply by 1 as from the complete set relation one can write

X |nihni = 1. (3.2.2) n

We assume that |Ωihas some overlap with |0i, thus the above equation can be written as

We get rid of all the n 6= 0 terms by sending T → ∞(1−i), so our interaction factor will be a bounded norm by 1, that will make it dies faster than the other non-interacting terms so after isolating the ground state we write the above equation as:

|Ωi = lim (e−iE0T hΩ|0i)−1e−iHT |0i. (3.2.4) T →∞(1−i)

As T is very large we shift it by a small amount t0, so |Ωi will be:

|Ωi = lim (e−iE0(T +t0)|Ω|0i)−1e−iH(T +t0 |0i T →∞(1−i) |Ωi = lim (e−iE0(t0−(−T ))|Ω|0i)−1e−iH(t0−(−T )e−iH0(−T −t)|0i, T →∞(1−i)

−iH (−T −t ) for the new term in the second line e 0 0 we use the fact that H0|0i = 0 to get |Ωi by evolving |0i from time −T to time t0 with the operator U as follows:

X h0|e−iHt = hΩ|h0|Ωie−iE0T + e−iEnT hn|hn | 0i, (3.2.6) n6=0

The summation will vanish for the same reason as above, so we have:

hΩ| = lim h0|e−iHT (e−iE0T h0|Ωi)−1. (3.2.7) T →∞(1−i) Section 3.2. The Gell-Mann-Low theorem Page 18

As T is very large we shift it by a small amount t0, so hΩ| will be:

−iE0(t0−(−T )) −1 hΩ| = lim h0|U(t0, −T )i(e h0|Ωi) . (3.2.8) T →∞(1−i)

Now we put the bra and the ket together to calculate the correlation function hΩ|φ(x)φ(y)|Ωi by taking 0 0 x > y > t0

−iE0(T −t0) −1 hΩ|φ(x)φ(y)|Ωi = lim (e h0|Ωi) h0|U(T, t0) T →(1−i)

0 † 0 0 † 0 −iE0(t0−(−T )) −1 ×[U(x , t0)] φ1(x)U(x , t0)[U(y ), t0] φ1(y)U(y , t0) × U(t0, −T )|0i(e hΩ|0i) , (3.2.9)

The expression in equation (3.2.9)can be simpliﬁed by using the following identities that are satisﬁed by the time evolution operator:

U(t1, t2)U(t2, t3) = U(t1, t3) † U(t1, t3)[U(t2, t3)] = U(t1, t2).

These identities can be applied to the terms in the correlation function to obtain the terms :

0 † 0 U(T, t0)[U(x , t0)] = U(T, x ), 0 0 † 0 0 U(x , t0)[U(y , t0)] = U(x , y ), 0 0 U(y , t0)U(t0, −T ) = U(y , −T ).

Substituting these terms back into the correlation function, and collecting the two terms with the −1 power we get:

2 −iE0(2T ) −1 0 0 0 0 hΩ|φ(x)φ(y)|Ωi = lim (|h0|Ωi| e ) × h0|U(T, x )φ1(x)U(x , y )φ1(y)U(y , −T )|0i. T →∞(1−i) (3.2.10) We get rid of the term (|h0|Ωi|2e−iE0(2T ))−1 by dividing by 1 in the form

2 −iE0(2T ) −1 1 = hΩ|Ωi = h0|ΩihΩ|0i = (|h0|Ωi| e ) h0|U(T, t0)U(t0, −T )|0i.

So the ﬁnal expression for the correlation function is

0 0 0 0 h0|U(T, x )φ1(x)U(x , y )φ y)U(y , −T )|0i hΩ|φ(x)φ(y)|Ωi = lim ( . T →∞(1−i) h0|U(T, −T )|0i Section 3.3. The Keldysh formalism Page 19

Using equation (2.4.28) we simplify the above equation to obtain the ﬁnal expression

R T h0|T φ1(x)φ1(y)exp[−i dtH1(t)]|0i hΩ|T φ(x)φ(y)|Ωi = lim −T . (3.2.11) T →∞(1−i) R T h0|T exp[−i −T dtH1(t)]|0i Note here we consider the time ordered product to order the operators in one large T operator (Peskin and Schroeder, 1995). This formula also holds for the correlation functions of arbitrarily many ﬁelds; we should add a factor of φ on the left for every factor of φ1 on the right.

3.3 The Keldysh formalism

As the Gell-Mann-Low is a way to relate the interaction state to the ground state only, in this section we introduce a formalism that turn to the interaction picture by using the closed time path formulation and generalizing the time-ordering to contour-ordering (Rammer, 2007).

We consider the operator φ(t) to be determined by the full Hamiltonian H = H0 +Hint, the relation between the interacting picture and the Heisenberg picture can be expressed by the unitary transformation as in equation 2.4.22: † φ(t) = U (t, t0)φI (t)U(t, t0). (3.3.1)

† The time evolution operator is donated by U(t, t0) and the hermitian conjugate of it is donated U (t, t0), can be written in it’s exponential form similarly to equation (2.4.22).

t −i R dtH (t) t 1 U(t, t0) = T [e 0 ] , (3.3.2) and the conjugate as t † i R dtH (t) † t 1 U (t, t0) = T˜ e 0 ] , (3.3.3) where T˜ is the anti-chronological time ordering function. That is where operators that later happen go right, (φ(y)φ(x) when x0 > y0 T˜[φ(x)φ(y)] = (3.3.4) φ(x)φ(y) when y0 > x0. Now we rewrite equation (3.3.1) in terms of equation (3.3.2) and equation (3.3.3) to ﬁnd:

t R t −i R dtH (t) −i dtH1(t) t 1 φ(t) = T˜ e 0 T φI (t)e 0 , (3.3.5) which is the same as equation (2.4.21). That shows a unitary transformation between the Heisenberg picture and the interaction picture (Rammer, 2007). Now we introduce the closed time path contour which starts at t0 going along the real time axis to time t and come back again to t0, shaping a closed contour ct as illustrated in Figure 3.2. Now we write the unitary transformation between the interaction picture and the Heisenberg picture in terms of the closed contour form as (Rammer, 2007) R −i HI (τ)dτ ct φ(t) = TCt {e φI (t)}, (3.3.6) Section 3.3. The Keldysh formalism Page 20

Figure 3.2: The closed time path contour ct

where τ is the contour variable that is the variable on ct and TCt is the contour ordering along Ct, indicated by the arrows in Figure 3.2. The expression of the expectation value of φ(t) on any state Ω in terms of the contour Ct is:

R −i dτH1(τ) hφ(t)i = hΩ | Tct e ct φI (t) | Ωi, (3.3.7)

† = hΩ|U (t, t0)φI (t)U(t, t0)|Ωi. (3.3.8)

By sending t0to −∞(1 + i) the second term goes to zero so the above equation becomes

lim U(t, t0)|0i = lim U(t, t0)|ΩihΩ|0i. (3.3.12) t0→−∞(1+i) t0→−∞(1+i)

After isolating |Ωi we get:

U(t, t )|0i |Ωi = lim 0 . t0→−∞(1+i) U(t, t0)hΩ|0i

Following the same steps we need to ﬁnd hΩ|, ﬁrst by writing hΩ| in terms of h0|:

h0| = hΩ|h0|Ωi + Xhn|h0|ni, (3.3.13) n>0 applying equation (3.3.10) on the above equation we ﬁnd:

By sending t0to −∞(1 + i) the second term goes to zero. Hence equation (3.3.14) will be:

lim h0|U(t, t0) = lim U(t, t0)hΩ|h0|Ωi, (3.3.15) t0→−∞(1+i) t0→−∞(1+i)

h0|U †(t, t )φ (t)U(t, t )|0i hΩ|φ(t)|Ωi = lim 0 I 0 , (3.3.17) t0→−∞(1+i) h0|ΩihΩ|0i

Since we know that h0|0i = 1 and h0|ΩihΩ|0i = 1, we ﬁnd:

† hΩ|φ(t)|Ωi = h0|U (t, −∞)φI (t)U(t, −∞)|0i. (3.3.18)

As we have sent t to −∞ the expectation value of φ(t) can be expressed in terms of a new contour that goes from −∞ to time t and back to−∞ again shaping a is so called Schwinger-Keldysh closed time path C illustrated in Figure 3.3 below. The expectation value in terms of the contour C is:

Figure 3.3: The Schwinger-Kekdsh contour C

R −i HI (τ)dτ hφ(t)i = h0|TC (e C φI (t))|0i, (3.3.19)

Where TC is the contour ordering on C. 4. Conclusion

The quark-gluon plasma is a new phase of matter that drew the attention of the high energy scientiﬁc community because of its nearly prefect properties as a ﬂuid, and because it describes the state of the universe directly after the big bang. Studying QGP will give us much insight into the structure of the universe. One way to study the strong interactions between quarks and gluons is to think about it in terms of some properties of black holes in a higher dimensional space-time. In this essay we reviewed some methods of calculating the correlation functions of particles generalized to (n + 1) dimensional space-time. First we discussed Green’s function in (n + 1) dimensions and re-derived the cross section formula in (n + 1) dimensions. Then we re-derived the Gell-Mann-Low theorem as a method of calculating the expectation values and we re-derived the Keldysh formalism to calculate the expectation values. Finally we can conclude that studying the correlations of the particles provide us information about their distributions and collisions from the initial equilibrium state through the QGP phase and after. Further work is to calculate the expectation values of operators that are not subject to the ground state like the energy momentum tensor operator and deriving the LSZ reduction theorem as an additional formalism for calculating the in/out and in/in formalism.

22 Acknowledgements

Loads of thanks and appreciation for the AIMS family for their ultimate support. To my supervisor, Dr. W.A Horowitz, I am very grateful for your patience and helpful spirit. A special thanks to my friends at AIMS for being by my side during the tough times during this essay phase. To my biggest supporter ,father and mother, I am very thankful.

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