On the Regularization of Phase-Space Path Integral in Curved Manifolds

Home , First quantization, Regularization (physics)

Marco Falconi

June 1, 2010

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Introduction

n this work we discuss path integrals both in flat and curved space-time: in particular I we describe how to write the integral kernel of the time evolution operator (in a curved space time we describe the evolution with respect to a geodesic affine parameter, such as the proper time if the particle has a non-zero mass) in a path integral form, both in configuration and phase space. We briefly introduce a path integral formulation of Quantum Field Theo- ries, however we focus our attention mainly on quantum mechanical models. We develop the idea, originally due to Schwinger, to describe effective actions in quantum field theory as quantum mechanical path integrals of a fictitious particle with evolution dictated by a suitable Hamiltonian function. We will see that one could have obtained the same result as in field theory by first quantizing the particle which actually makes the loop of the Feynman graph corresponding to the effective action. In the second part we focus on a particle of mass m ≥ 0 that classically moves along a geodesic of curved space-time. Quantizing this model we will see that even if it is still possible to describe its evolution in the affine parameter by means of a path integral, the results obtained would be ambiguous unless we introduce suitable regularization schemes and related counterterms. This is, in fact, the key point of path integrals used in physics. Configuration space path integral needs different counterterms, depending on the regularization scheme used, in order to give the same results at any perturbative order for the integral kernel of the evolution operator. These counterterms are finite and univocally fixed by renormalization conditions arising in two-loop calculations. No higher loop calculation are needed since if we consider quantum mechanics in curved space as a field theory, it would be super-renormalizable, which means that possible divergences and ambiguities can appear only up to a finite number of loops. Different is the case of phase space path integral, as we show in this work: in such functional integral perturbative calculations are not ambiguous nor divergent in the continuum limit, so as far as we limit ourselves to phase space perturbative calculations the introduction of regularization schemes is not particularly useful: they just serve the purpose of explicitly defining the path integral measure, but in many perturbative calculations the regularization can be removed at once. Nevertheless we cannot avoid the regularization procedure completely, for example if we want to integrate out momenta, because in the continuum limit such measure has not a definite meaning. Introducing a cutoff in the Fourier modes of momentum and coordinate paths we calculate explicitly the “counterterms” of phase space path integral in mode regularization.

Contents

Introduction i

I FLAT SPACE-TIME 1

1 Path Integrals 5 1.1: Some History ...... 5 1.1.1: Probability Amplitudes ...... 5 1.1.2: Wave Function ...... 7 1.1.3: Equivalence of Formulations ...... 8 1.2: A modern formulation ...... 9 1.2.1: Time-independent Hamiltonian ...... 11 1.2.2: Time-dependent Hamiltonian ...... 13 1.2.3: Transition to Lagrangian formulation ...... 16 1.3: Applications: Free Theories, Harmonic Oscillators & Perturbative Theory . . 17 1.3.1: Systems of free particles ...... 20 1.3.2: The Harmonic Oscillator ...... 22 1.3.3: Perturbative Methods ...... 24 1.3.4: Some Perturbative Calculations ...... 27

2 Quantum Fields or Quantum Particles? 37 2.1: QFT Path Integrals ...... 37 2.1.1: Fock Spaces ...... 38 2.1.2: Q-Space ...... 40 2.1.3: Derivation of Path integral ...... 43 2.2: Worldline Methods ...... 46 2.2.1: The Quantum Effective Action ...... 46 2.2.2: Fictitious Quantum Mechanics ...... 48

A Trotter formula and other systematic approximants 53 A.1: Trotter product formula ...... 53 A.2: Zassenhaus formula ...... 54

iii B the Dyson expansion 55 B.1: Proof of the Theorem ...... 55

C Gaussian integrals 59

II CURVED SPACE-TIME 61

3 Path Integrals 65 3.1: Particles in curved manifolds of space-time ...... 65 3.1.1: Relativistic Action ...... 65 3.1.2: Quantization ...... 69 3.2: Derivation of path integral ...... 73 3.2.1: Time slicing ...... 73 3.2.2: Lagrangian formulation ...... 77 3.3: Continuum Manipulations ...... 78 3.3.1: Ghosts ...... 78 3.3.2: Analysis of divergences ...... 79 3.3.3: Perturbative Expansions ...... 81 3.4: Regularization schemes ...... 83 3.4.1: Mode Regularization ...... 84 3.4.2: Dimensional regularization ...... 87 3.4.3: Time slicing ...... 88

4 Regularizations in phase space 91 4.1: Worldline methods on curved space ...... 92 4.2: Phase Space ...... 93 4.2.1: Mode expansion ...... 94 4.2.2: Propagators ...... 96 4.2.3: Perturbative expansion ...... 100

4.2.4: Fixing A and Vph ...... 102 4.2.5: Regularization is not needed for perturbative calculations ...... 104

D Weyl Ordering 107

E Distributional integration in phase space 109

Conclusions 111

Bibliography 113

iv PartI

FLAT SPACE-TIME

he aim of this first part is to introduce the reader to the concept of “Path-integral”, T providing historical remarks as well as its modern derivation; and then to illustrate some of its application, both in Quantum Mechanics and Quantum Field Theory. We will confine ourselves here to flat space-time, for it is easier to develop basic concepts in this environment, and we will see in detail the generalization to an arbitrary curved space-time in the second part of this work. We start exposing briefly the formulation of R.P. Feynman, who first introduced in 1949 the idea of Path integral inspired by some remarks of P.A.M. Dirac concerning the relation of classical action to quantum mechanics; then we derive Feynman formulas from operator Quantum Mechanics, following a more modern and more general point of view. Techni- cal discussions about functional integration, Trotter formula, Feynman-Kac formula, time- dependent Hamiltonians and Gaussian integrals are developed in the appendices. outline of first part After the theoretical derivation of path integrals we develop some applications: we analyze the analytically solvable free and harmonic theories, as well as some perturbative methods for a quite general class of systems. In the following chapter we develop the mathematical tools necessary to introduce quantum field theory and its formulation through path integrals. We show that the introduction of a particular space, called Q-space, allows us to derive quantum fields path integral in a way perfectly analogous to the one followed in quantum mechanics. Some mathematical technicalities are relegated to the appendices. Finally, we introduce the effective action for a field interacting with an external background source, and discuss briefly the worldline approach that leads to a description of its one-loop part by means of a quantum mechanical path integral. However we will give a deeper insight of this topic in the second part of this work.

with square absolute the is afﬁrmative be the will distributions, result complex the of that sum probability a the of then space-time, of R region a 1. Postulate postulate: ﬁrst its presenting start We Amplitudes Probability 1.1.1: work[2]. Dirac 1933 a and by Schrödinger the inspired both and to one, equivalent theory, Heißenberg quantum of formulation” “third a provides It A History Some 1.1: P ϕ ehnc,bsdo pc-iepts sh rsne to aoswr[]i 1948. in work[1] famous a on it presented he as paths, Space-Time on based Mechanics, a itrclnt,w eiwhr ena praht o-eaiitcQuantum Non-Relativistic to approach Feynman here review we note, historical an s ( ( R R = ) = ) fa da esrmn spromdt eemn hte atcehsapt yn in lying path a has particle a whether determine to performed is measurement ideal an If e lim | → ϕ 0 ( Z R R ) φ | 2 ( , . . . ( , x i , x i + 1 . . . , rbblt amplitudes probability ) · · · d x i d x i + 1 · · · n o ahpt nteregion: the in path each for one , ( , oefesa aair uthv felt have must Cavalieri as feels “one acltn h oueo pyramid a of volume the calculating rbblt mltd o einR) region for amplitude probability ahIntegrals Path eoeteivnino calculus” of invention the before rbblt o einR) region for probability

R.P.Feynman C h a p t e r 1 postulate ﬁrst Feynman’s 5 Part I: FLAT SPACE-TIME

and

xi+1 particle’s position at time ti+1 = ti + e .

This postulate prescribes the type of mathematical framework required by Heißenberg uncertainty principle for the calculation of probabilities. Now we can imagine that, as e

approaches zero, the probability amplitude φ(..., xi , xi+1 ,... ) essentially depends on the entire path x(t) rather than on the discrete points xi of the path. We might call Φ[x(t)] the probability amplitude functional of paths x(t). We can now introduce the second postulate, that gives a prescription to compute this functional for each path:

Postulate 2. The paths contribute equally in magnitude, but the phase of their contribution is the classical action (in units of }h ); i.e., the time integral of the Lagrangian taken along the path:

i h S[x(t)] Feynman second Φ[x(t)] ∝ e } , postulate where

Z S[x(t)] = (x˙(t), x(t))dt , L and the proportionality is intended up to a normalization global factor.

The Lagrangian is a function of position and velocity, and if we suppose it to be a quadratic function of the velocities we can show the equivalence of these postulates and usual quantum mechanics formulations. But we have ﬁrst to interpret this postulate for a discontinuous path, i.e. to calculate

ϕ(R): we shall assume that x(t) in the interval between ti and ti+1 is the classical path given by Lagrangian , which starting from xi at ti, reaches xi+1 at ti+1, at least in the limit e→0 L ti −−→ ti+1. Lagrangian function must therefore depend on no higher time derivatives of the position than the first, in order to be sufficient the specification of start-point and end-point

to determine classical path. So for a discontinuous path we can write S = ∑i S(xi+1, xi), where

Z ti+1 S(xi+1, xi) = min (x˙(t), x(t))dt , (1.1.1) paths ti L and the minimum is taken to evaluate action on classical trajectory. So now we can combine the two postulates and ﬁnd:

Z i ∑ S(xi+1 , xi) dxi+1 dxi ϕ(R) = lim e }h i ··· ··· , (1.1.2) e→0 A A R

where we have split the normalization factor into 1/A factors, one for each time slice.

6 1.1. SOME HISTORY CHAPTER 1. PATH INTEGRALS

1.1.2: Wave Function

In order to check equivalence we have to define the wave function, and show that its time development is given by Schrödinger equation. The wave function is a quantity having a well defined (by Schrödinger equation) time development, so we have to define it by means of successive sections of the path, with respect to time. In order to do that we choose a particular time, say t, and split region R of (1.1.2) into pieces, future and past relative to t. The split is done in this way: a) a region R0, restricted arbitrarily in space, but lying entirely earlier than some time t0, such that t0 < t; b) a region R00 defined in the same way as in a), but lying later than time t00, such that t00 > t; c) the region between t0 and t00, in which all values of x coordinates are unrestricted, i.e. all of space-time between t0 and t00.

Region c) is not necessary at all, for it can be taken as narrow in time as desired. Nevertheless it is convenient in letting us vary t a little without having to redeﬁne a) and b) . So we can deﬁne the probability |ϕ(R0, R00)|2 that if a path had been in region R0, then will be found in region R00. This is the crucial measured quantity in most experiments: we prepare a system in a certain way (e.g., it was in region R0) and measure some other property (e.g., will it be found in region R00?). Now we can use a (1.1.2)-like equation to compute the quantity ϕ(R0, R00). Assuming the time t to be one particular point k of the subdivision into e time steps, i.e. t = tk, we can split the action exponential in the product of two exponentials:

∞ k−1 i i ( ) i h ∑ S(xi+1 , xi) h ∑ S xi+1 , xi ∑ S(xi+1 , xi) } } =− e }h i ≡ e i=k · e i ∞ .

This operation is possible essentially because the Lagrangian is a function only of positions and velocities, and so we could split the action into a sum of discretized parts. Now, calling xk ≡ x and tk ≡ t we obtain: Z ϕ(R0, R00) = χ∗(x, t)ψ(x, t)dx , (1.1.3)

where Probability amplitudes and k−1 i wave functions Z ∑ S(xi+1 , xi) }h dx dx ψ(x, t) = lim e i=−∞ k−1 k−2 ··· , (1.1.4) e→0 A A R0 and

∞ i Z ∑ S(xi+1 , xi) ∗ }h dx dx χ (x, t) = lim e i=k 1 k+1 k+2 ··· . (1.1.5) e→0 A A A R00

7 Part I: FLAT SPACE-TIME

The quantity ψ depends only on past information on the system, with respect to t, and it is completely defined when the region R0 is known. The information on the future lies entirely on the quantity χ. We have thus defined the concept of wave functions ψ(x, t) , χ(x, t) defining the state of a system. The quantity | R χ∗(x, t)ψ(x, t)dx|2 is the probability that a system in state ψ will be found by an experiment whose characteristic state is χ, in agreement with ordinary quantum mechanics.

1.1.3: Equivalence of Formulations

If the wave functions just deﬁned satisfy Schrödinger equation we have completed our task to show the equivalence between Feynman formulations and the other ones. In order to do that we have to ask that is a quadratic, but perhaps inhomogeneous, form in the velocities x˙(t). L If we compute ψ in (1.1.4) at the next time slice we obtain:

k i Z ∑ S(x + , x ) Z }h i 1 i i i=−∞ dxk dxk−1 h S(xk+1,xk) dxk ψ(xk+1, t + e) = e A A ··· = e } ψ(xk, t) A . (1.1.6) R0

This equation is not exact, it is only true in the limit e → 0 and we shall derive Schrödinger equation by assuming it is valid to first order in e. In order to do this we need first of all to give an approximate value of (1.1.1), provided the error of the approximation be of an order smaller than the first in e. The Lagrangian is a quadratic form in the velocities, and 1 we will see that the paths important in the calculation are those for which xi+1 − xi ∼ e 2 . Under these circumstances, it is sufficient to calculate the integral in (1.1.1) over the classical path taken by a free particle. If we chose the coordinates of the system to be Cartesian1,2, the path of a free particle is a straight line. So in this case it is sufficiently accurate to replace the integral by3: xi+1 − xi S(xi+1, xi) = e , xi+1 . (1.1.7) L e 1more generally, coordinates for which the terms quadratic in the velocities in the Lagrangian function appear with constant coefficients. 2if one seeks the differential equation in a different coordinate system, the easiest way is to find the equation in Cartesian coordinates and then to transform the coordinate system to the one desired. 3this approximation is not valid if we have a vector potential or other terms linear in the velocity in the Lagrangian, if those are the cases we need to use an approximation of the type e xi+1 − xi e xi+1 − xi S(xi+1, xi) = , xi+1 + , xi 2 L e 2 L e

or xi+1 − xi xi+1 + xi S(xi+1, xi) = e , . L e 2

8 1.2. A MODERN FORMULATION CHAPTER 1. PATH INTEGRALS

Thus, in the simple unidimensional case of a particle of mass m moving under a potential V(x), we have:

2 me xi+1 − xi S(x + , x ) = − eV(x + ) . i 1 i 2 e i 1

So if we use this last equation and we call xk+1 ≡ x , (xk+1 − xk) ≡ ξ so that xk = x − ξ, then (1.1.6) becomes: Z − i eV(x) 1 im ξ2 ψ(x, t + e) = e }h e 2e}h ψ(x − ξ, t)dξ , (1.1.8) A so if we Taylor expand around ξ the wave function, perform Gaussian integrations, and consider terms up to ﬁrst order in e1, we obtain:

1 2πhei 2 1 ie hei ∂2 ( ) + = } − ( ) ( ) + } ψ x, t e∂tψ 1 V x ψ x, y 2 ψ . m A }h 2m ∂x In order that both sides may agree both to zero and to ﬁrst order in e, we must have:

1 2πhei 2 A = } m and of course 1 ih∂ ψ = (−ih∂ )2ψ + V(x)ψ , } t 2m } x which is the Schrödinger equation we sought. The generalization to arbitrary degrees of freedom is straightforward. For what it concerns χ∗, a differential equation can be developed in the same way, but it appears to have the time reversed. However taking the complex conjugates we can see that χ satisﬁes the same equation as ψ. We have ﬁnally shown that Feynman formulation is equivalent2 to the usual formulation of quantum mechanics provided by Schrödinger or Heißenberg. We shall conclude here our historic note, and begin to deal with modern formulations of path integral.

1.2: A modern formulation

ath integral emerges naturally in usual non-relativistic quantum mechanics as one P deals with the dynamics, i.e. as one considers evolution of a state through time. In order to know such an evolution in wave functions space, one has to seek the kernel of the following integral equation: Z ψ(x f , t f ) = dxi K(x f , t f ; xi, ti) ψ(xi, ti) , (1.2.1)

1Taylor expanding the ﬁrst member of (1.1.8) and the exponential of the potential up to ﬁrst order in e. 2at least if the system is described by a Lagrangian that is function only of time, position, and velocities and is a quadratic form of the latter (perhaps inhomogeneous).

9 Part I: FLAT SPACE-TIME

where x is intended to be a vector of the system’s configuration space, and dx is the infinitesimal volume element of the same space. If the Hamiltonian is time independent is straightforward to show that K is a solution 0 0 1 of the Schrödinger equation, i.e. i}h∂t − H(x, −i}h∇) K(x, t; x , t ) = 0 . Let’s see how this kernel can be calculated when we deal with a generic vector of the Hilbert space2: suppose that we have a state |ψ(t)i ∈ , its time evolution is governed by the evolu- 0 H tion operator U(t, t ) in such a way that |ψ(t f )i = U(t f , ti)|ψ(ti)i. We know that position eigenvectors form a complete set of commuting observables, so we can use the relation R R 1 = dxi |xiihxi| to obtain |ψ(t f )i = dxi U(t f , ti)|xiihxi|ψ(ti)i; now finally projecting each member on |x f i and remembering the usual definition hx|ψ(t)i ≡ ψ(x, t) we obtain R ψ(x f , t f ) = dxi hx f |U(t f , ti)|xiiψ(xi, ti), so from (1.2.1) we immediately identify:

K(x f , t f ; xi, ti) = hx f |U(t f , ti)|xii . (1.2.2)

Now U is a unitary operator (in order to conserve probabilities), and can be written in the infinitesimal form U(t + e, t) = 1 − ieH(t), provided H, called the infinitesimal generator of the transformation, to be a self-adjoint operator. In fact we can identify H(t) as the total Hamilto- nian function of the system. We can also obtain a finite transformation iterating infinitesimal ones, the result being the exponentiation of the generator, provided to be time-ordered to preserve the causality condition3: Hamiltonian- ( ) induced time Z t f evolution U(t f , ti) = T exp −i dt H(t) where T stands for time-ordered product ; (1.2.3) ti

such a relation is known as Dyson series, and has to be intended as an integral series:

∞ Z t Z t1 Z tn−1 n f U(t f , ti) = 1 + ∑ (−i) ··· dt1 ··· dtn H(t1) ···H(tn) . (1.2.4) n=1 ti ti ti If, as it happens in most cases, the Hamiltonian function does not depend explicitly on time, 4 equation (1.2.3) takes the well known form U(t f , ti) ≡ U(t f − ti) = exp −i(t f − ti)H , as the time ordering of operators is no longer necessary and the integration is straightforward5.

Now that the background is set, we can see how to put the propagator K(x f , t f ; xi, ti) in a path integral form. We will start analyzing the case of a time independent Hamiltonian H, for it happens to be the case in a very wide range of physical situations, in particular those we will treat in greater detail in the rest of the work; nevertheless afterwards we will show how to obtain a path integral propagator even in the case of a time dependent Hamiltonian, although not in the very detail.

1for a prove see [3]; for an analogous prove in relativistic quantum mechanics, i.e. for Klein-Gordon and Dirac equations, see [4]. 2 from now on we will set }h = 1. 3See [5, 6] for ulterior details. 4the two-parameter family of unitary operators U(s, t) reduces in that case to the unitary one-parameter group U(t), usual representation of time-translations group on a Hilbert space . 5H does not bear any time dependence and could be carried out. H

10 1.2. A MODERN FORMULATION CHAPTER 1. PATH INTEGRALS

1.2.1: Time-independent Hamiltonian

In that case relation (1.2.2) becomes:

−i(t f −ti)H K(x f , t f ; xi, ti) = hx f |e |xii . (1.2.5)

For the sake of simplicity we will consider an usual Hamiltonian H(x, p) = 0(p) + V(x), T where 0 is a quadratic function of only the momenta and V is an arbitrary function of the coordinatesT 1. Now we can use a formula introduced by H.F.Trotter [7]2:

A+B 1 A 1 B n e = lim (e n e n ) (1.2.6) → n ∞ Trotter Product that in our case becomes Formula

−i(t f −ti)[ 0(p)+V(x)] −ie 0(p) −ieV(x) −ie 0(p) −ieV(x) e T = lim (e T e ) ··· (e T e ) , (1.2.7) e→0 | {z } n−1 times with e = (t f − ti)/(n − 1). Now using the two decompositions of identity operator 1x = R | ih | 1 = R dp | ih | dx x x and p (2π)D p p (the subscript x or p has no meaning than to distinguish between different decompositions), and deﬁning xn ≡ x f , x0 ≡ xi , we obtain:

−ie 0(p) −ieV(x) −ie 0(p) −ieV(x) K(x f , t f ; xi, ti) = limhxn| e e ··· e e |x0i (1.2.8) e→0 ↓ T ↓ ↓ T 1 1 · 1 1 · 1 pn xn−1 pn−1 x1 p1 acting with exp[−ie 0(p)] on hp| and with exp[−ieV(x)] on |xi T n n n−1 Z dp −ie ∑ 0(pk)+V(xk−1) k = = lim dxk e k 1 T hxn|pni · · · hp1|x0i e→0 ∏ ( )D ∏ k=1 2π k=1

ﬁnally recalling that hx|pi = hp|xi∗ = expi(p · x)

n h ·( − ) i Z n n−1 pk xk xk−1 − ( )+V( ) dp ie ∑ e 0 pk xk−1 k = = lim dxk e k 1 T . (1.2.9) e→0 ∏ ( )D ∏ k=1 2π k=1

We have obtained a result for the integral kernel K(x f , t f ; xi, ti) which is formally equivalent to the one formulated by Feynman in (1.1.2), despite the fact that it is deﬁned in phase space rather than in conﬁguration space. However we will see that in most cases3 the integral in the momenta can be easily performed reducing effectively (1.2.9) to (1.1.2).

1we will analyze later the case of more general Hamiltonian functions. 2this formula is discussed in Appendix A. However, as better explained in the appendix, it gives us a valid approximant only to ﬁrst order, and for our particular needs it turns out to be sufﬁcient; however if we wanted a more precise approximation (with respect to some parameter, perhaps the commutator itself), we had to use the Zassenhaus formula[8, 9]:

λ(A+B) λA λB λ2 1 [B,A] λ3 1 [[B,A],A+2B] e = e e e 2 e 6 ··· .

See Appendix A of this part for further details. 3obviously, even in the particular case we are analyzing.

11 Part I: FLAT SPACE-TIME

Our result, eq. (1.2.9), can be put in a much more elegant form effectively taking the

limit e → 0. If we consider the paths in phase space x(t) and p(t), such that x(tk) ≡ xk and p(tk) ≡ pk, they deﬁne a functional space, and in the limit the argument of the exponential in (1.2.9) becomes just an integral over t1:

n h pk · (xk − xk−1) i e ∑ − 0(pk) + V(xk−1) k=1 e T n h i 2 = e ∑ p(tk) · x˙(tk) − H x(tk), p(tk) + O(e ) k=1 Z t f → dt p(t) · x˙(t) − Hx(t), p(t) . (1.2.10) ti

Furthermore, the integration over all the points x(tk) and p(tk) in the limit leads to a functional integration R [x(t)] [p(t)] over all the possible paths in phase space2,3 that start from D D Functional xi at ti and end at x f at t f . Eq. (1.2.9) then becomes an integral over all constrained paths Integration x(t), as well as over all unconstrained p(t)4: Z Z t f K(x f , t f ; xi, ti) = [x(t)] [p(t)] exp i dt p(t)x˙(t) − H x(t), p(t) . (1.2.11) D D ti x(ti)=xi x(t f )=x f

We discuss now the case of a general Hamiltonian operator of two variables x and p5 (that 6 can possibly contain mixed xi pj terms) . We would like to repeat the same procedure we have just done through Eq.s (1.2.5)– (1.2.11), but we cannot use Trotter formula (1.2.6) any-

more, since it is difﬁcult to treat exponentials of mixed terms xi pj, because we don’t know how to act on them with |pi and |xi, in order to extract the Hamiltonian function from the Hamiltonian operator. If it would be possible to manage expieH to have all the ps to the left of all the xs7, then we could repeat the procedure done in (1.2.8), and then continue in 8 (x,p) (x,p) the same previous way to obtain (1.2.11) .The big problem is that eO 6= : eO : , in 1about O(e2) in the second line, refer to Appendix A for a more detailed discussion on errors and approximations in path integrals. 2this integration is intended to be deﬁned by a regularization process, e.g.:

Z Z n n−1 dpk [x(t)] [p(t)] ≡ lim dxk (time-slicing). e→0 ∏ ( )D ∏ D D k=1 2π k=1 However time slicing is not the only regularization possible for this functional measure, as we will see in the following. 3There is in fact a mathematical notion of integration over paths due to Wiener[10, 11], but unfortunately this notion cannot be used in this case (we need to perform an analytic continuation in time, the Wick rotation, in order to use Wiener measure). 4that is why it is called path integral. 5in this general formulation, the variables x and p are two complete sets of independent observables, maybe not the ones related to position and momentum coordinates. 6that doesn’t represent a big deal as far as we limit ourselves to order e in the approximations, as it is done in most cases, however it becomes tougher to use more precise approximations (see Appendix A). 7we call such an operator ordering normal ordering, and it will be denoted by : . . . : , i.e. given an arbitrary operator (x, p), we will denote with : (x, p) : its normal ordered form, with all ps to the left of all xs. 8 O O considering the whole : exp ieH : instead of exp ie 0 exp ieV . T

12 1.2. A MODERN FORMULATION CHAPTER 1. PATH INTEGRALS

: (x,p): : (x,p): particular either e O 6= : e O : ! But in the latter case, we are able to write a relation between the two different exponentials. In particular if the exponential depends on some parameter (e.g. e in exp{−ieH} ), the relation will be a power series of such parameter. So when we are dealing with a general time-independent Hamiltonian, all we have to do is to put it in normal ordered form H (x, p), and use the following relation[12]: N ∞ (−ie)n −ieH (x,p) = −ieH (x,p) − 2 Hn+2( )− Hn+2( ) e N : e N : e ∑ x, p : x, p : ; (1.2.12) n=0 (n + 2)! N N

ﬁnally, since we need to consider only terms up to order e in the limit e → 0, one can proceed −ieH (x,p) through Eq.s (1.2.8)–(1.2.9) using : e N : to obtain Z Z t f Path Integral for K(x f , t f ; xi, ti) = [x(t)] [p(t)] exp i dt p(t)x˙(t) − H x(t), p(t) . (1.2.13) time-independent t D D i N Hamiltonians x(ti)=xi x(t f )=x f

1.2.2: Time-dependent Hamiltonian

The most general Hamiltonian function could bear an explicit time dependence, so it would dH ∂H be not a constant of the motion anymore, since dt = ∂t . Nevertheless we know that evolution operator obeys Dyson series (1.2.3), and we will show that a path integral expression of the propagator is still possible1. Consider an Hamiltonian function in normal ordering H(x, p, t) bearing an explicit time dependence. The integral kernel we have to calculate has the form:

K(x f , t f ; xi, ti) = hx f |U(t f , ti)|xii . (1.2.14)

Up to this point we need to describe the two-parameter family of unitary operators U(t, s) a little more in depth. We know that for a time-independent Hamiltonian it reduces to a one-parameter family of unitary operators that satisﬁes the properties of a group, and being time its parameter it turns out to be the unitary representation of time translations group. We would like to know the properties of the two-parameter family U(t, s) as well.

Deﬁnition 1.2.1 (Unitary Propagator). A two-parameter family of unitary operators A(s, t), s , t ∈ R which satisﬁes:

(i) A(r, s)A(s, t) = A(r, t)

(ii) A(t, t) = 1

(iii) A(s, t) is jointly strongly continuous in s and t is called a unitary propagator.

1this derivation follows the one given in [5].

13 Part I: FLAT SPACE-TIME

Theorem 1.2.1 (the Dyson expansion1). Let t → H(t) be a strongly continuous map of R into the bounded self-adjoint operators on a Hilbert space . Then there is a unitary propagator U(t, s) on so that, for all ψ ∈ , H H H

ϕs(t) = U(t, s)ψ

satisﬁes

d ϕ (t) = −iH(t)ϕ (t) , ϕ (s) = ψ . dt s s s Unfortunately in most cases the Hamiltonian H(x, p, t) does not satisfy the conditions of Theorem 1.2.1, since it often has an unbounded kinetic part. Nevertheless time dependent

Hamiltonians physically relevant are usually of the form H(t) = H0 + V(t), where both H0 and V(t) are normal ordered operators depending on coordinates and momenta, and H0 is possibly unbounded, while V(t) satisﬁes the conditions of Theorem 1.2.1. In such a case we 2 are able to express K(x f , t f ; xi, ti) in a functional integral form . In fact we can use the last theorem, passing to the “interaction representation”. We deﬁne

− Ve(t) = eiH0tV(t)e iH0t ; Evolution in a time-dependent system then t → Ve(t) also satisﬁes the hypotheses of Theorem 1.2.1, and we call the corresponding propagator Ue(t, s). We recall that it is determined by a Dyson series (see Appendix B for a proof):

∞ Z t Z t1 Z tn−1 n iH0t1 −iH0(tn−1−tn) −iH0tn Ue(t, s) = 1 + ∑ (−i) ··· dt1 ··· dtn e V(t1) ··· e V(tn)e . n=1 s s s (1.2.15)

Now setting

− U(t, s) = e itH0 Ue(t, s)eisH0 , (1.2.16)

we see that, at least formally, U(t, s) satisﬁes

d −itH0 isH0 −itH0 isH0 U(t, s) = −iH0e Ue(t, s)e + e −iVe(t) Ue(t, s)e dt = −i H0 + V(t) U(t, s) ,

so ϕs(t) = U(t, s)ψ should be a strong solution of

d ϕ (t) = −i H + V(t) ϕ (t) , ϕ (s) = ψ . dt s 0 s s

1for a proof, and a discussion on its implications see Appendix B. 2see Appendix B for the most general case.

14 1.2. A MODERN FORMULATION CHAPTER 1. PATH INTEGRALS

The problem arises from the fact that H0 U(t, s)ψ may not take sense, since Ue(t, s)ψ may not be in the domain of H0 even if ψ is. Now a particular case of the general Theorem B.1.2 of Appendix B assures us that if t → [H0, V(t)] is strongly continuous, then ϕs(t) is a strong solution; however it is always a “weak” solution, in the sense that for any η ∈ D(H0), 1 (η, ϕs(t)) is differentiable and d (η, ϕ (t)) = −i(H η, ϕ (t)) − i(V(t)η, ϕ (t)) . dt s 0 s s

Now the evolution operator U(t f , ti), if exists, is indeed a unitary propagator, since Ue(t, s) is a unitary propagator. Property (i) in particular turns out to be very useful since it allows us to write, deﬁning t f ≡ tn , ti ≡ t0 :

U(tn, t0) = U(tn, tn−1) ···U(t1, t0) (1.2.17) t f − ti with t ≡ t − e , e = so obviously t > t − > ... > t . k k+1 n n n 1 0 So Eq. (1.2.14) becomes:

K(xn, tn; x0, t0) = hxn|U(tn, tn−1) ···U(t1, t0)|x0i . (1.2.18)

Now using the decompositions of the identity operator 1x and 1p deﬁned in section 1.2.1 we have:

All we have to do is evaluate hpk|U(tk, tk−1)|xk−1i , and put it back in (1.2.19). But expanding tk around tk−1 in Eq. (1.2.15) we obtain:

iH0tk−1 −iH0tk−1 2 Ue(tk, tk−1) = 1 − ie e V(tk−1)e + O(e ) ; then

−itk H0 itk−1 H0 U(tk, tk−1) = e Ue(tk, tk−1)e −i(tk−tk−1)H0 2 = e 1 − ieV(tk−1) + O(e )

= e−ieH0 e−ieV(tk−1) + O(e2) = e−ieH(tk−1) + O(e2) =: e−ieH(tk−1) : +O(e2) , recalling Eq. (1.2.12) in the last equality. So if we take the limit e → 0 we can forget about second-order e-terms, and acting with hpk| and |xk−1i on the normal ordered exponential we get:

−ieH(xk−1,pk,tk−1) limhpk|U(tk, tk−1)|xk−1i = lim e hpk|xk−1i ; (1.2.20) e→0 e→0 1 D(H0) is the domain of H0 and (η, ϕs(t)) is the scalar product between η, ϕs(t) ∈ . H

15 Part I: FLAT SPACE-TIME

ﬁnally inserting this result in (1.2.19) we obtain the discretized path integral:

n h ·( − ) i Z n n−1 pk xk xk−1 −H( ) dp ie ∑ e xk−1,pk,tk−1 k = K(xn, tn; x0, t0) = lim dxk e k 1 . (1.2.21) e→0 ∏ ( )D ∏ k=1 2π k=1

Following the same reasoning of (1.2.10), we obtain the ﬁnal path integral formula: Functional Integral Z Z t f for time-dependent K(x f , t f ; xi, ti) = [x(t)] [p(t)] exp i dt p(t)x˙(t) − H x(t), p(t), t . (1.2.22) t Hamiltonians D D i x(ti)=xi x(t f )=x f

We see that this formula is perfectly analogous to Eq. (1.2.11), provided with the right time dependence of the Hamiltonian function.

1.2.3: Transition to Lagrangian formulation

We see that in Equations (1.2.11), (1.2.13) and (1.2.22) the integrand in the exponential looks like the Lagrangian function associated with the Hamiltonian H; path integral thus becoming the integral over pathsL of the exponential of i1 times the classical action associated with the system. But this appearance is misleading, since the momenta p(t) are independent variables, not yet related to x(t) or their derivatives. However, if the Hamiltonian is a quadratic function of the p(t), then the integral over this variables can be performed just substituting them with their values dictated by canonical formalism, thus the integrand in the exponential really is the Lagrangian function[13]. Let’s see how this passage is performed: we deal with a quadratic form of the p(t), i.e.2

1 H(x(t), p(t), t) = Aij[x(t), t]p (t)p (t) + Bi[x(t), t]p (t) + C[x(t), t] ; 2 i j i so the exponential becomes 1 ij i exp −i pi pj + pi + , 2A B C

where we have used a shorthand notation where an index like i stands for a multi-index, both discrete and continuous, and the contraction of equal indices stands for summation over discrete indices and integration over continuous ones3 and

ij ≡ Aij[x(t), t]δ(t − t0) , A i ≡ Bi[x(t), t] − x˙i(t) , B Z ≡ dt C[x(t), t] . C

1 −1 (times }h ). 2making explicit the components of the x(t) and p(t) vectors, and using Einstein summation convention. 3see Appendix C for further informations.

16 1.3. APPLICATIONS: FREE THEORIES, HARMONIC OSCILLATORS & PERTURBATIVE THEORY CHAPTER 1. PATH INTEGRALS

R 1 ij i Now we know how to solve Gaussian integrals like [p(t)] exp −i 2 pi pj + pi + D A B 1 −1 j : it is sufficient to substitute to the variables pi their value p¯i ≡ − ij , i.e. the sta- C A B tionary point of the exponent2. But the stationary point of R dt p(t)x˙(t) − Hx(t), p(t), t obeys the equation: δH x(t), p(t), t x˙i(t) = , δpi(t) p=p¯ i.e. the usual formula dictated by canonical formalism. So performing the integration over momentum paths we see that the exponent of Equations (1.2.11) , (1.2.13) and (1.2.22) is indeed i times the classical action of the system; therefore finally we can write the path integral in configuration space3: Z Z t f Path Integral in K(x f , t f ; xi, ti) = [x(t)] exp i dt x(t), x˙(t), t . (1.2.23) Lagrangian t D i L formulation x(ti)=xi x(t f )=x f

However this derivation is just formal and does not take into account the factor 1 r , h i [x(t)] i Det A2π that contains a functional determinant that has to be suitably managed. In particular if does not depend on x(t), such a determinant could be adsorbed in the definition of the mea-A sure [x(t)], however if it depends on x(t) could give rise to modifications to the Lagrangian Integrating out D momenta is a that one has to take into account, in particular we will see that in curved space-time the touchy business integrationL of momenta modifies introducing a counterterm that depends on the regularization scheme we have chosen toL properly define the functional measure [x(t)] [p(t)]. D D 1.3: Applications: Free Theories, Harmonic Oscillators & Perturbative Theory

t is now time to explicit our general results in some useful cases. Let’s start from the I simplest case: a system consisting of n free4 particles of equal mass m in a N dimensional 1see Appendix C 2the value p¯ obeying:

δ 1 ij i −i pi pj + pi + = 0 ; δpi 2 A B C p=p¯ where the derivative is a functional derivative. 3we present here only the most general case of a time-dependent Lagrangian function, the other are particular cases. 4as free particles we mean distinguishable particles that don’t interact neither mutually, neither with some external potential.

17 Part I: FLAT SPACE-TIME

space. In that case we can describe our system with D ≡ nN coordinates and momenta i.e. with a D-dimensional conﬁguration space described by a Lagrangian function

m 2 f ree x(t), x˙(t) = x˙(t) (1.3.1) L 2 where we use a shorthand notation for a D-dimensional vector

D x(t)2 ≡ ∑ xi(t)2 , i=1

and the dot stands for a derivative with respect to time as usual. Before dealing with the path integral of such a system we will see that if the Lagrangian

function is a quadratic form, the kernel K(x f , t f ; xi, ti) is proportional to the exponential of i times the action evaluated on the classical path, i.e. on the path satisfying the classical equations of motion and the boundary conditions1: consider a general quadratic Lagrangian function

1 i j i j i j i i = (aijx˙ x˙ + bijx x ) + cijx x˙ + dix˙ + eix , L 2 where the implicit time dependence of coordinates and their time derivatives has been omitted, and a, b, c are D × D dimensional time-dependent matrices and d, e are D-dimensional time-dependent vectors, in the usual component notation with summation over repeated in-

dices. Now it is immediate to see that aij = aji and bij = bji , i.e. both a and b are symmetric matrices2. With matrix c we have to be more careful: a priori it has not a deﬁnite symmetry, but as usual we can split it in a symmetric and an antisymmetric part deﬁning

s a cij = cij + cij ,

with

c + c cs ≡ ij ji , ij 2 c − c ca ≡ ij ji ; ij 2

s i j but the term cijx x˙ can be written as a total derivative (that we can cancel out of the La- i j grangian, not contributing to the equations of motion) plus a term with the structure αijx x that can be adsorbed redeﬁning bij in a suitable way:

d 1 1 cs xix˙ j = cs xixj − c˙s xixj . ij dt 2 ij 2 ij 1the action evaluated on the classical path is not to be confused with the classical action: the latter is the action of a classical system corresponding to the quantum mechanical Hamiltonian function of the system under examination; the former is the classical action evaluated on the path dictated by classical equations of motion and boundary conditions. 2in fact if they have not a deﬁnite symmetry, they can be split in a symmetric and antisymmetric part, but the antisymmetric part contracted with the product of two coordinates (that is symmetric) vanishes.

18 1.3. APPLICATIONS: FREE THEORIES, HARMONIC OSCILLATORS & PERTURBATIVE THEORY CHAPTER 1. PATH INTEGRALS

i In an analogous way we can see that the term dix˙ equals a total derivative plus a term adsorbed in redeﬁning vector e; so ﬁnally a general quadratic Lagrangian can be written as

1 s i j s i j a i j i = (aijx˙ x˙ + bijx x ) + cijx x˙ + eix , (1.3.2) L 2 where the symmetry or antisymmetry of the matrices has been made explicit with indices s and a that will be henceforward omitted. Equations of motion

d ∂ ∂ = dt ∂Lx˙i ∂Lxi read in that case

j j j j j i a˙ijx + aijx¨ − c˙ijx − 2cijx˙ − bijx − d = 0 . (1.3.3)

If we now perform a split x(t) = xcl(t) + q(t), where xcl(t) is a vector satisfying equations of motion and the boundary conditions, i.e.1:

xcl(ti) = xi , xcl(t f ) = x f , j + j − j − j − j − i = a˙ijxcl aijx¨cl c˙ijxcl 2cijx˙cl bijxcl d 0 : Classical Path & Quantum and q(t) has vanishing boundary conditions, i.e.2: Fluctuations split

q(ti) = 0 , q(t f ) = 0 ; we immediately see that the Lagrangian can be split as:

i j i j i j i j i (x, x˙) ≡ (xcl, x˙cl) + 2(q, q˙) + aijq˙ x˙cl + bijq xcl − cijq˙ xcl + cijq x˙cl + diq , L L L i where 2(q, q˙) ≡ (q, q˙) − diq (it is the purely quadratic part of the Lagrangian). In order to L L evaluate the kernel K(x f , t f ; xi, ti) we need the action S[x, t f , ti], i.e. the time integral between the initial and ﬁnal time of the Lagrangian function. But S[x, t f , ti] ≡ S[xcl, t f , ti] + S2[q, t f , ti] where

Z t f S2[q, t f , ti] = dt 2(q, q˙) : ti L in fact the time integral

Z t f i j i j i j i j i dt aijq˙ x˙cl + bijq xcl − cijq˙ xcl + cijq x˙cl + diq ti vanishes. In order to prove the last assertion is sufﬁcient to integrate by parts the q˙-terms, remembering that the q function vanishes at the boundaries and xcl satisﬁes equations of

1 xcl (t) is often called the “classical path”. 2q(t) is often called the “quantum ﬂuctuation”.

19 Part I: FLAT SPACE-TIME

motion. Finally since we usually deﬁne the functional measure [x(t)] to be invariant under translations, so that [x(t)] ≡ [q(t)], formula (1.2.23) becomesD in that case: D D Z Z t f iS[xcl,t f ,ti] K(x f , t f ; xi, ti) = e [q(t)] exp i dt 2 q(t), q˙(t), t , (1.3.4) D ti L VBC where VBC stands for vanishing boundary conditions. Now deﬁning Z Z t f [q(t)] exp i dt 2 q(t), q˙(t), t ≡ F(t f , ti) , D ti L VBC we see explicitly that the kernel is proportional to the exponential of the action on the classical path:

iS[xcl,t f ,ti] K(x f , t f ; xi, ti) = F(t f , ti) e . (1.3.5)

1.3.1: Systems of free particles

We will now calculate explicitly K(x f , t f ; xi, ti) in the case of a system of free particles. We have seen at the beginning of this section that such a system has in general a Lagrangian function:

m 2 f ree x(t), x˙(t) = x˙(t) (1.3.6) L 2 where x(t) is a D-dimensional vector. We have to make a sense out of the functional measure [x(t)], and in that simple case it is useful to remember the time discretization that led to ourD path integral formulation, so we will write

D N−1 m 2 N [x(t)] ≡ lim dxi , e→0 ∏ D 2πie i=1

t f −ti where dxi is the D dimensional Lebesgue measure, e ≡ N and the constant multiplying the measures is due to the integration over momenta (see section 1.2.3). The time discretized m xi−xi−1 2 1 action becomes e ∑ 2 ( e ) , so we have to evaluate the following expression :

Z N−1 D N N 2 m 2 m xi − xi−1 K(x f , t f ; xi, ti) = lim dxi exp ie . (1.3.7) e→0 ∏ ∑ i=1 2πie i=1 2 e But if we deﬁne D 2 m 2 imx F (x) ≡ exp , e 2πie 2e we see that with the usual convolution product2 Z ∞ Fe ∗ Fe0 (x) ≡ dy Fe(y)Fe0 (x − y) −∞ 1 identifying xN ≡ x f and x0 ≡ xi. 2being the Lebesgue measure translationally invariant, this deﬁnition is equivalent to: Z ∞ Fe ∗ Fe0 (x) ≡ dy Fe(y − a)Fe0 (x − y) , −∞ where a is an arbitrary factor.

20 1.3. APPLICATIONS: FREE THEORIES, HARMONIC OSCILLATORS & PERTURBATIVE THEORY CHAPTER 1. PATH INTEGRALS holds the following property: Fe ∗ Fe0 (x) = Fe+e0 (x) .(ﬁrst convolution property)

The demonstration of the (ﬁrst convolution property) involves some boring calculations completing squares and performing a Gaussian integration on y1. The usefulness of this property is seen considering the integration over dxi in Eq. (1.3.7), isolating xi-depending terms we see that this integration reduces to:

D D Z 2 2 m 2 m 2 im(x − x − ) im(x + − x ) dx exp i i 1 exp i 1 i = 2πie 2πie i 2e 2e remembering the deﬁnition of convolution product

= Fe ∗ Fe (xi+1 − xi−1) = F2e(xi+1 − xi−1) .

Now the integration over dxi+1 is perfectly analogous but involves the convolution product F2e ∗ Fe (xi+2 − xi−1); so it is straightforward to perform all consecutive N − 1 integrations, R 2 starting from dx1 , obtaining :

D 2 m 2 im(x f − xi) FNe(xN − x0) ≡ exp ; 2πi(t f − ti) 2(t f − ti) so ﬁnally

D 2 m 2 im(x f − xi) K f ree(x f , t f ; xi, ti) = exp . (1.3.8) 2πi(t f − ti) 2(t f − ti)

As we expected, the Kernel in Eq. (1.3.8) has the structure predicted in (1.3.5), being the 3 free Lagrangian effectively a quadratic form . Furthermore, we could have found K f ree n o starting from Eq. (1.3.5): we know that K f ree(x f , t f ; xi, ti) = F(t f , ti) exp iS[xcl, t f , ti] , so all we need to evaluate is F(t f , ti); but since Eq. (1.2.1) tells us that K obeys Schrödinger

1for examples on such a method as “completing squares” in Gaussian integration see Appendix C. 2 remembering that xN ≡ x f , x0 ≡ xi and Ne ≡ t f − ti. 3the solution of equations of motion

x¨ = 0 respecting the boundary conditions x(ti) = xi and x(t f ) = x f is

x f − xi xit f − x f ti x(t) = t + , t f − ti t f − ti so the action on the classical path is effectively

2 m (x f − xi) S[xcl, t f , ti] = . 2 (t f − ti)

21 Part I: FLAT SPACE-TIME

1 Equation , we can fix F(t f , ti) to be the value for which such equation is satisfied by K f ree. A straightforward calculation fixes obviously:

D m 2 F(t f , ti) = , 2πi(t f − ti)

as we expected.

1.3.2: The Harmonic Oscillator

Another system of physical interest is a system of n uncoupled and distinguishable Har- monic Oscillators of equal mass m, in N dimensions, that is characterized in a D = nN dimensional space by a Lagrangian function

m 2 2 2 osc(x(t), x˙(t)) = x˙(t) − ω x(t) , L 2

with usual vectorial notations. It is a discomforting task to evaluate directly K(x f , t f ; xi, ti), since the path satisfying equations of motion and these general boundary conditions is:

x x f tan(ωt ) + xi tan(ωt ) f − xi cos(ωt f ) i cos(ωti) f cos(ωt f ) cos(ωti) xcl(t) = cos(ωt) + sin(ωt) ; tan(ωt f ) − tan(ωti) tan(ωt f ) − tan(ωti)

but since m and ω are constants, the Lagrangian function doesn’t depend explicitly on

time, so time translations are a symmetry of the system, and therefore K(x f , t f ; xi, ti) ≡ 2 K x f , (t f − ti); xi, 0 , with the advantage that with these boundaries xcl(t) reduces to : x f − xi cos ω(t f − ti) xcl(t) = xi cos(ωt) + sin(ωt) . (1.3.9) sin ω(t f − ti)

The action evaluated on Eq. (1.3.9) is

mω 2 2 Sosc[xcl, (t f − ti)] = (x f + xi ) cos ω(t f − ti) − 2x f · xi . (1.3.10) 2 sin ω(t f − ti)

Since as the free Lagrangian function, osc is a quadratic form, we can use formula (1.3.5) and evaluate only L

Z Z (t f −ti) m 2 2 2 F(t f − ti) = [q(t)] exp i dt q˙(t) − ω q(t) . D 2 0 VBC

In order to do that we will use a particular technique, called “mode expansion”: the idea is quite simple, we will expand paths q(t) in a Fourier series, and deﬁne the integration over all possible paths as the integration of the Fourier coefﬁcients over all their possible values,

1 since K(x f , t f ; xi, ti) is the only x f and t f -dependent object on the right member of Eq. (1.2.1), and Schrödinger Equation has to be satisﬁed for ψ(x f , t f ). 2 we will limit ourselves to the case 0 < ω(t f − ti) < π in order to determine xcl (t) univocally.

22 1.3. APPLICATIONS: FREE THEORIES, HARMONIC OSCILLATORS & PERTURBATIVE THEORY CHAPTER 1. PATH INTEGRALS since every path vanishing at the boundaries could be written as a series of sines with real Fourier coefﬁcients1. So we perform the expansion

∞ q(t) = ∑ Cnφn(t) , n=1

D where Cn ∈ R and φn(t) is a complete set of orthonormal functions on [0, (t f − ti)], vanishing at the boundaries, i.e. the sine functions s 2 nπt φn(t) = sin . t f − ti t f − ti

A simple integration by parts allow us to rewrite the action S2[q, (t f − ti)] as a function of the Fourier coefﬁcients, precisely

∞ m Z (t f −ti) m nπ 2 S [q, (t − t )] ≡ dt q˙(t)2 − ω2q(t)2 = C2 − ω2 . 2 f i ∑ n − 2 0 2 n=1 t f ti So, setting

N [q(t)] = M lim dCn , N→∞ ∏ D n=1 we need to calculate

Z N N 2 m 2 nπ 2 F(t f − ti) = M lim dCn exp i Cn − ω , N→∞ ∏ ∑ − n=1 2 n=1 t f ti after performing N Gaussian integrations we obtain

− D D N 2 2 2 0 ω(t f − ti) 0 ω(t f − ti) F(t f − ti) = M lim 1 − = M ; N→∞ ∏ ( − ) n=1 nπ sin ω t f ti

ND 0 t f −ti 2πi 2 where M ≡ nπ m M and we have used the formula

sin x N x 2 = lim 1 − . N→∞ ∏ x n=1 nπ In order to ﬁx M0, we remember that in the limit ω → 0 the system under consideration reduces to a system of free particles, that has a well known kernel K f ree that we calculated in the previous subsection. But since

ω→0 Sosc[xcl, t f , ti] −−→ S f ree[xcl, t f , ti] , it has to be

ω→0 Fosc(t f − ti) −−→ F f ree(t f , ti) ; 1more precisely we will limit to say that the functional measure [q(t)] is proportional to the measure of all Fourier coefﬁcients, the proportionality factor being ﬁxed later to satisfyD the free particles limit ω → 0.

23 Part I: FLAT SPACE-TIME

so being

ω→0 0 Fosc(t f − ti) −−→ M ,

0 1 we can identify M = F f ree(t f , ti), so ﬁnally we can write

D mω 2 mω K (x , t ; x , t ) = exp i (x2 + x2) osc f f i i 2πi sinω(t − t ) 2 sinω(t − t ) f i f i f i (1.3.11) · cos ω(t f − ti) − 2x f · xi .

1.3.3: Perturbative Methods

Unfortunately it is quite difﬁcult to explicitly solve path integrals for more complicated systems, but we will see soon that we can develop a quite powerful tool to calculate perturbative expansions with path integrals introducing the so-called “correlation functions”. We recall that there are two equivalent realizations of time evolution on a Hilbert Space : one is often referred to as Schrödinger picture, the other as Heißenberg picture. The for- merH makes the states, i.e. the elements of Hilbert Space, carry any implicit time dependence (in fact they have to obey Schrödinger equation) while operators might have only explicit time dependence2; in the latter elements of are constant in time, and operators carry all time dependence3. Now, since they must describeH the same physical system, they have to be

Heißenberg & related by a unitary transformation, so S and H would be isomorphic. This is possible if H H Schrödinger time evolution is unitary, i.e. realized by a unitary operator U(t, s) ∈ L( ). As explained in pictures Appendix B, this happens in most cases of physical interest. Furthermore,H if time evolution is not unitary, the entire Schrödinger picture would be almost unmanageable, since in fact

it consists in a mapping t → t, where t ∈ R is time, and t is the Space of states associ- H H ated with the system at time t; now since the evolution is not unitary t and t0 could be H H non-isomorphic, or worse even if t is a Hilbert Space, t0 could not! In fact in cases where we areH not able to tell whetherH time evolution is unitary or not we must rely on Heißenberg picture, as in the case of interacting Quantum Field Theories, since in that picture the Hilbert Space describing the system is constant through time. If time 4 evolution is unitary, we can use U(t, s) to associate H and S(t) , with arbitrary t ∈ R; H H 1 being, as we have already seen, Kosc(x f , t f ; xi, ti) ≡ Kosc(x f , (t f − ti); xi, 0). 2in fact dA (t) ∂A (t) S = S , dt ∂t

where AS is an arbitrary operator in L( S). 3we recall that their time evolutionH is given by dAH(t) ∂AS(t) = − i[AH(t), HH(t)] , dt ∂t H

where AH is an arbitrary operator in L( H), and HH is the Hamiltonian function. 4 H0 0 0 since time evolution is unitary, S(t ) ≡ U(t , t) S(t) is isomorphic to S(t) for any t , t ∈ R, so S has the same structure at any time. H H H H

24 1.3. APPLICATIONS: FREE THEORIES, HARMONIC OSCILLATORS & PERTURBATIVE THEORY CHAPTER 1. PATH INTEGRALS

† in particular H ≡ U (t, t0) S(t), where t0 is a ﬁxed parameter. So S(t0) ≡ H, while H H H H any other S(t) is isomorphic to it. L( H), the family of linear operators in Heißenberg H H picture is related to L( S) as usually done when we perform a unitary transformation, i.e. † H L( H) ≡ U (t, t0)L( S)U(t, t0). H H So we immediately see that K(x f , t f ; xi, ti) ≡ hx f (t f )|U(t f , ti)|xi(ti)iS is the transition element between position eigenstates in Heißenberg picture:

† K(x f , t f ; xi, ti) = hx f (t f )|U(t f , t0)U (ti, t0)|xi(ti)iS = hx f , t f |xi, tiiH .

Now on we will use Heißenberg picture, omitting index H, so K(x f , t f ; xi, ti) = hx f , t f |xi, tii. Up to this point, we would like to be able to calculate hx f , t f |O(t1)|xi, tii, where O(t1) ∈ L( ) and t1 ∈ [ti, t f ], with a path integral. This is easily done: H hx f , t f |O(t1)|xi, tii = hx f (t f )|U(t f , t1)O(t1)U(t1, ti)|xi(ti)iS Z 0 0 0 = dxdx K(x f , t f ; x, t)hx(t)|O(t1)|x (t)iSK(x , t; xi, ti) ; if O(t)S is diagonal in the coordinates representation, i.e.

0 D 0 hx(t)|O(t1)|x (t)iS = O(x(t1), t1)δ (x − x ) , then above expression reduces to a single path integral1: Z iS[x,t f ,ti] hx f , t f |O(t1)|xi, tii = [x(t)] e O[x(t1)] . (1.3.12) D x(ti)=xi x(t f )=x f This construction can be generalized to a time-ordered product of operators n o T O1(t1)O2(t2) ··· with t1, t2,... ∈ [ti, t f ] , calculation of obtaining, if the operators are all diagonal in the coordinates representation, transition elements with path integrals n o Z iS[x,t f ,ti] hx f , t f |T O1(t1)O2(t2) ··· |xi, tii = [x(t)] e O1[x(t1)]O2[x(t2)] ··· . (1.3.13) D x(ti)=xi x(t f )=x f

i If O1(t1) ≡ x 1 (t1), and so on, formula (1.3.13) reduces to: n o Z i1 i2 iS[x,t f ,ti] i1 i2 hx f , t f |T x (t1)x (t2) ··· |xi, tii = [x(t)] e x (t1)x (t2) ··· ; D x(ti)=xi x(t f )=x f so we can define the n-point propagator, or n-point correlation function2 n o i in hx f , t f |T x 1 (t1) ··· x (tn) |xi, tii i1 in h x (t1) ··· x (tn) i ≡ (1.3.14) hx f , t f |xi, tii 1 to make this possible it is necessary that t ∈ [ti, t f ], in order to identify the integration over x as one of the slices of the path at an intermediate time t. 2with this definition the zero point correlation function, i.e. h 1 i = 1; but we will use a different notation n o i1 in i1 in here, we define h 1 i ≡ hx f , t f |xi, tii: so hx f , t f |T x (t1) ··· x (tn) |xi, tii = h 1 ih x (t1) ··· x (tn) i.

25 Part I: FLAT SPACE-TIME

through a path integral. We will see in a moment that there is a beautiful way to calculate such correlations functions with n arbitrary knowing only the 2-point propagator. Deﬁne a “generating functional”

Z R i i S[x,t f ,ti]+ dt Ji(t)x (t) Generating Z[J] = [x(t)] e , (1.3.15) Functional D x(ti)=xi x(t f )=x f

where the “sources” J(t) are D-dimensional vectors. It is immediate to see that the n-point propagator can be written in terms of functional derivatives of generating functional1:

1 δ δ h xi1 (t ) ··· xin (t ) i = (−i)n ··· Z[J] . (1.3.16) 1 n [ ] ( ) ( ) Z J δJi1 t1 δJin tn J=0

t t Now if S[x, t , t ] reduces to S [x, t , t ] = − R f R f dtdt0 1 K (t, t0)xi(t)xj(t0), where K (t, t0) f i 2 f i ti ti 2 ij ij is a differential operator, Z[J] reduces to a Gaussian Integral2,3

Z 1 i j i 1 h iK i 1 hi i i −1 ij −i Kij x x −Ji x − − (K ) Ji Jj Z[J] = [x(t)] e 2 = Det 2 Det 2 A e 2 . D 2π 2π x(ti)=xi x(t f )=x f

So we immediately deduce by derivation the so called “Wick Theorem”, i.e. that any odd- point propagator must vanish, and any even-point propagator is calculated by means of all possible products of 2-point propagators4:

i h x i2 = 0 ,

i1 i2 −1 ij h x x i2 = −i(K ) ,

i1 in+1 h x ··· x i2 = 0 , (Wick Theorem) i1 i2n −1 ip1ip2 −1 ip(2n−1)ip2n h x ··· x i2 = ∑ −i(K ) ··· −i(K ) . permutations of i1...i2n

Unfortunately only when we deal with quadratic Lagrangian functions we can write the t action as S [q, t , t ] = − R f dt 1 K (t)qi(t)qj(t), following the method described at the be- 2 f i ti 2 ij ginning of Section 1.3. Nevertheless we can use these tools in the case of a general action

S[x, t f , ti] if we are able to write it as S2[x, t f , ti] + Sint[x, t f , ti]: in fact K(x f , t f ; xi, ti) thus

1the functional derivative is deﬁned (in the unidimensional case, the generalization is straightforward) as the following: given a functional F[ f (x)] its functional derivative is the fraction

δF[ f (x)] F[ f (x) + eδ(x − y)] − F[ f (x)] ≡ lim . δ f (y) e→0 e

2using the usual compact multi-index notation. 3 −1/2h i i see Appendix C and Section 1.2.3 for an explanation of the factor Det 2Aπ . 4 we have used 2 to make explicit that this Theorem holds if the averages are calculated with an action S2. Since now on we will always calculate averages of that kind, the index 2 will be omitted.

26 1.3. APPLICATIONS: FREE THEORIES, HARMONIC OSCILLATORS & PERTURBATIVE THEORY CHAPTER 1. PATH INTEGRALS becomes Z iS2[x,t f ,ti] iSint[x,t f ,ti] iSint[x,t f ,ti] K(x f , t f ; xi, ti) = [x(t)] e e = h 1 ih e i ; D x(ti)=xi x(t f )=x f and since the exponential can be expanded in a power series, and the average of a sum is the sum of the averages we obtain:

∞ n i n K(x f , t f ; xi, ti) = h 1 i 1 + ∑ h Sint i . n=1 n! If this series depends on a small parameter, this expression could be seen as the perturbative expansion we sought1. We stress that with this construction all we need to know to perform perturbative calculations is the 2-point propagator, i.e. the inverse of the kinetic operator 0 Kij(t, t ) appearing in S2[x, t f , ti].

1.3.4: Some Perturbative Calculations

Let’s see how to apply the powerful tools of the previous section to some simple case. In fact we will use generating functionals, propagators and perturbative expansions almost everywhere throughout this work, but some simple example will help us get acquainted with such techniques. Let’s start with a well known problem, a system of uncoupled harmonic oscillators: we will calculate the 2-point propagator for such a theory (recall that Sosc is a S2-type action), in a particular case. It is particularly useful to analyze this propagator even if we already solved the path integral for such a system, because a quite crucial problem will emerge quite naturally in performing that calculation, related with the impossibility to define coherently path integrals in real time without recurring to analytic continuation of calculations in imaginary time. We already encountered a first problem: path integrals in real time involve “imagi- 2 nary” Gaussian integrals, like R [p]eip /2, also known as Fresnel integrals, that are analytic 2 continuations of well-known realD Gaussian integrals, like R [p]e−p /2, see Appendix C for D further details. This underlies a deeper problem: we are not able to define a functional Euclidean time & measure in the space of paths in real time due to oscillatory factors eiS, but we can do that Wick rotation if we continue time analytically to purely imaginary values; this procedure is called “Wick Rotation”, and consists in a redefinition of time t → −it: with that procedure the oscillatory −S factor becomes a real exponential e E , where SE is the so-called Euclidean action, and we can define Wiener measures in path space in such a case. So if we want to be more rigorous and solve ambiguities, we have to perform calculations in Euclidean (imaginary) time, and then “rotate back” to real time through analytic continuation. Returning to the propagator for a theory of harmonic oscillators, we will see that in real time we have poles on the integration path, and we have an ambiguity on how to “pass by”

1 recall that such a parameter could be }h that we set to 1 throughout our derivation but else would have −n appeared as a factor }h in the perturbative expansion; also a suitable choice of t f − ti could serve as the small parameter as it appears in the power series (only if the Hamiltonian is time-independent).

27 Part I: FLAT SPACE-TIME

them, but in Euclidean time there are no poles on the integration path so the ambiguity is solved, and analytic continuation of the result gives us back the real time propagator. We 1 are not able to perform calculations in the case of an arbitrary Kosc(x f , t f ; xi, ti) , so we will R restrict to the calculation of Z(t f , ti) ≡ dx hx, t f |x, tii: hx, t f |x, tii is a path integral, with a periodic boundary condition x(t f ) = x(ti) = x, and such boundary value is then integrated over all its possible values; this integration over the boundary value is equivalent to modify the space of paths we are considering in defining the functional measure: we pass from integration over all possible paths with fixed endpoints to integration over all possible paths with the initial point equivalent to the final one, and we will call this constraint “Periodic Boundary Conditions”, or PBC. So we can write Z iS[x,t f ,ti] Z(t f , ti) = [x(t)] e . (1.3.17) D PBC It is straightforward that all the considerations about generating functionals, propagators

etc. we did for Kosc(x f , t f ; xi, ti) remain valid for Z(t f , ti): it is sufficient to substitute in any formula fixed endpoints conditions with PBC. We will see the usefulness of Z immediately as we will consider the Euclidean case. By now we will limit ourselves to find the 2-point propagator for Z in the case of Harmonic Lagrangian function. First of all since time translation is a symmetry of Harmonic systems, we can write: Z iS[x, β ,− β ] Z(t f , ti) ≡ Z(β) = [x(t)] e 2 2 with β ≡ t f − ti . D PBC Then we note that periodic boundary conditions allow us to integrate by parts in the action, since the boundary term vanishes, so we can write Harmonic action, using multi-index notation as usual: 1 S = K xixj , osc 2 ij ( 0) = ( d2 + 2) ( − 0) with Kij t, t m dt2 ω δ t t δij. So in order to find the propagator all we have to do 0 is find the inverse of Kij(t, t ); we have to solve the distributional equation Z 00 00 −1 jk 00 0 k 0 dt Kij(t, t )(K ) (t , t ) = δi δ(t − t ) ,

i.e.: Z d2 d2 dt00 mδ δD(t − t00)( + ω2)(K−1)jk(t00, t0) = mδ ( + ω2)(K−1)jk(t, t0) = δ kδ(t − t0) , ij dt2 ij dt2 i δjk ⇒ (K−1)jk(t, t0) = K−1(t − t0) , m

1because even if the Lagrangian function of such a system is a quadratic function, we are not able to find a solution for the inverse of the Harmonic kinetic operator that satisfies either general boundary conditions or vanishing ones, as we would have to if we wanted to solve the general case. We will see that for a free system we know a solution of the inverse equation that satisfies vanishing boundary conditions, so we are able to find the propagator for a general Kernel.

28 1.3. APPLICATIONS: FREE THEORIES, HARMONIC OSCILLATORS & PERTURBATIVE THEORY CHAPTER 1. PATH INTEGRALS here we guess that since K(t, t0) ≡ K(t − t0) the same property holds for K−1(t, t0); if we can ﬁnd a solution we prove that guess is right, since the solution is unique. So we have to seek a solution of:

d2 ( + ω2)K−1(x) = δ(x) , dx2 so if we consider the limit β → ∞ we can apply Fourier transform, so the equation for Kˆ −1(k) reads

−(k2 − ω2)Kˆ −1(k) = 1 , 1 ⇒ Kˆ −1(k) = − , k2 − ω2 Z dk 1 ⇒ K−1(x) = − eikx ; 2π k2 − ω2 here we see that there are poles on the integration path, as we anticipated, and we don’t know what is the right prescription to manage these singularities. We are thus forced to perform calculations in Euclidean time and recover the real time result through analytic continuation. But what is the difference between real and Euclidean time calculations? Since t S[x, t , t ] = R f dt (x(t), x˙(t), t) we have, performing the transformation t0 = it: f i ti L

Z t0 f 0 0 0 0 iS[x, t , t ] = i d(−it ) (x(t ), ix˙(t ), −it ) = −SE[x, t , t ] f i 0 f i ti L

t where S [x, t , t ] = − R f dt (x(t), ix˙(t), −it), recalling t0 = t. So Euclidean time path E f i ti integral becomes L

Z −SE[x,t f ,ti] K(x f , t f ; xi, ti) = [x(t)] e , (1.3.18) D x(ti)=xi x(t f )=x f Z −SE[x,t f ,ti] Z(t f , ti) = [x(t)] e ; (1.3.19) D PBC and it is straightforward to generalize generating functionals, correlation functions and

29 Part I: FLAT SPACE-TIME

Wick Theorem1 in Euclidean time:

Z R i Z[J] = [x(t)] e− SE[x,t f ,ti]− dt Ji(t)x (t) , (1.3.20) D boundaries

1 δ δ h xi1 (t ) ··· xin (t ) i = ··· Z[J] , (1.3.21) 1 n [ ] ( ) ( ) Z J δJi1 t1 δJin tn J=0

i1 i2 −1 ij h x x i2 = (K ) ,

i1 in+1 h x ··· x i2 = 0 , (Eucl. Wick Th.) i1 i2n −1 ip1ip2 −1 ip(2n−1)ip2n h x ··· x i2 = ∑ (K ) ··· (K ) . permutations of i1...i2n In the case of Harmonic oscillators Euclidean action becomes:

β m Z 2 S [x, t − t ] = dt (x˙(t)2 + ω2x(t)2) , E,osc f i β 2 − 2 2 = 1 i j ( 0) = (− d2 + so integrating by parts we obtain the usual form S2 2 Kijx x with Kij t, t m dt2 2 0 ω )δ(t − t )δij. Now repeating the same procedure we did in real time for the propagator of Z(β) we ﬁnd that

δjk (K−1)jk(t, t0) = K−1(t − t0) m with Z dk 1 K−1(x) = eikx ; 2π k2 + ω2 it’s clear that we don’t have any pole in the integration path, so a usual integration in the complex plane gives us

ij δ 0 (K−1)ij(t, t0) = e−ω|t−t | . 2mω Clearly the Euclidean propagator is

ij δ 0 h xi(t)xj(t0) i = e−ω|t−t | , E 2mω the real time propagator is given by analytic continuation of the Euclidean one3:

ij δ 0 h xi(t)xj(t0) i = e−iω|t−t | . R 2mω 1 R 0 1 0 i j 0 we recall that in Euclidean time we have SE2 = dtdt 2 Kij(t, t )x (t)x (t ). 2in the calculation of Z, where the integration by parts is allowed without boundary terms. 3in that case absolute value is used to write the propagator in a compact way, but when making analytic continuation we have to consider separately the cases t − t0 < 0 and t − t0 > 0, and then restore the compact notation.

30 1.3. APPLICATIONS: FREE THEORIES, HARMONIC OSCILLATORS & PERTURBATIVE THEORY CHAPTER 1. PATH INTEGRALS

We note that the result obtained through analytic continuation is the same we would have obtained with the so-called Feynman prescription Z dk 1 K−1(x) = − eikx 2π k2 − ω2 + ie for the treatment of poles on integration path. Let’s see now the physical meaning of Z(β) in Euclidean time: we have seen in Eq. (1.3.19) that it can be written in a path integral form, but if we remember its own deﬁnition1 and perform Wick rotation we obtain: Z Z(β) ≡ dx hx|e−βH|xi = Tr e−βH ; it is clear now that Z(β) is the partition function of a statistical system with an Hamiltonian H at temperature T = (kβ)−1, where k is the Boltzmann constant. So, at least in the asymptotic limit β → ∞, we know the 2-point propagator for the R β β partition function of Harmonic systems, and also h 1 i: calculating dx Kosc(x, 2 ; x, − 2 ), we obtain (performing Wick rotation in Eq. (1.3.11) and then integrating),

D 1 2 h 1 i = . 2(cosh(ωβ) − 1)

We have thus all we need to make perturbative calculations for a system with action SE = Sosc,E + Sint,E : consider for example a system of Harmonic oscillators subjected to a perturbation potential

i j k i j k l Vint(x) = gijkx x x + λijkl x x x x ; we can see the effect of that perturbation on the partition function for large β by means of R i j k i j k l a perturbative series in g and λ: in that case Sint,E = dt (gijkx x x + λijkl x x x x ), so we have2

β Z 2 Z (β)= h 1 i 1 − dt g h xi(t)xj(t)xk(t) i + λ h xi(t)xj(t)xk(t)xl(t) i + O(g2 + λ2) int β ijk ijkl − 2 but since odd-point propagators vanish we see that the only ﬁrst-order contribution to per- i j turbation is given by the quartic potential, in particular recalling that h x (t)x (t) iE = ij δ /2mω, that λijkl has to be completely symmetric in the exchange of two indices (the eventual antisymmetric part would vanish when contracted with four xs) and that h 1 i ≡

Zosc(β), we obtain:

i k 3λ i k 3βλi k 2 2 −β i k 2 2 Z (β) = Z (β) 1 − + O(g + λ ) = Z (β) e 4m2ω2 + O(g + λ ) , int osc 4m2ω2 osc 1in the case of a time-independent Hamiltonian, i.e. when time translation is a symmetry of the system and Z(t f , ti) ≡ Z(t f − ti, 0). 2remembering that in Euclidean time: Z ∞ (−)n − SE2+Sint,E) −Sint,E n [x(t)] e = h 1 ih e i = h 1 i 1 + ∑ h Sint,E i . D n=1 n! boundaries

31 Part I: FLAT SPACE-TIME

i k ij kl with λi k ≡ λijklδ δ . A useful application of this result is to calculate the ﬁrst order effect of the anharmonic perturbation on the ground-state energy of the system. In fact if we calculate the trace in the partition function using a basis of Hamiltonian eigenfunctions we obtain:

β→∞ Z( ) = h | −βH| i = −βEn = −βE0 + −β(En−E0) ∼ −βE0 β ∑n n e n ∑n e e 1 ∑n e e ;

so the ground-state energy shift is:

i k int osc 1 3 λi k ∆E0 = E0 − E0 = lim − ln(Zint/Zosc) = . β→∞ β 4 m2ω2

As a last simple example we will ﬁnd propagators for a free theory, and use them to make perturbative calculations in the case of a system of particles with the same mass subjected to an arbitrary external potential (using evolution time as the small parameter). In order to ﬁnd propagators we will use directly Euclidean time, and in that case the ac- [ ] = R β m ( )2 ≡ ( − ) tion reads S f ree,E x, β 0 dt 2 x˙ t , with β t f ti and recalling that since free La- grangian does not depend explicitly on time, time translation is a symmetry of the system

so K(x f , t f , xi, ti) ≡ K(x f , β; xi, 0). Since the free Lagrangian is also a quadratic function, we can do the usual xcl-q split, remembering that in the Euclidean case we obtain: Z −S[xcl,E ,t f ,ti] −SE2[q,t f ,ti] K(x f , t f ; xi, ti) = e [q(t)] e ; (1.3.22) D VBC

that reads in this case:

m (x − x )2 S[x , β] = f i , cl,E 2 β Z β 1 d2 [ ] = 0 (− ) ( − 0) i( ) j( 0) SE2 q, β dtdt mδij 2 δ t t q t q t . 0 2 dt

1 i j −1 ij So we recognize again the usual form S2 = 2 Kijq q . But now in order to ﬁnd (K ) we will proceed in a slightly different way: we perform the mode expansion we did in Section 1.3.2, 1 so we can perform the integration over t in SE2, obtaining a form

1 i j S2 = lim KijC C , N,P→∞ 2

where Kij depends only on discrete indices (it is a real matrix), and so it is much easier to invert. If one deﬁnes the Fourier expansion:

N s 2 nπt D q(t) = lim Cn sin where Cn ∈ R , (1.3.23) N→∞ ∑ n=1 β β

1 i i C ≡ Cn are Fourier coefﬁcients of expansion, recalling notation of Section 1.3.2.

32 1.3. APPLICATIONS: FREE THEORIES, HARMONIC OSCILLATORS & PERTURBATIVE THEORY CHAPTER 1. PATH INTEGRALS

1 the integration over t in SE2[q, β] is straightforward, obtaining the usual form

1 i j S2 = lim KijC C N,P→∞ 2 with C ∈ RDN and 2 2 np mπ n K = δ δnp , ij β2 ij so the inverse is immediately found as:

2 ij β (K−1) = δijδ , np mπ2n2 np and since integration over all possible paths vanishing at the boundaries is equivalent to an i integration over all Fourier coefﬁcients Cn, we can ﬁnd propagators of the Cs in a standard way obtaining:

2 j β h Ci C i = δijδ , n p E mπ2n2 np so using Eq. (1.3.23) we ﬁnd

N P 0 ij i j 0 2 πnt πpt i j δ 0 h q (t)q (t ) iE = lim sin sin h CnCp iE = ∆(t, t ) , (1.3.24) N,P→∞ ∑ ∑ n=1 p=1 β β β m with ∞ h 2β πnt πns i ( ) = ∆ t, s ∑ 2 2 sin sin . (1.3.25) n=1 π n β β In order to perform calculations, we need to know the distributional meaning of ∆(t, s) in the space of continuous functions on [0, β] vanishing at the boundaries, i.e. the algebra of continuous functions with compact support κ([0, β]). Since the propagator (1.3.24) has to ( − 0)(− d2 ) ( ) ∈ ([ ])∗ 2 be the inverse of the kinetic operator mδijδ t t dt2 we expect that ∆ t, s κ 0, β satisﬁes the equation

d2 − ∆(t, s) = δ(t − s) . (1.3.26) dt2 Let’s see directly if it is the case: ﬁrst of all we note that ∆(t, s) is symmetric in t and s, so if Eq. (1.3.26) holds, it holds even

d2 − ∆(t, s) = δ(t − s) ; ds2 denoting a derivation of ∆ with a •, to the left of ∆ if we derive with respect to the ﬁrst argument, to the right if we derive with respect to the second, we have: ∞ 2 πnt πns −••∆(t, s) = −∆••(t, s) = ∑ sin sin , (1.3.27) n=1 β β β 1the cutoff N in the Fourier expansion is used in order to make K a ﬁnite DN × DN dimensional matrix. 2κ([0, β])∗ is the dual space of κ([0, β]).

33 Part I: FLAT SPACE-TIME

is it δ(t − s)? Yes, indeed: the deﬁnition of δ(t − s) as a functional in κ([0, β])∗ is

Z β δ(t − s)[ f (t)] ≡ dt δ(t − s) f (t) = f (s) with f (t) ∈ κ([0, β]) ; 0

so if we search the coefﬁcients Dn(s) of a Fourier expansion s ∞ 2 πnt δ(t − s) = ∑ Dn(s) sin n=1 β β

we have to solve1 2 Z β πnt πpt ( ) = ( ) ∑n,p Dn s Fp dt sin sin ∑n,p Dn s Fpδnp β 0 β β s 2 πns ⇒ D (s) = sin , n β β ∞ 2 πnt πns ⇒ δ(t − s) = sin sin = −••∆(t, s) . Analysis of ∑ n=1 β β β Distributions in ∗ κ([0, β]) by So a general solution of (1.3.26), symmetric in t and s and in κ([0, β])∗ is: means of Fourier series 1 ∆(t, s) = [(β − t)sθ(t − s) + (β − s)tθ(s − t)] , β

where θ(t − s) ∈ κ([0, β])∗ is deﬁned as

Z β Z β θ(t − s)[ f (t)] ≡ dt θ(t − s) f (t) = dt f (t) with f (t) ∈ κ([0, β]) . 0 s From its own deﬁnition θ(t − s) has a Fourier expansion

∞ 1 h π2ns i π2nt ∞ 1 h π(2n + 1)s θ(t − s) = 2 cos −1 sin + cos ∑ ∑ ( + ) n=1 π2n β β n=0 π 2n 1 β i π(2n + 1)t +1 sin ; β

so it is directly checked that θ(0) ≡ θ(t − t) satisﬁes for f (t) ∈ κ([0, β]) :

Z β 1 Z β 1 θ(0)[ f (t)] ≡ dt θ(0) f (t) = dt f (t) ≡ [ f (t)] , 0 2 0 2 where 1/2 is an element of κ([0, β])∗, so we can say that in κ([0, β])∗, θ(0) = 1/2 ; therefore we obtain that 1 ∆(t, t) = t − t2 (1.3.28) β

as an element of κ([0, β])∗. Now that we know the behavior of ∆ in time integrals we can perform perturbative calculations for a wide class of physical systems. In fact in most cases

1 expanding f (t) in a Fourier series with coefﬁcients Fn.

34 1.3. APPLICATIONS: FREE THEORIES, HARMONIC OSCILLATORS & PERTURBATIVE THEORY CHAPTER 1. PATH INTEGRALS

m 2 of physical interest the quantum Hamiltonian function of the system is H = − 2 ∇ + V(x), 2 D where ∇ is the Laplace operator in D dimensions and V(x) ∈ C∞(R ), the inﬁnitely differentiable functions in RD. The Euclidean action corresponding to such a system is simply:

Z β SE(x, t f , ti) = S f ree,E(x, β) + dt V x(t) , 0 where as usual β = t f − ti; so we are able to write

∞ n (−) Z β K(x f , t f ; xi, ti) = h 1 i 1 + ∑ dt1 ··· dtn h V q(t1) + xcl(t1) ··· V q(tn) + xcl(tn) i n=1 n! 0

i j 0 ij 0 with h q (t)q (t ) i = (δ /m)∆(t, t ) and xcl(t) = [(x f − xi)/β]t + xi . This is not yet a perturbative series, because we don’t have deﬁned a controlling parameter that permits to approximate the series when it is small. If we use β as a perturbative parameter we are interested to write a series

2 K(x f , t f ; xi, ti) = a0(x f , xi) + a1(x f , xi)β + a2(x f , xi)β + ...; we will not perform all the boring calculations here, we will limit to describe the methodol- ogy: ﬁrst of all we expand the potential V(x) in a Taylor series around the initial point xi, obtaining a series1

ξi 1 ξi ξ j V(x) = V(x ) + (qi(t) + t)∂ V(x ) + (qi(t) + t)(qj(t) + t)∂ ∂ V(x ) + ..., i β i i 2 β β i j i then we perform averages using the 2-point propagator h qi(t)qj(t0) i and ﬁnally solve integrals remembering the distributional meaning of ∆(t, s); for example one has to solve the integral

Z β Z β 1 1 dt ∆(t, t) = dt t − t2 = β2 . 0 0 β 6

As a mention we write here the values of the ﬁrst three coefﬁcients if x f = xi = x, see[14] for further details:

D m 2 a (x, x) = h 1 i = , 0 2πβ D m 2 a (x, x) = − V(x) , 1 2πβ D m 2 1 1 a (x, x) = V(x)2 − ∇2V(x) . 2 2πβ 2 12m

1remembering that

x f − xi x(t) = x (t) + q(t) = q(t) + t + x ; cl β i and deﬁning ξ = x f − xi.

h ocle scn uniain,s ewl nrdc tbrieﬂy. it introduce will we so quantization”, “second so-called the I Integrals Path QFT 2.1: Quan- perform can space-time in we moving representation proper-time calculations Schwinger Theory environment. the Field an tum using such that in see rules Feynman alter- we derive an Then provide and and propagators Theories calculate Field to Quantum method to native easily quite extended be can integrals path P ohv eprisgto h ahmtclfaeokdsrbn eaiitcsystem, relativistic a describing framework mathematical the on insight deeper a have to 2 1 odrt osrc ucinlitga omlto fQatmFedTer eneed we Theory Field Quantum of formulation integral functional a construct to order n hs ehd r fe ald“ol-iept nerlrpeettos o fetv Actions. Effective for representations” integral path “world-line called often are methods Actions. these Effective of calculation the particular in vlto fasse fQatmmcaia atce.W ilsei hscatrthat chapter this in see will We particles. mechanical Quantum of system a of evolution voscatraaye h sfleso ucinlitgaindsrbn h time the describing integration functional of usefulness the analyzed chapter revious unu ilso unu Particles? Quantum or Fields Quantum 2 . 1 hog h ahitga faQatmmcaia particle mechanical Quantum a of integral path the through ia ed[ a erltdt the to related be can ] . . [. ﬁeld Dirac yaia rpriso ‘particle’ a of properties dynamical Tevcu urn facharged a of current vacuum “The ihsaetm oriae that coordinates space-time with eeduo proper-time a upon depend uinSchwinger Julian C h a p t e r parameter.” 2 37 Part I: FLAT SPACE-TIME

2.1.1: Fock Spaces

The Hilbert space in the Quantum mechanical description of a single particle depends on the system we are considering: as an example a Schrödinger particle of spin one half has = L2(R3, dx; C2) ≡ L2(R3) ⊗ C2, where ≡ means natural isomorphism; i.e. the set of H pairs {ψ1(x), ψ2(x)} of square-integrable functions (dx is the Lebesgue measure). If we want to deal with a relativistic system, in which the number of particles is not ﬁxed, we need to deﬁne a different framework, called Fock space[15]:

Deﬁnition 2.1.1 (Fock Space over ). Let be a Hilbert space and n = ⊗ · · · ⊗ an n-fold tensor product. Setting 0H= C weH deﬁne the Fock Space overH as:H H H H ∞ M ( ) = n . F H n=0 H

However ( ) is too large, since quantum particles are indistinguishable, so we have to restrict to twoF H of its subspaces, the symmetric and antisymmetric Fock spaces. Let’s see

how to construct such subspaces: let n be the permutation group on n elements and {φk} P be a basis for . For each σ ∈ n, we deﬁne an operator (also denoted by σ) on basis elements of nHby P Fock Spaces H ( ⊗ · · · ⊗ ) = ⊗ · · · ⊗ σ φk1 φkn φkσ(1) φkσ(n) ;

n σ extends by linearity to a bounded operator of norm one on so we can deﬁne Sn = ( ) H 1/n! ∑σ∈ n σ . Sn is an orthogonal projection and its range is called the n-fold symmetric P 2 3 n 2 3n tensor product of . If = L (R ), Sn is just the subspace of L (R ) of all functions left invariant underH anyH permutation of theH variables. So we can deﬁne

Deﬁnition 2.1.2 (Symmetric (or Boson) Fock Space over ). H ∞ M n s( ) = Sn . F H n=0 H

Let e : n → {1, −1} which is one on even permutations and minus one on odd per- P = ( ) ( ) n mutations. Define An 1/n! ∑σ∈ n e σ σ ; then An is an orthogonal projection on . n P H An is called the n-fold antisymmetric tensor product of . So we define H H Definition 2.1.3 (Antisymmetric (or Fermion) Fock Space over ). H ∞ M n a( ) = An . F H n=0 H

Now that we have constructed a suitable Hilbert space for a relativistic quantum system, we have to introduce the fundamental tools in describing such a system: annihila-

tion, creation operators and quantum ﬁelds. We will do that in the Boson subspace s( ). F H

38 2.1. QFT PATH INTEGRALS CHAPTER 2. QUANTUM FIELDS OR QUANTUM PARTICLES?

n 1 Let f ∈ be fixed. For vectors in of the form η = ψ1 ⊗ · · · ⊗ ψn we define a map b−( f ) : Hn → n−1 by2 H H H − b ( f )η = ( f , ψ1)(ψ2 ⊗ · · · ⊗ ψn) ; b−( f ) extends by linearity to a finite linear combination of such η, and b−( f )η ≤ k f k kηk. Thus b−( f ) extends to a bounded map of n into n−1. Since it is true for each n (except for n = 0 in which case we define b−( f ) H: C → 0),H b−( f ) is in a natural way a bounded operator of norm k f k from ( ) to ( ). Obviously b+( f ) ≡ (b−( f ))∗ takes each n into n+1 with the action F H F H H H + b ( f )(ψ1 ⊗ · · · ⊗ ψn) = f ⊗ ψ1 ⊗ · · · ⊗ ψn on product vectors. Notice that the map f 7→ b+( f ) is linear, but f 7→ b−( f ) is anti-linear. L∞ n n − Define S = n=0 Sn; s ≡ Sn and call it the n-particle subspace of s( ). b ( f ) H + H (n) ∞ F H takes s( ) into itself, but b ( f ) does not. A vector ψ = {ψ }n=0 ∈ s( ) for which ψ(n) =F0 forH all but finitely many n is called a finite particle vector. We willF denoteH the set of finite particle vectors by F0. The vector Ω0 = {1, 0, 0, . . . } plays a special role; it is called the vacuum. Let A be any self-adjoint operator on with domain of essential self-adjointness D. Let (n) Nn H n DA = {ψ ∈ F0|ψ ∈ k=1 D for each n} and define dΓ(A) on D(A) ∩ s as H dΓ(A) = A ⊗ 1 ⊗ · · · ⊗ 1 + 1 ⊗ A ⊗ · · · ⊗ 1 + ··· + 1 ⊗ · · · ⊗ 1 ⊗ A . The Formalism of Second dΓ(A) is essentially self-adjoint on DA, and it is called the second quantization of A. If Quantization n A = 1, then N = dΓ(1) is essentially self-adjoint on F0 and for ψ ∈ s , Nψ = nψ. N is called the number operator. If U is a unitary operator on , we defineH Γ(U) to be the Nn H n unitary operator on s( ) which equals k=1 U when restricted to s for n > 0, and F H 0 itA H itA which equals the identity on s . If e is a continuous unitary group on , then Γ(e ) is the group generated by dΓ(AH), i.e. Γ(eitA) = eitdΓ(A). H − We can now define the annihilation operator a ( f ) on s( ) with domain F0 by √ F H a−( f ) = N + 1 b−( f ) , a−( f ) is called annihilation operator because it takes each (n + 1)-particle subspace into the n-particle subspace. For each ψ and η in F0, √ √ ( N + 1 b−( f )ψ, η) = (ψ, Sb+( f ) N + 1η) .

This implies that √ − ∗ + (a ( f )) F0 = Sb ( f ) N + 1 .

− ∗ − − ∗ The operator (a ( f )) is called a creation operator. Both a ( f ) and a ( f ) F0 are closable; we denote their closures also by a−( f ) and a−( f )∗.

1we recall that n has an orthonormal basis of such vectors. 2we denote withH(·, ·) the scalar product. H

39 Part I: FLAT SPACE-TIME

As an example consider the case = L2(M, dµ), the square integrable function on the measure space hM, dµi. We know thatH

n O 2 2 S L (M, dµ) = Ls (M × · · · × M, dµ ⊗ · · · ⊗ dµ) , j=1

2 2 where Ls is the set of functions in L invariant under permutation of the coordinates. Anni- hilation and creation operators are then given by √ Z − (n) (n+1) (a ( f )ψ) (m1,..., mn) = n + 1 dµ(m) f¯(m)ψ (m, m1,..., mn) , M n − ∗ (n) 1 (n−1) (a ( f ) ψ) (m1,..., mn) = √ ∑ f (mi)ψ (m1,..., mˆ i,..., mn) , n i=1

2 where mˆ i means that mi is omitted. If A operates on L (M, dµ) by multiplication by the real-valued function ω(m), then

n (n) (n) (dΓ(A)ψ) (m1,..., mn) = ∑ ω(mi) ψ (m1,..., mn) . i=1

1 It is now time to deﬁne the Segal ﬁeld operator ΦS( f ) on F0 as

1 − − ∗ ΦS( f ) = √ a ( f ) + a ( f ) ; 2

from the deﬁnition of a−( f )∗ we see that it is symmetric, in fact it is essentially self-adjoint. We will not analyze here the properties of Segal ﬁelds, see[16, 17].

2.1.2: Q-Space

Since we use to experience a world obeying to classical mechanics, it is usual to rephrase all physical systems in a classical-like framework. In particular we give a special role to time: we describe a system by means of “fixed-time degrees of freedom” which evolve in time. This leads to the Hamiltonian description of mechanics, both classical and quantum. We want to do that even in relativistic systems, but this breaks Lorentz invariance. The standard way to do it for a free field is to introduce the time-zero field and the canonical conjugate momentum2. As a matter of fact, in constructing interacting field theories by perturbing the free theory using the time-zero fields, it is quite difficult to recover Lorentz invariance. We will see that path integral formulation of field theory allows us to construct a Lorentz invariant perturbative theory in a natural and easy way. We introduce now the so-called canonical fields starting from Segal quantization. A conjugation on a Hilbert space is an anti-linear isometry C so that C2 = 1. H 1 we will call the mapping f 7→ ΦS( f ) the Segal quantization over . 2there is no direct connection between the canonically conjugateH momentum of a field and the physical momentum operator Pb.

40 2.1. QFT PATH INTEGRALS CHAPTER 2. QUANTUM FIELDS OR QUANTUM PARTICLES?

Definition 2.1.4 (canonical free field and canonical conjugate momentum). Let be a com- H plex Hilbert space, ΦS(·) the associated Segal quantization. Let C be a conjugation on H and define C = { f ∈ |C f = f }. For each f ∈ C we define ϕ( f ) = ΦS( f ) and H H H π( f ) = ΦS(i f ). The map f 7→ ϕ( f ) is called the canonical free field over h , Ci and the map f 7→ π( f ) is called the canonical conjugate momentum. We will drop theH h , Ci and just write , if the intended conjugation is clear. The set of elements of forH which the maps f 7→Hϕ( f ) and f 7→ π( f ) are defined depends on the conjugation C. H

We are now ready to introduce a very useful representation of the Fock space structure we have presented: the Q-space and L2(Q, dµ)[6, 16, 17]. Our task is to define an 1 2 −1 isomorphism S between s( ) and L (Q, dµ) so that for each f ∈ C, Sϕ( f )S acts on L2(Q, dµ) by multiplicationF byH a measurable function. It will allow usH to construct a com- ∞ plete orthonormal basis of field eigenstates. Let { fn}n=1 be an orthonormal basis for so N H that each fn ∈ C and let {gk}k=1 be a finite collection of the fn. Define N as the closure of the set H F

{P(ϕ(g1),..., ϕ(gN))Ω0|P a polynomial}

(N) in s( ) and deﬁne F = N ∩ F0. It follows that ϕ(g ) and π(g ) are essentially self- 0 k l The construction of F H (N) F adjoint on F0 and that Q-space

eitϕ(gk)eisπ(gl ) = e−istδkl eisπ(gl )eitϕ(gk) .

Thus we have a representation of Weyl relations[18] in which the vector Ω0 satisﬁes

2 2 (ϕ(gk) + π(gk) − 1)Ω0 = 0

N and is cyclic for the operators {ϕ(gk)}k=1. A Theorem by Von Neumann[19, 20] therefore (N) 2 N N guarantees us there is a unitary map S1 : N → L (R , dµ1) (where dµ1 ≡ d x, the Lebesgue measure of RN), so that F

(N) (N) −1 S1 ϕ(gk)(S1 ) = xk , ( ) ( ) 1 S N π(g )(S N )−1 = ∂ , 1 k 1 i k and the vacuum corresponds to the vector

N 2 (N) − N x = 4 − k S1 Ω0 π exp ∑ . k=1 2 Often it is convenient that the transform of the vacuum is the function identically one, so one introduces the Hilbert space

N 2 N − 2 2 N L (R , π exp{− ∑ xk /2}d x) 1 it is possible to do the same construction in the Fermionic case a( ), but since we will not make use fermionic ﬁelds in this work we do not develop such a construction. F H

41 Part I: FLAT SPACE-TIME

2 N −1/2 −x2 instead of L (R ) so let dµk,2 = π e k dxk and deﬁne

N 2 N x (T f )(x) = π 4 exp ∑ k f (x) . k=1 2 2 N 2 N N Then T is a unitary map of L (R , dµ1) onto L (R , dµ2) with dµ2 = ∏k=1 dµ2,k and if we (N) (N) (N) 2 N deﬁne S2 = TS1 , we have S2 : N → L (R , dµ2), and F (N) (N) −1 S2 ϕ(gk)(S2 ) = xk , ( ) ( ) x 1 S N π(g )(S N )−1 = − k + ∂ , 2 k 2 i i k (N) S2 Ω0 = 1 . (N) (N) Since each µ2,k has mass one, the following property holds for both maps S1 and S2 , so 1 we will omit the index 1 or 2 : Z N Z (Ω0, P1(ϕ(g1)) ··· PN(ϕ(gN))Ω0) = dµ P1(x1) ··· PN(xN) = ∏ dµk pk(xk) k= R RN 1 N = ∏(Ω0, Pk(ϕ(gk))Ω0) , k=1

where P1,..., PN are polynomials. ∞ We can now construct hQ, dµi. We deﬁne Q = ×k=1R. Taking the σ-algebra generated N∞ by countable products of measurable sets in R we set µ = k=1 µk. The points of Q will be denoted by q = hq1, q2,... i. Then hQ, µi is a measure space and the set of functions of the 2 form P(q1,..., qn), where P is a polynomial and n is arbitrary, is dense in L (Q, dµ). Let P be a polynomial in N variables:

P(x ,..., x ) = c xl1 ··· xlN , k1 kn ∑ l1...lM k1 kN l1,...,lN and deﬁne

S : P(ϕ( fk1 ),..., ϕ( fkN ))Ω0 → P(qk1 ,..., qkN ) . Then by the property above

2 + + P( ( f ) ( f )) = c c ( ( f )l1 m1 ··· ( f )lN mN ) ϕ k1 ,..., ϕ kN Ω0 ∑ l ¯m Ω0, ϕ k1 ϕ kN Ω0 l,m Z N Z l1+m1 lN +mN 2 = c c¯m dµ q ··· q = dµ P(q ,..., q ) . ∑ l ∏ ki k1 kN k1 kN l m i=1 , RN Q

Since Ω0 is cyclic for the polynomials in the ﬁelds, S extends to a unitary map of s( ) 2 −1 F H onto L (Q, dµ). Obviously Sϕ( fk)S = qk, and S2Ω0 = 1, while q2 S Ω ∝ exp − k . 1 0 ∑k 2 1we will do the same in the following: when we don’t explicit the index it means that the construction is valid in both cases.

42 2.1. QFT PATH INTEGRALS CHAPTER 2. QUANTUM FIELDS OR QUANTUM PARTICLES?

So we have proved that s( ) is isomorphic to a space of square integrable functions where the canonical fieldF operatorH reduces to multiplication by a coordinate and its conjugate momentum to differentiation with respect to the same coordinate: so they behave as position and momentum operators on the space of square integrable functions isomorphic to ; but since we can find an orthonormal basis of position eigenstates in , and also a basisH of momentum eigenstates, that means we can find an orthonormal basisH of canonical time zero field eigenstates, or conjugate momentum ones, in s( ). Explic- F H itly, with the usual Dirac notation, we have a continuous basis {|qi} ∈ s( ) so that R F H ϕ( fk)|qi = qk|qi, 1 = dµ1(q) |qi hq|; and a basis {|pi} ∈ s( ) so that π( fk)|pi = pk|pi, R F H 1 = dµ1(p) |pi hp|; furthermore we know that 1 hq|pi = eiqk pk , ∏k (2π)1/2 and if we call the vacuum vector |0i, q2 hq|0i ∝ exp − k . ∑k 2

2.1.3: Derivation of Path integral

In order to find a path integral formulation of perturbative quantum field theory we have to apply the useful mathematical results of previous sections on physical situations. Usually 1 2 3 3 in quantum field theory one uses the annihilation operator a(p) on s(L (R )), p ∈ R , the free scalar field and time zero fields thus becoming[4]: F 1 Z d3 p Φ (x, t) = ei(µ(p)t−p·x)a†(p) + e−i(µ(p)t−p·x)a(p) , m (2π)3/2 (2µ(p))1/2 R3 1 Z d3 p ϕ (x) = e−ip·xa†(p) + eip·xa(p) , m (2π)3/2 (2µ(p))1/2 R3 Z r i µ(p) 3 −ip·x † ip·x Fields and Q-space πm(x) = d p e a (p) + e a(p) , (2π)3/2 2 of QFT R3 p with µ(p) = p2 + m2. We can repeat the construction of the Q-space for these canonical

ﬁelds, but the Q-space has to be a functional space, in particular the action of S1(the unitary transformation we will use) would become

−1 S1 ϕm(x)S1 = q(x) , 1 δ S π (x)S−1 = , 1 m 1 i δq(x) and since we can write r s ! 1 Z µ(p) 1 δ S a(p)S−1 = d3x e−ip·x q(x) + , 1 1 (2π)3/2 2 2µ(p) δq(x)

1for the sake of simplicity we consider the case of an Hermitian scalar ﬁeld of mass m, the general to higher spin ﬁelds is straightforward.

43 Part I: FLAT SPACE-TIME

−1 and S1a(p)S1 S1Ω0 = 0, we obtain that Z n 1 3 3 o S1Ω0 = exp − d xd y (x, y)q(x)q(y) , N 2 E where is a constant and (x, y) = (2π)−3 R d3 p eip·(x−y)µ(p). We can again construct a basis ofN ﬁeld or momentumE eigenstates, but now they will satisfy the following relations1:

ϕm(x)|qi = q(x)|qi ,

πm(x)|pi = p(x)|pi , Z Z 1 = [q(x)]|qi hq| = [p(x)]|pi hp| , D D (2.1.1) n Z o hq|pi = exp i d3x q(x)p(x) , M n 1 Z o hq|0i = exp − d3xd3y (x, y)q(x)q(y) , N 2 E where and are constants. WeM know thatN to obtain elements of the S-matrix we have to know the vacuum expecta- tion values: h0, out|T ϕ(x1, t1)ϕ(x2, t2) ··· |0, ini h ϕ(x , t )ϕ(x , t ) · · · i ≡ ; 1 1 2 2 0 h0, out|0, ini but we can write both numerator and denominator in a path integral form using relations (2.1.1) and a derivation identical to the quantum mechanical one developed in Sec- tions 1.2 and 1.3. So we obtain a formula2: Z h0, out|T ϕ(x1, t1)ϕ(x2, t2) ··· |0, ini = [q(x, t)] [p(x, t)]q(x1, t1)q(x2, t2) ··· D D n hZ ∞ Z ········· exp i dt d3x q˙(x, t)p(x, t) − H(q(x, t), p(x, t)) −∞ i Z io + d3xd3y (x, y)q(x, +∞)q(y, +∞) + q(x, −∞)q(y, −∞) , 2 E where H(q(x, t), p(x, t)) is the ﬁelds Hamiltonian density, and the last term is due to the

projections h0, out|q+∞i and hq−∞|0, ini. For any reasonably smooth function g(x) holds the following equality:

Z ∞ g(+∞) + g(−∞) = lim e dt g(t)e−e|t| ; e→0+ −∞ so we have i Z Z ∞ d3xd3y (x, y)q(x, +∞)q(y, +∞) + q(x, −∞)q(y, −∞) = ie dt (t) , 2 E −∞ G 1 [x(t)] is the measure associated with the Q-space in such a case; since the Q-space is a functional space, the measureD would be a functional measure. 2the constants and are adsorbed in the deﬁnition of a proper functional measure for time-dependent ﬁelds. M N

44 2.1. QFT PATH INTEGRALS CHAPTER 2. QUANTUM FIELDS OR QUANTUM PARTICLES? with (t) = 1/2 R d3xd3y (x, y)q(x, t)q(y, t) a quadratic form in the fields. So finally we obtainG the phase space pathE integral for a quantum field theory:

Z h0, out|T ϕ(x1, t1)ϕ(x2, t2) ··· |0, ini = [q(x, t)] [p(x, t)]q(x1, t1)q(x2, t2) ··· D D (2.1.2) Z ∞ hZ i ··· exp i dt d3x q˙(x, t)p(x, t) − Hq(x, t), p(x, t) + ie (t) . −∞ G

The whole effect of the last term in Eq. (2.1.2) is to provide the Feynman prescription in the field propagator, solving the ambiguities due to poles on integration path. Following the procedure of Section 1.2.3 we can pass from phase space path integral to configuration space path integral, but if the factor Det [q] is field dependent it gives a correction mod- ifying the effective Lagrangian density, andA this modification has to be calculated case by case. However in most theories Det is field-independent, so it can be adsorbed in the definition of functional measure withoutA problems. Moreover all this construction, from the definition of Q-space to Eq. (2.1.2), is generalized to vectorial fields without any further technical difficulty, and also to fermionic fields, introducing the Grassman algebra of commuting variables and Berezinian calculus1; so we can write the configuration space path integral for an arbitrary field:

Z iS[ψ] h0, out|T Ψl1 (x1, t1)Ψl2 (x2, t2) ··· |0, ini = [ψl(x, t)]ψl1 (x1, t1)ψl2 (x2, t2) ··· e D (2.1.3) QFT Path Integral where S[ψ] is the ﬁeld action

Z ∞ Z h 3 i S[ψ] = dt d x e f f (ψl(x, t), ψ˙ l(x, t)) + ie (t) , −∞ L G and the covariant Lagrangian density e f f takes into account possible corrections due to Det [ψ]. It is possible now to derive freeL propagators, Wick theorem and Feynman rules in aA way analogous to the quantum mechanical one we developed in the previous chapter, however further calculations are beyond the scope of this work. Nevertheless we emphasize that since propagators and vertex contributions in the Feynman rules are obtained directly by inspection of the Lagrangian, explicitly covariant, we avoid all the problems due to non-covariant nature of Hamiltonian formulation, where propagator and interacting Hamil- tonian both contain non-covariant terms2. Finally we can say that path integrals provide a beautiful method to perform explicit quantum ﬁeld theory calculations that go directly from the covariant canonical Lagrangian to the Feynman rules in their ﬁnal, Lorentz-covariant form[13].

1since we will not use fermionic path integrals in quantum ﬁeld theory we will not analyze these mathematical structures here. 2obviously even in the usual operator formulation these terms cancel out in order to restore Lorentz- covariance of the theory.

45 Part I: FLAT SPACE-TIME

2.2: Worldline Methods

lthough we did not perform explicit calculations in the previous section, we men- A tioned that, as usual in path integrals, we find free propagators introducing external sources in the generating functional for the free action, and in field theory these source terms are equivalent to a coupling of the free field to external classical currents. We will see in a moment that it is very useful to define a generating functional even for the complete action, for it will lead to the introduction of the so-called “quantum effective action”. This effective action Γ[ψ] will allow us to take into account all multi-loop effects by summing ‘tree’ graphs, i.e. without loops, whose vertices and propagators are taken from Γ[ψ]. Further- more we will see that we can calculate the one-loop part of such effective actions by means of a path integral in a suitable “fictitious” quantum mechanics.

2.2.1: The Quantum Effective Action

Consider a field theory with action S[ψ], with ψl(x)1 a generic field (however we shall not bother to keep track of the minus signs that would appear in that case). We introduce a set l of classical currents Jl(x) coupled to the fields ψ (x). So we define the generating functional

Z R 4 r l iS[ψ]+i d x ψ (x)Jr(x) Z[J] ≡ h0, out|0, iniJ = [ψ (y)] e . (2.2.1) D Now, Z[J] is given by the sum of all vacuum-vacuum amplitudes in the presence of the current J, including disconnected as well as connected diagrams. In order to take into account only connected diagrams we introduce 1 W[J] ≡ ln Z[J] . i So in terms of the sum of connected graphs we have

l l h0, out|Ψ (x)|0, iniJ 1 δ δ ψj(x) ≡ = Z[J] = W[J] . h0, out|0, iniJ iZ[J] δJl(x) δJl(x) l Such a formula can be inverted, in the sense that we can invert the map Jl(x) 7→ ψJ(x) l l deﬁning a map ψ (x) 7→ Jl,ψ(x) | ψ (x) = (δ/δJl,ψ(x))W[J]. Now we are able to deﬁne the quantum effective action Γ[ψ] as the Legendre transformation Z 4 l Γ[ψ] ≡ W[Jψ] − d x ψ (x)Jl,ψ(x) .

Γ[ψ] is the sum of all connected one-particle-irreducible2 graphs. From its own deﬁnition we can explain why it is called the ‘effective action’: if we calculate its functional derivative δΓ[ψ] = −J (y) ; δψr(y) r,ψ 1we use a shorthand notation where x is the four-dimensional space-time vector. 2A one-particle-irreducible (1PI) graph is one that cannot be disconnected by cutting through any one internal line.

46 2.2. WORLDLINE METHODS CHAPTER 2. QUANTUM FIELDS OR QUANTUM PARTICLES? we see that the possible values for the external ﬁelds ψr(y) in the absence of currents J are given by the stationary points of Γ:

δΓ[ψ] = 0 . δψr(y)

Hence this may be regarded as the equation of motion for the external ﬁeld ψ, taking quantum corrections into account. However Γ[ψ] does not provide only the quantum-corrected ﬁeld equations; it can be shown that iW[J] can be calculated as a sum of connected tree graphs, with vertices calculated as if the action were Γ[ψ] instead of S[ψ]. We don’t give here the rigorous proof, however it is possible to show that[21]:

Z R 4 r iW[J] = [ψl(x)]eiΓ[ψ]+i d y ψ (y)Jr(y) , D connected, tree graphs where the boundary means that we have to keep only connected, tree graphs. In order for this last expression to be correct, iΓ[ψ] must be the sum of all one-particle-irreducible connected graphs with arbitrary number of external lines, each line corresponding to a factor Quantum Effective l ψ rather than a propagator or wave function. For this reason iΓ[ψ0] for some ﬁxed ﬁeld ψ0(x) Action and its may be expressed as the sum of one-particle-irreducible graphs for the vacuum-vacuum applications amplitude, calculated with the action S[ψ + ψ0]: Z l iS[ψ+ψ0] iΓ[ψ0] = [ψ (x)]e . D 1PI connected

Differentiating Γ with respect to ψ N times we obtain the N-point correlation function 1PI. If we consider the complete 2-point correlation function (or complete propagator), we have

δ2W[J] δψl (x) ∆lr(x, y) ≡ = J , δJl(x)δJr(y) δJr(y) while its 1PI part Πlr(x, y) is given by

δ2Γ[ψ] δJ (x) Π (x, y) = − l,ψ ; lr δψl(x)δψr(y) δψr(y) it follows immediately that the matrices ∆ and Π are related by ∆ = −Π−1, and this is the well-known relation between propagators and self-energy parts. It is useful to split the effective action in its contributions due to 1PI connected graphs with zero, one, . . . loops. In the case of a self-interacting real scalar ﬁeld φ(x) with action

Z h1 1 1 i S[φ] = − d4x ∂ φ∂µφ + m2φ2 + gφ4 , 2 µ 2 24

47 Part I: FLAT SPACE-TIME

and φ0(x) = φ0 a constant ﬁeld, we obtain

0 loop 1 2 2 g 4 iΓ [φ0] = −i 4 m φ0 + φ0 , V 2 24 1 iK iΓ1 loop[φ ] = − Tr ln , 0 2 π

where 4 is the (divergent) volume of space-time, and V 2 K = − + µ (φ0) − ie ,

2 2 2 2 2 with = ∇ − ∂t and µ (φ0) = m + (1/2g)φ0. Note that the one-loop effective action is written as the trace of the logarithm of an operator: the presence of such a logarithm will allow us to write it as a quantum mechanics path integral. Consider another example: if we have an action S[ψ, A], where A is an external background ﬁeld, that is a (purely) quadratic function of ψ the vertices of the theory would contain only two ψs and a certain number of As (usually one, or maybe two); but since A could appear only as an external leg in the graphs since it is a background ﬁeld, the only connected diagrams are one-loop ψ diagrams with an arbitrary number of A insertions. In the case of one-loop diagrams the connected part is also 1PI, and the insertion of A gives analytically a multiplicative factor A; so if we write in Euclidean time Z −SE[ψ,A] −W[A] Z[A] = h0, out|0, iniA = [ψ]e = e , D 1 loop we recognize that Γ [A] = W[A], the connected part of h0, out|0, iniA. But since SE[ψ, A] is a purely quadratic form we can write it in a condensed notation as Z 4 r A s SE[ψ, A] = d x ψ Krsψ ,

but in that case the path integral in Z[A] is a usual Gaussian integral, so we solve it obtaining

Z[A] = (Det KA)ρ , C where ρ = −1/2, −1 for a system of real or complex bosonic ﬁelds, and ρ = 1 for a system of fermionic ﬁelds. So since ln Det = Tr ln we can write1: A A Γ1 loop[A] = −ρ ln Det KA ; (2.2.2)

again we see that the one-loop effective action is written as the trace of the logarithm of an operator.

2.2.2: Fictitious Quantum Mechanics

We are now able to introduce the beautiful worldline method to calculate one-loop effective actions. The starting-point is the idea, due to Schwinger[22], that we can use the formula A Z ∞ dT ln = − e−AT − e−BT B 0 T 1dropping the non-inﬂuent, A-independent additive constant ln . C

48 2.2. WORLDLINE METHODS CHAPTER 2. QUANTUM FIELDS OR QUANTUM PARTICLES? to rewrite the one-loop effective action of Eq. (2.2.2)1 as:

∞ Z dT A Γ1 loop[A] = −ρ ln Det KA = ρ Tr e−TK , 0 T where we have dropped an additive, A-independent constant2. Now if we redeﬁne KA ≡ H[A], T ≡ β, we obtain, remembering Eq. (1.3.19)

Z ∞ β Z ∞ β Z One-loop Effective 1 loop d d −SE,A[x,β] Γ [A] = ρ Z[β] = ρ [x(t)]e , (2.2.3) Action as a path 0 β 0 β D PBC integral where SE,A[x, β] is the Euclidean action of a ﬁctitious quantum mechanical system with Hamiltonian function HA. So if we know the classical action corresponding to a quantum Hamiltonian function HA we are able to calculate the one-loop effective action as a quantum mechanical path integral. In the previous section we found that the one-loop effective action for the constant part of a self-interacting scalar ﬁeld with quartic interaction was

1 iK iΓ1 loop[φ ] = − Tr ln , 0 2 π in Euclidean time it becomes 1 H Γ1 loop[φ ] = − Tr ln E,φ0 , 0 2 π H = −∇2 + 2( ) ∇2 where E,φ0 4 µ φ0 , with 4 the 4-dimensional Laplacian operator. We know that this is the Hamiltonian function of a 4-dimensional particle of mass 1/2 moving in a 2 constant potential µ (φ0). So in that case Eq. (2.2.3) becomes

Z ∞ Z Z β 1 loop 1 dβ n 1 2 2 o Γ [φ0] = − [x(t)] exp − dt x˙(t) + µ (φ0) ; 2 0 β D 0 4 PBC thus we have seen that the one-loop effective action is calculated by means of a fictitious 4-dimensional spinless particle moving in a constant potential. We have certainly emphasized that the quantum mechanics we use to describe the effective action is a fictitious one, but a careful analysis of worldline models for relativistic particles will reveal a quite interesting feature: this fictitious quantum mechanics is not so fictitious, for it corresponds to the first quantization of the relativistic particle which makes the loop in the effective action3. However we will not analyze worldline models and their implications here; we will do it in the next chapter in the more general case of curved space- time. For a quite complete review of such methods see[23].

1or in general every time it is written as the trace of a logarithm. 2although this constant does not affect the calculation of correlation functions or of the equations of motion for A and is thus canceled, it is infinite! 3in the case, as analyzed in the previous section, of a field with a quadratic action coupled to an external background field; in that case the connected diagrams reduces to one-loop diagrams with an arbitrary number of background field insertions.

Appendices

setal efajito D on self-adjoint essentially A.1.1 Theorem assumption general the under it formulate T formula product Trotter A.1: exponentials. of product W oevr fAadBaebuddfo eo,then below, from bounded are B and A if Moreover, ihteepnnilo h omlodrdHmloin,adi pedxDw ild a do will we D of Appendix ordering Weyl in the and for Hamiltonian), approximation ordered formula, similar normal similar the a of ﬁnd exponential can the in we with order that showed ﬁrst we at 1.2.1 holds Section that in however parts; more or two hsfruaapist ahitga nyi ecudwieteHmloina u of sum a as Hamiltonian the write could we if only integral path to applies formula This fepnnil ic thlsfrawd ls foperators of class wide a for holds it since exponentials of rte rdc oml stems sdt xrs exp express to used most the is formula product trotter he s s − − edntpofti eea hoe ee see[24]. here, theorem general this proof don’t We te ehd oapoiaeteepnnilo u foeaosb en of means by operators of sum a of exponential the approximate to methods other itouehr h rte oml n t ag fapiaiiy n eto some mention and applicability, of range its and formula Trotter the here introduce e n n lim lim → → ∞ ∞ ( ( e e itA − Totrpoutformula) product (Trotter tA / / n n e e itB r te oml n te systematic other and formula Trotter − / tB n / ) n ( n ) A t = n / ) = n ∩ e o h omlodrn of ordering normal the for , i ( e D A − + ( ( B A B ) + ) t then , B ) t . . . fAadBaesl-don prtr n A and operators self-adjoint are B and A If e − e H . e i e H ta a eapproximated be can (that approximants { A it ( and A + B B nfc ecan we fact in ; ) } A p p e n d i x saproduct a as A + is B 53 Part I: FLAT SPACE-TIME

A.2: Zassenhaus formula

n the formulation of path integral Trotter formula or a first order approximation in t/n I is sufficient, but if we wanted to evaluate the discretized action at orders higher than the first in e, as it is sometimes done in statistical mechanics, we have to use the Zassenhaus formula[8, 9]:

2 m eλ(A+B) = eλAeλBeλ C2 ··· eλ Cm ··· ,

where n 1 ∂ − n−1 − − ( + ) = λ Cn−1 ··· λB λA λ A B Cn n e e e e , n! ∂λ λ=0 so as an example

1 1 C = [B, A] , C = [C , A + 2B] . 2 2 3 6 2

However this formula is proved to approximate the operator eλ(A+B) at the desired order in a rigorous way only if A and B are bounded, so it is not useful with an Hamiltonian with unbounded kinetic part.

54 pc oaohr fw en norm: a define we if another, to space satisfies H bounded B.1.1 Theorem F Theorem the of Proof B.1: on topology induced the x each for If, Y. space linear normed some to X from transformations linear bounded of B.0.1 Theorem I k Tx ota,frall for that, so plcblt.W ilsaehr sfltheorem: useful a here state will We applicability. ti pedxw ilgv ro fTerm121 n ics reyisrneof range its briefly discuss and 1.2.1, Theorem of proof a give will we appendix this n eas ealta ecncntutaBnc space Banach a construct can we that recall also We tesk fcmoiyw ilrpa eeTerm1.2.1: Theorem here repeat will we commodity of sake the or k k ϕ d d y T s t ( k ϕ

t efajitoeaoso ibr space Hilbert a on operators self-adjoint = ) s T = ( t ∈ = ) sup x U 6 = F teDsnexpansion) Dyson (the 0 ( Picpeo nfr boundedness) uniform of (Principle ψ −

t k , k s Tx i ∈ sbudd then bounded; is x H ) ψ k H k ( X t Y ) , ϕ L s , ( t ( ) X , , Y ) ϕ s scalled is ( k s T = ) . k e t Let

ψ T nfr prtrtopology operator uniform ∈ H → (B.1.2) . F H

hnteei ntr propagator unitary a is there Then . ( t sbounded. is ) h yo expansion Dyson the easrnl otnosmpof map continuous strongly a be . e eaBnc space, Banach a be X Let L ( X , Y ) foeaosfo Banach a from operators of .

A p p e n d i x F R U B family a ( nothe into t (B.1.1) ∈ , s ) , X on 55 Part I: FLAT SPACE-TIME

Proof. Let’s start deﬁning Dyson expansion:

∞ Z t Z t1 Z tn−1 n U(t, s)ϕ = ϕ + ∑ (−i) ··· dt1 ··· dtn H(t1) ···H(tn)ϕ ; (B.1.3) n=1 s s s

H(τ) is uniformly bounded on [s, t] by Theorem B.0.1, so the nth term on the right is bounded by

|t − s|n n sup kH(τ)k kϕk n! τ∈[s,t]

so the series on the right converges in the uniform operator topology to U(t, s). Thus U(t, s) is jointly strongly continuous in s and t since this is true of each term on the right. It is trivial to check that U(t, t) = 1 and U †(t, s) = U(s, t). The formula U(r, s)U(s, t) = U(r, t) is proven by multiplying out the series. Therefore

U(s, t)U †(s, t) = 1 = U †(s, t)U(s, t) ,

so U(t, s) is unitary. The ﬁrst statement of Eq. (B.1.2) is proven differentiating the Dyson series for U(t, s) term by term and noting that the resulting series converges uniformly.

Finally we state a more general theorem that extends (B.1.3) to the case of contraction semi-groups on a Banach space[6].

Deﬁnition B.1.1 (Contraction semi-group). A family {T(t)|0 ≤ t < ∞} of bounded operators on a Banach space X is called a contraction semi-group if

1. T(0) = 1 ,

2. T(s)T(t) = T(s + t) for all s, t ∈ R+ ,

3. for each ϕ ∈ X, t 7→ T(t)ϕ is continuous;

and moreover kT(t)k ≤ 1 for all t ∈ [0, ∞).

We can write T(r) = e−rA, where A is said to be the generator of the contraction semi- group. Now consider a generator A(t) of a contraction semi-group, and deﬁne

i − 1 U (t, s) = exp −(t − s)A with i, k integers, 1 ≤ i ≤ k, k k i − 1 i and t, s ∈ R+ with ≤ s ≤ t ≤ . k k

Finally deﬁne C(t, s) ≡ A(t)A(s)−1 − 1, and we have the following theorem:

Theorem B.1.2. Let X be a Banach space and let I be an open interval in R. For each t ∈ I, let A(t) be the generator of a contraction semi-group on X so that 0 is in the resolvent set of A(t) and

56 B.1. PROOF OF THE THEOREM APPENDIX B. THE DYSON EXPANSION

(i) the A(t) have a common domain D (from which it follows that A(t)A(s)−1 is bounded);

(ii) for each ϕ ∈ X, (t − s)−1C(t, s)ϕ is uniformly strongly continuous and uniformly bounded in s and t for t 6= s lying in any ﬁxed subinterval of I;

−1 (iii) for each ϕ ∈ X,C(t)ϕ ≡ lims%t(t − s) C(t, s)ϕ exists uniformly for t in each compact subinterval and C(t) is bounded and strongly continuous in t.

Then for all s ≤ t in any compact subinterval of I and any ϕ ∈ X,

U(t, s)ϕ = lim Uk(t, s)ϕ k→∞ exists uniformly in s and t. Further, if ψ ∈ D, then ϕs(t) ≡ U(t, s)ψ is in D for all t and satisﬁes

d ϕ (t) = −A(t)ϕ (t), ϕ (s) = ψ ; dt s s s and kϕs(t)k ≤ kψk for all t ≥ s.

o opeigtesur nteepnnil n eebrn httemauei trans- is measure the that obtain remembering we and invariant, exponential, lationally the in square the completing now T Since variables: eigenvalues The continuation. matrix analytic a by obtained be then can case general the where in case result the the positive; so consider We matrix non-singular. The and indices. repeated of convention summation usual the with where ymti elmti a edaoaie ya rhgnlmti,teeoeteeis there therefore matrix, orthogonal an by diagonalized be can matrix real symmetric A h otgnrlGusa nerlcnb rte as written be can integral Gaussian general most The ogottewr emk ag s fGusa nerl;w ics hmhere. them discuss we integrals; Gaussian of use large make we work the hroughout I I I ζ Q | Det r Q S ( = = = = T ζ ( S KS ζ = ) , S e e Z S ) S − − | − rs sagnrlqartcfnto of function quadratic general a is T ∞ M M = ∞ ζ rs 2 1 = s 0 ∏ ∏ ∏ K ,s eoti for obtain we so 1, = r r r rs S k . − r d r Z ζ δ − r rs 1 ζ r oiiedﬁie ecnuetematrix the use can We positive-deﬁnite. are ζ ∞ r 2 ∞ k uhthat such k s π e r r + d − e ζ Q 2 L . 0 1 ( k r e r ζ (( ζ ) − r S 1 2 − + k , 1 r ζ L M 0 ) 2 r − ) 2 I ( S − : , 1 ; L ) r ζ 0 , ζ : K rs , asinintegrals Gaussian L r and M S r l el with real, all are opromacag of change a perform to K a ob symmetric be to has

A p p e n d i x C K rs also 59 Part I: FLAT SPACE-TIME

and since

= −1 = rl sl −1 det K ∏ kr , Krs S S kl , r

ﬁnally becomes I 1 − 2 K 1 Lr LsK−1−M = det e 2 rs . (C.0.1) I 2π ¯r −1 s If we deﬁne ζ = −Krs L , we see that ∂Q = 0 ; ∂ζr ζ=ζ¯

so we can ﬁnally write

− 1 K 2 ¯ = det e−Q(ζ) . I 2π These Gaussian integrals can be generalized to the case of functional integrals, substituting r ∏r dζ with a functional measure [ζ(t)], considering matrix indices as a condensed multi- index with a discrete and a continuousD part, and considering the determinant det as a functional determinant Det. This could be justiﬁed by the fact that usually functional measures are translationally invariant and introducing a suitable regularization (such as time-slicing k or mode regularization) they effectively reduce to a product ∏k dζ . However for path integrals Eq. (C.0.1) is just a formal expression, since we have to give a meaning to the functional determinant Det K, where now K is an operator, and in general there would be additional multiplicative factors due to the deﬁnition of functional measure; nevertheless since in perturbative averages global multiplication factors disappear1, because they are present both in the numerator and denominator of the average, the generalization of (C.0.1) to functional integration is quite useful.

1but in general not the functional determinant Det K that could modify the path integral action, as discussed in Section 1.2.3; in particular in curved space functional integration of the momenta causes ambiguities and divergencies in perturbative calculations that lead to the introduction of ghosts and regularization-dependent counterterms.

60 PartII

CURVED SPACE-TIME

e start this second part introducing a relativistic model to describe the motion of W a particle in a curved space-time, and how we can perform a quantization of such model. Starting from the quantum self-adjoint and covariant Hamiltonian operator of our quantized theory we derive path integrals both in phase and configuration space. After a discussion of ambiguities and divergences we have to face when we make perturbative calculations in configuration space path integral, we introduce ghosts and regularization schemes that solve such divergences and ambiguities by the introduction of counterterms. Such counterterms that are fixed with a two-loop perturbative calculation and outline of second depend on the regularization scheme used. part In the last chapter we extend the worldline formalism we introduced in the first part (for the calculation of one-loop effective actions) to the case of a scalar field in a gravitational background. Then we analyze the phase space path integral for the relativistic quantum Hamiltonian operator we already mentioned, and show that in this case is not particularly useful to introduce regularization schemes, at least to perform perturbative calculations, since every calculation is nor ambiguous nor divergent in the continuum limit. Finally we calculate explicitly counterterms for mode regularization in phase space path integral.

ul xetgaiyisl) nfc,w nwta uhapril reyfalls, freely particle a such (obvi- that force know external we any fact, to In mass subjected of itself). particle gravity not test except space-time ously a curved for arbitrary principle an variational through a moving formulate to immediate not is It Action Relativistic 3.1.1: W space-time of manifolds curved in Particles 3.1: field models. a worldline for integral action path effective mechanical one-loop the quantum manage with can background we gravitational ana- that a see will in will we we so Also, distributions; detail. of in it products and lyze ill-defined parti- schemes and regularization of requires divergences description it treat integral example to for path counterterms trivial: a not of is development backgrounds curved the ( in that parameters cles see affine will We in we evolution and particles). particle, its massive a for for such formulation quantize integral to path methods a some give non-trivial develop a then by will characterized We manifold space-time tensor. a metric in worldline its along moving particle a D fl fsaetm.W ildvlpteuulntoso eea eaiiyt describe to relativity general of notions usual the develop will We space-time. of ifold frnilgoer ie stetost ecieapril oigi uvdman- curved a in moving particle a describe to tools the us gives geometry ifferential pc-ie n t quantization. its and space-time, wl td o h oest ecieacasclfe aln atcei curved a in particle falling free classical a describe to models the now study will e ahIntegrals Path Sn bic d’universo” ubriaco “Sono e.g. h rprtime proper the i.e. tflosa follows it G.Ungaretti C h a p t e r m , m 3 ≥ 0 65 Part II: CURVED SPACE-TIME

geodesic motion[25]: such a motion is described by the equation

d2xα dxµ dxν + Γα = 0 , (3.1.1) dλ2 µν dλ dλ where xα(λ) is the coordinate depending on some geodesic affine parameter and the Γs are the metric connection coefficients. This equation does not depend explicitly on mass, so if we want to obtain it from a Lagrangian variational principle δS = 0 the mass would have to be an overall multiplicative factor, i.e. S = mS0, but in that case we would have S ≡ 0 for massless particles. In fact this is the case even in special relativity, where we define S = −m R ds. However for massive particles this definition was a wise one: ds and m are invariant, and from geometric considerations we know that ds has extremal length on the physical trajectory so the variational principle δS = 0 is automatically satisfied. The problem is ds is identically zero for massless particles. However also geodesics of arbitrary space-time man- q µ ν ifolds has extremal ds = gµν(x)dx dx , and it is invariant under reparametrization of the affine parameter. So it appears to be a good choice to define the action of a massive particle, but its quantization would result quite difficult, due to the presence of the square root. We must proceed in a different way: we note that if we define a relativistic Hamiltonian function Geodesic Motion and the choice of 1 µν H(x, p) = g (x)pµ pν ; Relativistic Action 2 the action Z µ S[x, p] = (pµdx − H(x, p)dλ) ,

where λ is an afﬁne parameter, leads to Eq. (3.1.1) if we use the variational principle δS = 0 , δpµ δS = 0 . δxµ This action has three major problems:

1. it is not invariant under reparametrization of the afﬁne parameter λ;

2. it doesn’t fully characterize the free-falling particle, since its mass m does not appear;

3. it satisﬁes the condition that the particle moves along a geodesic, but it allows space- like geodesics as well as time-like or null ones, but the former have to be forbidden for a real particle (we can formulate this condition as the well-known mass-shell condition µν 2 1 g pµ pν + m = 0 ).

Last two problems are quite natural since we have already discussed that geodesic motion is mass-independent, so it is quite natural that an action could satisfy Eq. (3.1.1) without mass dependence or mass-shell condition.

1 we use here the signature (− + ++) for the ﬂat (Minkowskian) metric ηµν.

66 3.1. PARTICLES IN CURVED MANIFOLDS OF SPACE-TIME CHAPTER 3. PATH INTEGRALS

There is an interesting way to solve all these problems, and it is borrowed from Quantum Field Theory. Even if the action S is not invariant under general reparametrization of λ, i.e. under local transformations with δλ = ε(λ), it is invariant under the global transformation λ0 = λ + ε: Taylor-expanding p(λ0) and x(λ0) around λ and considering only ﬁrst order in ε (inﬁnitesimal transformation) we obtain, denoting derivation with respect to λ with a dot,

δxµ = εx˙µ ,

δpµ = εp˙ µ ; so the variation of the action becomes Z 1 δS = dλ ε p˙ x˙µ + p x¨µ − gµν x˙ρ p p − gµν p˙ p , µ µ 2 ,ρ µ ν µ ν for it to be zero for arbitrary ε, it must be zero the quantity inside parentheses. But since

δS µ µν = 0 ⇒ x˙ = g pν , (3.1.2) δpµ it is straightforward that the ﬁrst and the last term inside the parentheses cancel each other; then note that Eq. (3.1.1) allows to write the second term as

1 1 1 1 − p Γµ x˙νx˙ρ = − gµσ p Γ x˙νx˙ρ = − (g + g − g )x˙σx˙νx˙ρ = − g x˙σx˙νx˙ρ , 2 µ νρ 2 µ σνρ 2 σν,ρ σρ,ν νρ,σ 2 σν,ρ where we have used the definition of connection coefficients Γ as derivatives of the metric, the relation (3.1.2), the fact that the metric is symmetric and in the last equality the symmetry of x˙σx˙νx˙ρ in the permutation of indices; finally in the third term we use the relation

νρ ρ µν µσ ντ (gµνg ),σ = δµ ,σ = 0 ⇒ g ,ρ = −g g gστ,ρ and relation (3.1.2) to rewrite it as 1 g x˙σx˙νx˙ρ , 2 σν,ρ so effectively δS = 0 for arbitrary ε, thus λ → λ + ε is a global symmetry indeed. It corresponds to the conservation through geodesic motion of H(x, p), the Nöther current associated with the symmetry. In field theories one introduces gauge fields in order to localize global symmetries, we would like to do the same here: we introduce an auxiliary gauge field e(λ)1, usually gauge fields couple with the conserved current associated with the global symmetry, so we will try the same ansatz here, formulating a new action Z µ S[x, p, e] = dλ (pµx˙ − eH(x, p)) ; (3.1.3) this action is invariant under general reparametrizations δλ = ε(λ) if the gauge field has a variation under such a transformation δe = 2ε˙, so we have constructed an action invariant under general reparametrizations of the worldline, provided that such a transformation

1it is usually called the einbein.

67 Part II: CURVED SPACE-TIME

causes variations

δxµ = εx˙µ ,

δpµ = εp˙ µ , δe = 2ε˙ .

We have not yet analyzed the equations provided in that case by the variational method: δS = 0 ⇒ gµν p p = 0 , δe µ ν δS e = 0 ⇒ p˙ = − gνρ p p , δxµ µ 2 ,µ ν ρ δS µ µν = 0 ⇒ x˙ = eg pν ; δpµ we immediately see that the variational equation for e is the mass-shell condition for a massless particle, however we can derive the equation for geodesic motion from variational equations for x and p only if the einbein is constant. But the introduction of the gauge field introduced also a gauge freedom, underlying that there are more variables than physical degrees of freedom in the system; that freedom allows us to fix the einbein to be constant, so our The auxiliary field e model effectively describes a massless freely falling particle in a curved space-time. We have makes the action invariant under thus solved the problems of the action, that now is invariant under worldline reparametriza- reparametrizations tions and includes the mass-shell condition (in the case of a massless particle); gauge fixing of the worldline procedure explicit out the dynamics of the system (geodesic equation) restoring physical degrees of freedom, but destroys the covariance of the theory. The easiest way to introduce mass in such framework is to add a fictitious fifth space-time dimension x5, setting its

conjugate momentum p5 ≡ m, and setting   0    0   g(xµ, x5) = g(xµ) =  g(xµ) 0 ;      0 0 0 0 0 1

5 5 the term p5x˙ = mx˙ could be dropped, being a total derivative, so the action remains the 55 2 same as before apart for the additional term eg p5 p5/2 = em /2, ﬁnally obtaining an action with the same symmetries as before that reads: Z e S[x, p, e] = dλ p x˙µ − (gµν p p + m2) ; (3.1.4) µ 2 µ ν variational equations now becoming δS = 0 ⇒ gµν p p + m2 = 0 , δe µ ν δS e = 0 ⇒ p˙ = − gνρ p p , δxµ µ 2 ,µ ν ρ δS µ µν = 0 ⇒ x˙ = eg pν . δpµ

68 3.1. PARTICLES IN CURVED MANIFOLDS OF SPACE-TIME CHAPTER 3. PATH INTEGRALS

Now the equation for e gives the mass-shell condition in the general case of a mass m ≥ 0, and ﬁxing the gauge freedom with a constant einbein gives the geodesic equation. So we see that the covariant action in Eq. (3.1.4) describes a freely falling particle of mass m ≥ 0 in curved space-time with metric gµν(x). Using the variational equation for p we can get rid of conjugate momenta and obtain the conﬁguration space action

Z 1 S[x, e] = dλ e−1g (x)x˙µx˙ν − em2 . (3.1.5) 2 µν Getting then rid of the einbein e with its variational equation we obtain, for massive R p µ ν theories, S[x] = −m dλ gµνx˙ x˙ .

3.1.2: Quantization

Consider the phase space action (3.1.4), remembering the relativistic Hamiltonian H(x, p) we could write it as:

Z m2 S[x, p, e] = dλ p x˙µ − e H(x, p) + ; µ 2 and we see that variational equations could be written as

δS m2 = 0 ⇒ H(x, p) + = 0 , δe 2 δS ∂H = 0 ⇒ p˙ = −e , δxµ µ ∂xµ δS ∂H = 0 ⇒ x˙µ = e , δpµ ∂pµ the so-called Hamiltonian equations. Thus deﬁning Poisson brackets in a standard way

∂ f ∂g ∂ f ∂g [ ( ) ( )] ≡ − f x, p , g x, p p.b. µ µ , ∂x ∂pµ ∂pµ ∂x we obtain usual canonical relations (apart the presence of the einbein):

µ µ [x , pν]p.b. = δ ν , µ µ [x , eH(x, p)]p.b. = x˙ ,

[pµ, eH(x, p)]p.b. = −p˙ µ .

Since we don’t want the einbein in the quantization of the model we use a Gupta-Bleuler like quantization1: we ﬁx the gauge freedom, setting e(λ) = 1, obtaining Z µ S[x, p] = dλ pµx˙ − H(x, p) ,

1in analogy with the method introduced by Gupta and Bleuler for the quantization of the electromagnetic ﬁeld.

69 Part II: CURVED SPACE-TIME

since the additive factor proportional to m2 is now a constant unimportant for the equations µν 2 of motion, however we have to remember the constraint g pµ pν + m = 0 due to gauge ﬁeld. Poisson brackets now reduce effectively to the usual ones. We can now canonically quantize the system, deﬁning an Hilbert space , considering xˆµ , pˆµ and H(xˆ, pˆ) as operators in L( ), so the Poisson brackets becomeH commutators (with an additional i factor1), i.e.: H

µ µ [xˆ , pˆν] = iδ ν , [xˆµ, H(xˆ, pˆ)] = ixˆ˙µ ,

[pµ, H(xˆ, pˆ)] = −ipˆ˙ µ . p As usual is isomorphic to a space L2(R4, −g(x)d4x) (−g(x) is the absolute value of the determinantH of the metric2), where the unitary transform of operator xˆµ reduces to mul- µ tiplication by x and operator pˆµ reduces to −i∂µ. We have not considered the constraint Gupta-Bleuler 2 2 Quantization H(x, p) + m /2 = 0 yet. To impose it as an operator constraint, i.e. H(xˆ, pˆ) + m 1/2 = 0 is not realizable. Accordingly with the Gupta-Bleuler quantization of electrodynamics we will

restrict to ph, where H H 1 2 ψ ∈ ph ⇔ H(xˆ, pˆ) + m 1 ψ = 0 . (3.1.6) H 2 At this moment we can’t avoid any further to analyze the Hamiltonian operator H(xˆ, pˆ). The µν first ansatz we can try is the direct quantization H = g (xˆ)pˆµ pˆν/2, but it is not even self- adjoint! And self-adjointness is not all, since we want a relativistic theory for a quantum particle valid in any coordinate system, the states of Hilbert space must be invariant under general coordinate transformation, and if we impose that condition at a fixed value of the affine parameter, it must hold at any value: but since evolution is dictated by the Hamiltonian function (Schrödinger equation), it has to be also generally covariant. So we are looking for a self-adjoint Hamiltonian operator invariant under general coordinate transformations[26].

Theorem 3.1.1. The self-adjoint Hamiltonian operator

1 − 1 1 µν − 1 H(xˆ, pˆ) = (−g(xˆ)) 4 pˆ (−g(xˆ)) 2 g (xˆ)pˆ (−g(xˆ)) 4 (3.1.7) 2 µ ν is invariant under general coordinate transformations and classically equivalent to

1 H(x, p) = gµν(x)p p . 2 µ ν Proof. Classically an inﬁnitesimal general transformation is of the form x0µ = xµ + ξµ(x), i.e. δxµ = ξµ(x), where ξµ(x) is an inﬁnitesimal arbitrary function. The ps, being covariant, would transform as ∂xν p0 = p , µ ∂x0µ ν 1 we have set }h = 1. p 2 −g(x)d4x is the invariant measure in arbitrary curved space-time.

70 3.1. PARTICLES IN CURVED MANIFOLDS OF SPACE-TIME CHAPTER 3. PATH INTEGRALS

ν i.e. δpµ = −∂µξ pν. Performing two consecutive transformations we ﬁnd

µ nu µ ν µ [δ1, δ2]x = ξ2 (x)∂νξ1 (x) − ξ1(x)∂νξ2 (x) ; so the generator of quantum transformations has to satisfy the algebra G µ µ [ (ξ1), (ξ2)] = (ξ2 ∂µξ1 − ξ1 ∂µξ2) . G G G It is possible to see that splits into an orbital and a “spin” part satisfying the following relations: G

= o + s , with G G G i µ µ o = − (ξ pµ + pµξ ) , G 2 Z 4 ρ ρ ρ δ s = d x ξ gµν,ρ + (∂µξ )gρν + (∂νξ )gρµ . G δgµν In fact, calculating commutators with we obtain the right quantum transformations (from now on we will omit ˆ over x and pGsince we deal only with operators, so confusion is We choose the avoided): Quantum Hamiltonian to be δxµ = [xµ, ] = ξµ(x) , covariant and G self-adjoint 1 ν ν δpµ = [pµ, ] = − (pν∂µξ + ∂µξ pν) , G 2 ρ ρ δgµν = [gµν, ] = −(∂µξ )gρν − (∂νξ )gρµ ; G where the last transformation is the inﬁnitesimal counterpart of the ﬁnite transformation rule ∂xρ ∂xσ g0 (x0) = g (x) , µν ∂x0µ ∂x0ν ρσ so the determinant transforms as 2 0 0 ∂x −g (x ) = − g(x) , with ∂x0

∂x ∂xµ = det . ∂x0 ∂x0ν Since infinitesimally ∂xµ = δµ − ∂ ξµ , ∂x0ν ν ν we can generalize momentum infinitesimal transformation rule to a finite one in that way: ∂xν 1h ∂xν i p0 = p − , p ; µ ∂x0µ ν 2 ∂x0µ ν but since [ f (x), pµ] = i∂µ f (x) we obtain

h ∂xν i ∂x0ρ ∂2xν , p = i , ∂x0µ ν ∂xν ∂x0ρ∂x0µ

71 Part II: CURVED SPACE-TIME

so ﬁnally remembering that ln det A = tr ln A we have

ν 0 ∂x i ∂x p = p − ∂ ln . µ ∂x0µ ν 2 ν ∂x0 Using all these relations we see that ν 0 1 0 0 − 1 ∂x 1 − 1 (−g ) 4 p (−g ) 4 = (−g) 4 p (−g) 4 , µ ∂x0µ ν ν 0 − 1 0 0 1 − 1 1 ∂x (−g ) 4 p (−g ) 4 = (−g) 4 p (−g) 4 , µ ν ∂x0µ so since ∂x0ρ ∂x0σ g0µν(x0) = gρσ(x) , ∂xµ ∂xν we see that H(x0, p0) = H(x, p).

Now that we know the right quantum form of covariant Hamiltonian operator we would p like to see its action on L2(R4, −g(x)d4x): since in non-relativistic framework it is proportional to Laplacian operator, in our relativistic case we would expect it to be proportional p to the covariant Laplacian operator. It is indeed the case: if f (x) ∈ L2(R4, −g(x)d4x) we know that ! 1 − 1 1 µν − 1 (H(x, ∂ ) f )(x) = − (−g(x)) 4 ∂ (−g(x)) 2 g (x)∂ (−g(x)) 4 f (x) ; x 2 µ ν

the problem is to calculate the derivative of (−g(x))a, where a = 1/2, −1/4, . . . ; but

a a ∂αh(x) = ah(x) ∂α ln h(x)

−1 if a < 1, so using the property ∂α ln |det A| = tr(A ∂αA) valid for any matrix A, we have

a a νρ ∂µ(−g) = a(−g) g gνρ,µ ;

performing all the boring calculations and ﬁnally remembering that if ∇µ is the covariant derivative, acting on a scalar function like f (x) we have ∇µ f (x) ≡ ∂µ f (x), we obtain indeed 1 1 H f = − gµν∇ ∇ f ≡ − ∇2 f , 2 µ ν 2 i.e. the Hamiltonian operator acts like minus the covariant Laplacian times one-half, as we expected. Now it is clear what is the meaning of the mass-shell constraint (H + m2/2) f = 0, that restrict the Hilbert space to the physical one: it could be written as

(−∇2 + m2) f (x) = 0

i.e. physical states f ∈ ph must obey generalized Klein-Gordon equation. Conversely, we can suggest that imposingH Klein-Gordon equation on a particle is analogous to constraining it on the mass-shell, and this is indeed a necessary condition to avoid non-causal, unphysical behaviors.

72 3.2. DERIVATION OF PATH INTEGRAL CHAPTER 3. PATH INTEGRALS

3.2: Derivation of path integral

ovariant Hamiltonian describes evolution in λ, with Schrödinger equation

C d i Ψ = H(x, ∇ )Ψ , dλ x

So if the Hamiltonian is explicitly λ-independent1 the unitary operator that describes evolution in λ is

−i(λ f −λi)H U(λ f − λi) = e ; so we would like to calculate the integral kernel K(x f , λ f ; xi, λi), that satisﬁes equation Z q 4 ψ(x f , λ f ) = d x −g(xi) K(x f , λ f ; xi, λi)ψ(xi, λi) , in a path integral form. We derive this result in a more general setting: we extend the number of space-time dimensions to a value D, and we perform a Wick rotation both in λ and in the target time coordinates (in D space-time dimensions we could have more than one time coordinate), so we are considering an Euclidean D-dimensional space with a metric2 gij(x) that locally reduces to δij. We call the rotated afﬁne parameter t, so previous formulas would read3 d − Ψ = H(x, ∇ )Ψ , dt x −(t f −ti)H U(t f − ti) = e , Z q D ψ(x f , t f ) = d x g(xi) K(x f , t f ; xi, ti)ψ(xi, ti) .

All the considerations about the covariant Hamiltonian of previous section extend immediately to Euclidean afﬁne evolution in Euclidean target space-time. We will see that to calculate effective actions it is quite useful to extend the Hamiltonian introducing an external space-time dependent scalar potential V(x), so we are considering

1 − 1 1 ij − 1 H(xˆ, pˆ) = g(xˆ) 4 pˆ g(xˆ) 2 g (xˆ)pˆ g(xˆ) 4 + V(xˆ) ; 2 i j and since V(x) is a scalar function the covariance of the Hamiltonian is not affected.

3.2.1: Time slicing

As we saw in Part I, the integral kernel K reduces to the calculation of the transition element

−βH K(x f , t f ; xi, ti) = K(x f , β; xi, 0) = hx f |e |xii ,

1 and therefore it is also implicitly independent, since ∂λ d mathcalH = dλ H. 2we will use roman indices to indicate we are working in a space-time locally Euclidean. 3since the metric is now positive deﬁnite we avoid the minus sign in the absolute value of the determinant.

73 Part II: CURVED SPACE-TIME

where as usual β ≡ t f − ti. Since as we mentioned in the previous section the Hilbert p space we are considering is isomorphic to L2(RD, g(x)dDx), we need to use the following normalizations for position and momentum eigenstates:

so the hx|pi element becomes

D − 1 ip xi hx|pi = (2π) 2 g(x) 4 e i .

We can now proceed as in Section 1.2, setting β = Ne, and we obtain:

The problem is to evaluate the transition element

−eH hxk|e |pki ,

since H is not in normal ordering. However instead of rewriting it in normal ordered form, Weyl ordering we will use a different ordering, the Weyl ordering1, since a theorem proved in Appendix D

however if it might be difficult to write H in a closed Weyl ordered form, it is almost impos- sible to write exp{−eH} in Weyl ordering. However, as discussed in Appendix D, we can make the approximation, at first order in e (justified since we are taking the limit e → 0): e−eH = e−eHW . W 1We say that, given an operator O(x, p) (we omit hats over x and p, but we are dealing with operators), its Weyl ordered form OW (x, p) is the same operator where all the xs and ps are arranged in a symmetrical form, plus terms due to [x, p]. As an example consider the operator xp [26]:

1 1 1 1 xp = (xp + px) + (xp − px) = (xp) + [x, p] = (xp) + i = (xp) ; 2 2 S 2 S 2 W

where we have called (xp)S the symmetrical part (xp + px) of xp.

74 3.2. DERIVATION OF PATH INTEGRAL CHAPTER 3. PATH INTEGRALS

We also note that it could be quite difﬁcult to write a symmetric form for H if it contains an additive, only x-dependent term as V(x). So in the evaluation of

−eH hxk|e |pki , we use Trotter formula to write it as

−eH0 −eV −eV(xk) −eH0 hxk|e e |pki = e hxk|e |pki , with

1 − 1 1 ij − 1 H (xˆ, pˆ) = g(xˆ) 4 pˆ g(xˆ) 2 g (xˆ)pˆ g(xˆ) 4 . 0 2 i j

The problem is that H0,W is evaluated at the midpoint x¯k−1/2, while V is evaluated at xk. That 1 is not a problem, since we can show that at order zero in e, V(xk) = V(x¯k−1/2) : expanding V(xk) around x¯k−1/2 we obtain

∞ (n) n V (x¯k−1/2) xk − xk−1 V(xK) = V(x¯k−1/2) + ∑ ; n=1 n! 2 but xk − xk−1 ≡ x(tk−1 + e) − x(tk−1), so expanding x(tk−1 + e) about x(tk−1) we obtain  n ∞ (n) ∞ V (x¯k−1/2) 1 1 (j) j V(xK) = V(x¯k−1/2) + ∑  ∑ x (tk−1)e  ; n=1 n! 2 j=1 j! so we have shown that at zeroth order in e the relation is satisﬁed. From now on when we write HW (x¯k−1/2, pk) we mean

HW (x¯k−1/2, pk) = H0,W (x¯k−1/2, pk) + V(x¯k−1/2) .

Some effort and a lot more calculation permits to find H0,W: 1 ij 1 ij k l H0,W (xˆ, pˆ) = g (xˆ)pî pˆj + (R(xˆ) + g (xˆ)Γ il(xˆ)Γ jk(xˆ)) ; 2 S 8 where R(x) is the scalar curvature, and Γ(x) the metric connection coefficients. We can summarize our results, calculating the integral kernel K as:

N D N−1 1 Z − d pk D −SN K(x f , t f ; xi, ti) = lim(g(x f )g(xi)) 4 d xk e , (3.2.2) e→0 ∏ ( )D ∏ k=1 2π k=1 with

N (xk − xk−1) SN = e ∑ HW (x¯k−1/2, pk) − ipk · , k=1 e 1we stop to order zero because V multiplies e, and since our entire approximation is at ﬁrst order in e we can approximate V at order zero.

75 Part II: CURVED SPACE-TIME

where we identify xN ≡ x f and x0 ≡ xi. Taking the formal limit we obtain the phase space path integral formula: Z −SE[x,p,β] K(x f , β; xi, 0) = [p(t)] [x(t)] e , (3.2.3) D D x(0)=xi x(β)=x f where the phase space Euclidean action is Z β Z β 1 ij SE[x, p, β] = dt (−ip · x˙ + HW (x, p)) = dt −ip · x˙ + g (x)pi pj + V(x) + VTS(x) , 0 0 2

where we have called the time-slicing counterterm VTS the Weyl ordering term 1 V (x) = (R + gijΓk Γl ) . TS 8 il jk In that case the meaning of the functional measure [x(t)] [p(t)] is not clear, so the formal expression (3.2.3) is meaningless unless we introduceD aD proper regularization scheme

Path integral in to manage the otherwise ambiguous functional integration: in particular we will see that curved space constructing a perturbative theory in continuum limit leads to divergences and undefined needs products of distributions that assume significance only in a regulated environment. Time counterterms dependent on the slicing procedure[27, 28] that led to (3.2.3) is one example of such regularization schemes. regularization At the discretized level everything is well defined and the perturbative theory produces un- scheme chosen ambiguous results. As we will see in next sections, we can say that K has a well defined path integral provided we introduce a regularization scheme, and add a regularization- R dependent counterterm dt VCT to the classical Euclidean action. However, as we will discuss, this model is “super-renormalizable”, so a single counterterm is sufficient. An- other regularization-dependent factor, called a global renormalization factor, arise in order to properly define the path integral functional measure, however it is fixed by consistency conditions1. Therefore every time we write the path integral formulation Z −SE[x,p,β] K(x f , t f ; xi, ti) = [p(t)] [x(t)] e , (3.2.4) D D x(0)=xi x(β)=x f we must keep in mind we have to fix a regularization scheme in order to solve ambiguities, and therefore the action reads Z β i 1 ij SE[x, p, β] = dt −ipix˙ + g pi pj + V(x) + VCT,ph(x) ; 0 2 where the counterterm depends on the regularization scheme we have chosen, and on the fact we are dealing with a phase space formulation. In the case of time slicing we have seen ij k l that the counterterm VTS,ph(x) = (R + g Γ ilΓ jk)/8. 1as an example by the condition that Z q D ψ(x f , t f ) = d xi g(xi)K(x f , t f ; xi, ti)ψ(xi, ti)

satisﬁes the Schrödinger equation.

76 3.2. DERIVATION OF PATH INTEGRAL CHAPTER 3. PATH INTEGRALS

3.2.2: Lagrangian formulation

Although we have to be careful when we deal with the continuum expression (3.2.4), we can extend some considerations we have done in flat space-time. In particular we would like to integrate out the momenta, in order to formulate a Lagrangian version of path integral. In Section 1.2.3 we saw that it is possible for a wide range of Hamiltonian functions, provided such function is a quadratic form of the momenta. Well it is indeed the case, so we can repeat the procedure described in that section and integrate out the momenta1: at a first glance the price to pay is, apart some constant factors that are easily adsorbed in the functional measure [x(t)], a functional determinant that now, unlike most flat-space cases, is x(t)-dependent. InD particular it is the determinant of the inverse of the metric to the power of minus one-half, but since the determinant of the inverse is the inverse of the determinant we obtain a factor p g(x(t)). So we have to deal with the formula

Z q −SE[x,β] K(x f , t f ; xi, ti) = [x(t)] g(x(t)) e , D x(0)=xi x(β)=x f where the Euclidean action is

Integrating out β Z 1 momenta is a tricky [ ] = i j + ( ) + ( ) SE x, β dt gijx˙ x˙ V x VCT,con f x . procedure: 0 2 ambiguities and divergencies arise First of all we expect that the formal functional integration over momenta is not able to preserve the regularization introduced in phase space, so the counterterm VCT,con f (x) would differ from the phase space one, even in the same regularization environment. Furthermore p the additional g(x(t)) could be seen as the factor needed to make the functional measure covariant, by generalization of the usual vector space measure: in fact while the measure element in phase space dD pdDx is relativistically covariant, in configuration space the co- p variant measure element is dDx g(x). Since as we will see shortly we are able to manage that functional determinant in a quite elegant way, it seems that integrating out the momenta was not a big deal, unfortunately it is not the case: in fact through a careful analysis of phase space path integrals, as we will do in the next chapter, emerges an interesting fact, i.e. that the phase space model does not present divergences or ambiguities, and perturbative calculations could be done directly in the continuum limit. So the price to pay when we pass to configuration space is higher than what it seems: in getting rid of the momenta we introduce divergences and ambiguities in the continuum limit and thus the need of regularization schemes and suitable counterterms for perturbative calculations. However a more detailed analysis and further considerations will be done in the next chapter, by now we focus on the analysis of configuration space path integral.

1the calculations in Section 1.2.3 are in real time, however the generalization to Euclidean case is straightforward and doesn’t present any theoretical difﬁculty.

77 Part II: CURVED SPACE-TIME

3.3: Continuum Manipulations

e proceed with our analysis of conﬁguration phase path integral for a quantum me- W chanical model in curved space-time developing some general concepts in the continuum limit. However we recall that such manipulations are only formal, they are justiﬁed and meaningful only when we introduce a regularization scheme.

3.3.1: Ghosts p Even if we could identify [x(t)] g(x(t)) as the properly covariant functional measure with proper regularization,D surely it is not translationally invariant. Since in order to calculate free propagators and perform perturbative calculations we would like to use generating functionals and functional derivatives as we did in Section 1.3.3, we need the measure to be translationally invariant in order to perform Gaussian integrals. We need some trick to p rewrite the factor g(x(t)), and restore translational invariance. But since such factor is a functional determinant, we can rewrite it as a functional integral in some auxiliary ﬁelds, that will be called “ghosts”, since they don’t appear in our original model, but as we will see their diagrams are fundamental to eliminate the divergences of the model. We see again that the transition from phase space to conﬁguration space is quite awkward: integrating

Introduction of out momenta we recover Lagrangian formulation of the theory, but introduce otherwise Ghost Auxiliary non present divergences, and break translational invariance of the measure; in order to set Fields to exponentiate the up things we need to introduce some auxiliary ﬁelds (three, as we will see) otherwise un- metric determinant predicted in our model. We know very well that Gaussian path integrals of a real function

Z Z 1 0 00 0 00 i 0 j 0 − 1 [a(t)] exp − dt dt Aij(t , t )a (t )a (t ) = (Det A) 2 , D 2

if the functional measure is properly defined1. But in our case we have g1/2, so a real field a(t) is not sufficient. However, Gaussian integration of two Grassmann functions b(t) and c(t) defined as

Z Z 0 00 0 00 i 0 j 0 [b(t)] [c(t)] exp − dt dt Aij(t , t )b (t )c (t ) = Det A , D D so we can rewrite q Z g(x(t)) ∼ [a(t)] [b(t)] [c(t)] e−Sgh[a,b,c,β] , D D D where a(t) is a real function, b(t) and c(t) Grassmann functions and

Z β 1 i j i j Sgh[a, b, c, β] = dt gij(x) a a + b c , 0 2 1in general the result is proportional to (Det A)−1/2, however the constant factor can be adsorbed or in the deﬁnition of the ghost functional measure or in the global renormalization constant of the regularization scheme.

78 3.3. CONTINUUM MANIPULATIONS CHAPTER 3. PATH INTEGRALS and the ∼ symbol means that the eventual constant factor resulting from integrating ghosts is adsorbed in the global renormalization constant of the chosen regularization scheme. We can summarize our results writing: Z −SE[x,a,b,c,β] K(x f , t f ; xi, ti) = [x(t)] [a(t)] [b(t)] [c(t)] e , (3.3.1) D D D D x(0)=xi x(β)=x f where the total action SE[x, a, b, c, β] is:

Z β 1 i j i j i j SE[x, a, b, c, β] = SE + Sgh = dt gij(x˙ x˙ + a a + b c ) + V(x) + VCT(x) . 0 2

3.3.2: Analysis of divergences

We have already mentioned that since this theory is super-renormalizable a single counterterm is sufficient to give correct results at any perturbative order[26]: we will now justify our assertion analyzing our model with power counting techniques borrowed from quantum field theory. The basic assumption we make is that we can see the path x(t) as a classical field of the single time variable t, or in QFT terminology, that we are dealing with a (0 + 1)-dimensional quantum field theory1: so if we call the usual field ϕn(xµ), we identify xµ with the one dimensional t ∈ [0, β], and ϕn with xi. We see that in that case the 1-dimensional “space- time” t is finite, so the Fourier expansion of the D massless2 scalar fields xi will be a series that takes into account properly the boundary conditions of the xs at t = 0, β. Since, as discussed in the next section, we will usually shift the field to qi(t) in a way that q vanishes at the boundaries, the proper Fourier series will be a sine series, and the free propagators become proportional to

∞ 2 ( 0) = − ( ) ( 0) ∆ t, t ∑ 2 2 sin πmt sin πmt , m=1 π m as we will see in detail. So in the momentum space Feynman rules any internal line would be associated to a discrete sum rather than to an integral over “momenta”. Furthermore since t is ﬁnite, exists a minimal frequency3: if we factorize eventual zero modes (modes with null frequency), we don’t have any infrared divergency, caused by p → 0 in massless QFT theories, since minimal frequency is ﬁxed and 6= 0. We have ultraviolet problems instead: they are due to arbitrarily great momenta, and in 1 1 the limit m → ∞ the sum of m2 is equivalent to an integral over k2 . Physically, this is due to the fact that ultraviolet divergencies should not feel the boundaries, since they are related to the microscopical structure of the theory. So in our discussion of ultraviolet divergencies we

1we write 0 + 1 since we have no spatial dimensions and one temporal dimension, however since we are in Euclidean time this distinction is quite meaningless. 2a massive term would be determined by the coupling constant of a factor ∝ x2 in the action, but unless it appears in the potential V(x), and we will see that usually it doesn’t, such ﬁeld theory is a massless one. 3i.e. momentum, since in that model we identify momentum p ∼ πm, the latter being indeed a frequency.

79 Part II: CURVED SPACE-TIME

will consider Feynman graphs with usual integration rules and propagators proportional to 1 k2 , obviously in momentum space. The action we have to analyze is

Z 1 S = dt g(q)q˙q˙ + V(q) , 2

corresponding to our model, omitting for simplicity vectorial indices. Since we have }h = c = 1 the action is a pure number, time has the dimension of the inverse of a mass: [t] = M−1; and obviously [q] = M−1/2. If we expand the metric and the potential in a Taylor series we have:

∞ ∞ n n V(q) = ∑ Vnq , g(q) = ∑ gn+2 q , n=0 n=0

where the subscript indicates how many ﬁelds a given vertex contains. So easily we deduce that

n +1 n −1 [Vn] = M 2 ; [gn] = M 2 .

Interaction corresponds to terms with n ≥ 3, so all coupling constants have positive mass dimensions. We know that such a theory in QFT is called super-renormalizable: namely, from a certain loop level on there are no more superficial divergencies. Furthermore we will see The model is Super- that divergencies are canceled by ghosts, even if in an ambiguous way, so renormalization Renormalizable involves only finite terms: the VCT(x) already introduced. Let’s develop our considerations in detail: as usual in QFT we say that a diagram has a superficial degree of divergency D if it corresponds to an integral

Z ∞ dk kD−1 .

Obviously D > 0 corresponds to a behavior ∼ kD near infinity; D = 0 corresponds to ∼ ln k, 1 and we have superficial convergence for D < 0. Since any propagator behaves like k2 , each internal line in the diagram provides a factor −2 to D. Any loop corresponds to R dk , so to a factor +1 to D. Any interaction term with field derivatives also corresponds to an additional factor +1 to D, as in the case of g interaction terms that contain two derivatives. To summarize, if we call L the number of loops in the diagram, I the number of internal

lines, and gn the number of vertices with that coupling constant in the diagram considered, its maximum degree of divergency would be

D = L − 2I + 2 ∑ gn . n

Since for a connected diagram holds the following topological identity:

L = I − ∑(Vn + gn) + 1 , n

80 3.3. CONTINUUM MANIPULATIONS CHAPTER 3. PATH INTEGRALS we obtain

D = 2 − L − 2 ∑ Vn ; n and it is clear that such formula makes sense only for L > 0, since at the tree level no integrations survive after we impose momentum conservation. We see immediately that any diagram with one or more V vertex converges; while if Vn = 0 ∀n, we can have superfi- cially divergent diagrams only when L ≤ 2, as expected in a super-renormalizable theory. Divergencies are eliminated by the divergent contributions of ghosts: any interaction n n gnq q˙q˙ is balanced by a term gnq (aa + bc). The remaining logarithmic divergencies usually could be eliminated with symmetric integration, however finite results remain ambiguous: they depend on the regularization scheme we are considering. In particular Feynman graphs contain products of distributions, that are not define mathematically, and this is the ambiguity that has to be solved at the regularized level.

3.3.3: Perturbative Expansions

As we did in flat space-time, we would like to find free propagators, in order to perform perturbative calculations. In order to do that we have to split the action in Eq. (3.3.1) into a free and an interacting part. Furthermore, we need a controlling parameter for the perturbative series, and we choose β, so we single it out from the action doing the reparametrization τ = t/β − 1, and suitably redefining ghosts obtaining1:

Z 0 1 1 1 i j i j i j 2 S[x, a, b, c] = dτ gij(x)(x˙ x˙ + a a + b c ) + β V(x) + VCT(x) . β β −1 2

In order to do perturbative calculations, we have to split S = S2 + Sint, where the free part is used to ﬁnd propagators and thus has to be quadratic in coordinates and ghosts. In particular we ﬁnd:

Z 0 1 i j i j i j S2 = dτ gij(x f )(x˙ x˙ + a a + b c ) , (3.3.2) −1 2 Z 0 1 i j i j i j 2 Sint = dτ [gij(x) − gij(x f )](x˙ x˙ + a a + b c ) + β (V(x) + VCT(x)) ; (3.3.3) −1 2 where g(x f ) is the metric calculated at the final point and is therefore a constant. It is easier to interpret the meaning of propagators if we have vanishing boundary con- i i i i i i i ditions, so we make the translation q (τ) ≡ x (τ) − x f + ξ τ, where ξ ≡ xi − x f ; we see immediately that q(0) = q(−1) = 0, and since our functional measure [x(τ)] is transla- D 1ghosts redefinition will be adsorbed in the definition of their functional measure, i.e. fixed be the global renormalization constant of the model.

81 Part II: CURVED SPACE-TIME

tionally invariant, we obtain:

1 i j Z 1 − 2β gij(x f )ξ ξ − β S[q,a,b,c,β] K(x f , t f ; xi, ti) = e [q(τ)] [a(τ)] [b(τ)] [c(τ)] e , D D D D VBC

S[q, a, b, c] = S2[q, a, b, c] + Sint[q, a, b, c] , Z 0 1 i j i j i j S2[q, a, b, c] = dτ gij(x f ) q˙ q˙ + a a + b c , −1 2 Z 0 1 i j i j i j Sint[q, a, b, c] = dτ [gij(x(q)) − gij(x f )](x˙ (q)x˙ (q) + a a + b c ) −1 2 2 +β V(x(q)) + VCT(x(q)) .

So now we can calculate propagators from S2 adding sources, performing Gaussian integrations, and making functional derivatives with respect to the sources as we did in Sec- tion 1.3.3; again we ﬁnd that n-point propagators satisfy (Wick Theorem), and taking into account that we are in Euclidean and not real time and we have an additional 1/β factor that multiplies the action we ﬁnd the following formulas1:

1 i j 1 2β gij(x f )ξ ξ − β Sint K(x f , t f ; xi, ti) = h 1 ie h e i , i j ij h q (τ)q (σ) i = −βg (x f )∆(τ, σ) , i j ij h a (τ)a (σ) i = βg (x f )∆gh(τ, σ) , i j ij Conﬁguration h b (τ)c (σ) i = −2βg (x f )∆gh(τ, σ) . Space Propagators in the continuum We have to explain the meaning of ∆(τ, σ) and ∆ (τ, σ): as usual we read from S2 that they limit gh are the inverse respectively of the kinetic operators

d2 K(τ, σ) = δ(τ − σ) , dτ2

Kgh(τ, σ) = δ(τ − σ) ;

while the additional minus two factor in the h bc i propagator is due to Gaussian integration with anti-commuting variables. General solutions of equations

d2 ∆(τ, σ) = δ(τ − σ) , dτ2 Z 0 0 0 δ(τ − τ )∆gh(τ , σ) = δ(τ − σ) −1 in κ([−1, 0])∗ are:

∆(τ, σ) = τ(σ + 1)θ(τ − σ) + σ(τ + 1)θ(σ − τ) ,

∆gh(τ, σ) = δ(τ − σ) .

We immediately see that ghost propagator is proportional to a delta function, and thus it is divergent, as it is the propagator h q˙q˙ i, also proportional to δ(τ − σ), as a matter of

1using the same notation as in Section 1.3.3.

82 3.4. REGULARIZATION SCHEMES CHAPTER 3. PATH INTEGRALS fact ghost diagrams sum up to diagrams with q˙q˙ lines, and the resultant is a convergent diagram, as we will see. Using Fourier transform, that in κ([−1, 0]) is a sine series, we can ﬁnd the distributional meaning of θ(0) as an element of κ([−1, 0])∗, repeating the procedure of Section 1.3.4, and we ﬁnd θ(0) = 1/2, so we can say that

∆(τ, τ) = τ(τ + 1) .

However even if we have found a distributional meaning of ∆ and ∆gh, we can not make explicit calculations yet, since we don’t know how to manage products of distributions, and how to sum up divergent contributions to ﬁnite values (and the ﬁnite values they take): these calculations have to be made at the regulated level, and depend on the regularization chosen.

3.4: Regularization schemes

ow that we have developed the general structure of perturbative calculations in the N continuum limit we can brieﬂy review the results obtained in various regularization schemes. We are not going to develop algebraic calculations in the very detail, since a detailed analysis will be given in the next chapter for phase space path integrals. We perform calculations up to two loops, since at this order we ﬁx univocally all the counterterms, as discussed in Section 3.3.2. In the controlling parameter β, a two-loop calculation is up to order β, so since in the exponential

− 1 S e β int

2 we have a factor 1/β, we need to expand Sint up to order β : explicitly we write Sint = 5/2 S3 + S4 + O(β ), where the index 3 or 4 means the square of the order in β of this part of the expansion. Now expanding the exponential

− 1 S 1 1 e β int = 1 − S + S2 + ..., β int 2β int and retaining only terms up to order β we obtain

− 1 S 1 1 e β int = 1 − (S + S ) + S2 + O(β3/2) , β 3 4 2β 3 so performing the average and remembering that the average of a sum is the sum of averages we have

− 1 S 1 1 − 1 (h S i+h S i)+ 1 h S2 i h e β int i = 1 − (h S i + h S i) + h S2 i + O(β3/2) = e β 3 4 2β 3 c + O(β3/2) β 3 4 2β 3

2 2 where h S3 ic is the average of S3 with only connected diagrams, i.e. without diagrams that 2 2 2 are written as the product of two h S3 i diagrams, in particular h S3 ic ≡ h S3 i − (h S3 i) , 2 since (h S3 i) diagrams are counted in the second order expansion of the exponential.

83 Part II: CURVED SPACE-TIME

Previous considerations, that are still valid for any regularization scheme, lead to an expression:

1 i j 1 1 2 − 2β gij(x f )ξ ξ − β (h S3 i+h S4 i)+ 2β h S3 ic 3/2 K(x f , t f ; xi, ti) = h 1 ie e + O(β ) ;

we know that the principal counterterm we have to fix is VCT(x), but since it is multiplied 2 p by β in Sint, and each x(τ) is of order β, as we will see in detail, we expand it to zeroth order around x f , obtaining a function VCT(x f ), that has to be calculated imposing a renormalization condition on K; furthermore, we already said that we have to fix also a global renormalization factor that has to take into account the proper definition of functional measures, as well as the rescaling of affine parameter t, and the determinants arising from Gaussian integration: this global factor is A ≡ h 1 i, and also it has to be fixed by the

renormalization condition. A suitable renormalization condition is that ψ(x f , t f ), written as

Z q D ψ(x f , t f ) = d xi g(xi)K(x f , t f ; xi, ti)ψ(xi, ti) , (3.4.1)

has to obey Schrödinger equation d 1 − (x t ) = − ∇2 + V(x ) (x t ) ψ f , f x f f ψ f , f . dt f 2 Finally we can analyze the order in β of the variables q, a, b, c, ξ: from the propagators p h qq i, h aa i and h bc i we see immediately that any q, a, b or c counts as a factor ∼ β; D D and since in Eq. (3.4.1) we have to perform an integration in d xi ≡ d ξ and K provides a Gaussian factor

− 1 g (x )ξiξ j e 2β ij f p we can construct a propagator h ξξ i again of order β, so also ξ ∼ β. So in fact we can

expand g(x), V(x) and VCT(x) about x f , obtaining for S3 and S4 the following results:

Z 0 1 k k i j i j i j i j i j S3 = dτ gij,k(x f )(q − ξ τ)(ξ ξ − 2ξ q˙ + q˙ q˙ + a a + b c ) , −1 2 Z 0 1 k l k l 2 k l i j i j i j i j i j S4 = dτ gij,kl(x f )(q q + ξ ξ τ − 2q ξ τ)(ξ ξ − 2ξ q˙ + q˙ q˙ + a a + b c ) −1 4 2 i +β V(x f ) + VCT(x f ) .

3.4.1: Mode Regularization

The basic idea of mode regularization[29, 30, 31, 32] is to rewrite every τ-dependent quantity in Fourier transform, that due to the boundaries is a sine expansion1:

∞ i i ϕ (τ) = ∑ ϕm sin(πmτ) , m=1 1since ghosts were introduced to exponentiate the determinant of the metric, we don’t need them at endpoints, so we can impose also on them vanishing boundary conditions.

84 3.4. REGULARIZATION SCHEMES CHAPTER 3. PATH INTEGRALS with ϕ is one of the fields q, a, b or c. We can say that integrating over all possible paths vanishing at the boundaries is equivalent to an integration over all values of Fourier coefficients, and as usual additional factors will be adsorbed in the global factor A. In that way we can explicitly integrate over τ, and obtain an action depending only on Fourier coefficients. Adding sources and performing Gaussian integration as usual determines propagators, that are written in the well-known form:

1 i j 1 2β gij(x f )ξ ξ − β Sint K(x f , t f ; xi, ti) = h 1 ie h e i , i j ij h q (τ)q (σ) i = −βg (x f )∆(τ, σ) , i j ij h a (τ)a (σ) i = βg (x f )∆gh(τ, σ) , i j ij h b (τ)c (σ) i = −2βg (x f )∆gh(τ, σ) ; but now we have

∞ 2 ( ) = − ( ) ( ) ∆ τ, σ ∑ 2 2 sin πmτ sin πmσ , m=1 π m ∞ ∆gh(τ, σ) = ∑ 2 sin(πmτ) sin(πmσ) . m=1

An explicit calculation conﬁrms that these are indeed the Fourier expansions of the distributions:

∆(τ, σ) = τ(σ + 1)θ(τ − σ) + σ(τ + 1)θ(σ − τ) ,

∆gh(τ, σ) = δ(τ − σ) .

This far we haven’t introduced a regularization yet, we only rewrote previous results in the space of Fourier modes. The regularization is effectively performed imposing a cutoff M in Fourier modes, and then taking the limit M → ∞: explicitly we have

M i i ϕ (τ) = lim ϕm sin(πmτ) , M→∞ ∑ m=1 and the regulated ∆ and ∆gh are now:

M 2 ( ) = − ( ) ( ) ∆ τ, σ ∑ 2 2 sin πmτ sin πmσ , m=1 π m M ∆gh(τ, σ) = ∑ 2 sin(πmτ) sin(πmσ) . m=1

Let’s see how to use such regulated propagators to perform perturbative calculations. As we did in Section 1.3.4 we write the derivative of a distribution ∆(τ, σ) with respect to its ﬁrst variable with •∆, a derivation with respect to the second variable with ∆•. Using Wick 2 theorem to perform the averages h S3 i, h S4 i and h S3 ic one ﬁnds expressions that involve integrals in dτ, dσ or dτdσ of the ∆ and ∆gh distributions. Such integrals are ambiguous, or

85 Part II: CURVED SPACE-TIME

divergent in the continuum limit, so have to be manipulated at the regulated level: even if we did not perform all the explicit calculations here we give an explicit example, the integral

Z 0 • • I = dτ ∆|τ( ∆ + ∆gh)|τ ; −1

• • in the continuum limit it is divergent, since both ∆ and ∆gh are proportional to a delta •• function. However at the regulated level we immediately see that ∆gh = ∆, and that

M • • •• h 2 2 i ( ∆ + ∆)|τ = ∑ −2 cos (πmτ) + 2 sin (πmτ) m=1 M 2 = ∂τ ∑ − sin(πmτ) cos(πmτ) m=1 πm • = ∂τ ( ∆|τ) ;

so we can write

Z 0 Z 0 • • I = lim dτ ∆|τ∂τ ( ∆|τ) = lim dτ ∂τ (∆|τ) ∆|τ , M→∞ −1 M→∞ −1

• and the integration by parts is justiﬁed since at regulated level both ∆|τ and ∆|τ vanish at endpoint τ = −1, 0. Finally the last integration can be computed directly in the continuum limit since it does not present any more ambiguities, and thus

Z 0 Z 0 2 • 1 1 I = dτ ∂τ (∆|τ) ∆|τ = −2 dτ τ + = − , −1 −1 2 6

a perfectly unambiguous and finite result. However we emphasize the fact that this result depends on the tricks we used at regulated level, and therefore on the regularization scheme we use, in other schemes I could have a different result. Performing all calculation and solving ambiguities as we just explained, and finally imposing the renormalization condition one find the result

− D 1 1 ij mn k l A = (2πβ) 2 , V (x) = R − g g g Γ Γ . MR 8 24 kl im jn

2 We report the explicit results of h S3 i, h S4 i and h S3 ic[26] since they will be useful to make

86 3.4. REGULARIZATION SCHEMES CHAPTER 3. PATH INTEGRALS a confrontation with mode regularization results in phase space:

1 1 − h S i = − ξkξiξ j∂ g , β 3 4β k ij 1 1 2 − h S i = ∂ ∂ g βgijgkl − gikgjl − ξkξl gij − ξiξ jgkl + 2ξiξl gjk − ξiξ jξkξl β 4 24 k l ij β − β V + VMR , 1 1 h S2 i = ∂ g ∂ g −β 6gkl gimgjn − 4gkmgil gjn − gkl gijgmn + 4gikgjl gmn 2β 3 c 96 k ij l mn − 4gkiglmgjn + 4ξiξm gjngkl − gjl gkn + 2ξkξl gimgjn + 2ξiξ j 2gkmgln − gkl gmn + 4ξiξk gjl gmn − 2gjmgln 1 + ξiξ jξmξngkl − 4ξiξkξmξngjl + 4ξiξkξlξmgjn . β

3.4.2: Dimensional regularization

Dimensional regularization[33, 34, 35] is performed introducing S extra inﬁnite regulating dimensions t = (t1,..., tS) that extends the “temporal” (afﬁne) variable τ to tµ(τ, t) with µ = 0, 1, . . . , S and dS+1t = dτdSt, so the action in S + 1 dimensions reads

Z S+1 1 i j i j i j 2 S = d t gij(∂µx ∂µx + a a + b c ) + β (V + VDR) , Ω 2 where Ω = [−1, 0] × RS. Using VBC variables qs instead of xs and performing the usual split S = S2 + Sint we obtain Z S+1 1 i j i j i j S2 = d t gij(x f )(∂µq ∂µq + a a + b c ) , Ω 2 Z S+1 1 i j i j i j 2 Sint = d t (gij(x) − gij(x f ))(∂µx ∂µx + a a + b c ) + β (V + VDR) . Ω 2

In this case the Fourier transform will be a series for t0 ≡ τ, but an integral for t, so at a regulated level the propagators would read

i j ij h q (t)q (s) i = −βg (x f )∆(t, s) , i j ij h a (t)a (s) i = βg (x f )∆gh(t, s) , i j ij h b (t)c (s) i = −2βg (x f )∆gh(t, s) ; where

Z dSk ∞ −2 ∆(t, s) = sin(πmτ) sin(πmσ) eik·(t−s) , ( )S ∑ 2 2 2π m=1 π m Z dSk ∞ ∆ (t, s) = 2 sin(πmτ) sin(πmσ) eik·(t−s) = δ(τ − σ)δS(t − s) ≡ δS+1(t, s). gh ( )S ∑ 2π m=1

87 Part II: CURVED SPACE-TIME

In the limit S → 0 these propagators reduce to the usual

∆(τ, σ) = τ(σ + 1)θ(τ − σ) + σ(τ + 1)θ(σ − τ) ,

∆gh(τ, σ) = δ(τ − σ) .

However we note that at the regulated level holds the following equality:

2 S+1 ∂µ∆(t, s) = ∆gh(t, s) = δ (t, s) ;

this allows us to solve ambiguities and divergencies using manipulations at the regulated level and then taking the limit S → 0, as we see calculating the following integral1:

Z 0 Z 0 J = dτ dσ •∆∆••∆• −1 −1 Z Z S+1 S+1 → d t d s µ∆∆ν µ∆ν 1 Z Z = dS+1t dS+1s ∆ (∆ )2 2 µ µ ν 1 Z Z = − dS+1t dS+1s ∆(∆ )2 2 µµ ν 1 Z Z = − dS+1t dS+1s δS+1(t, s)(∆ )2 2 ν Z 1 S+1 2 = − d t (∆ν) 2 t 1 Z 0 1 → − dτ (∆•)2 = − ; 2 −1 τ 24 while in mode regularization this integral would result −1/12, so indeed ﬁxing ambiguous results is a procedure regularization-dependent. Performing all perturbative calculations up to order β one ﬁnds that dimensional regularization counterterm is generally covariant; explicitly

− D 1 A = (2πβ) 2 , V (x) = R . DR 8

3.4.3: Time slicing

In the time-slicing scheme we don’t need renormalization conditions, since is the time slicing procedure that led to a path integral formulation in the ﬁrst time: in particular if we recover expression (3.2.2), integrate out momenta with a Gaussian integration for each k, add ghosts

to exponentiate the metric determinant; then split discretized action SN into S2,N + Sint,N, add sources and perform perturbative calculations up to order β always remaining at the discretized level, everything is unambiguous and we don’t need any counterterm, in fact it is sufﬁcient the Weyl ordering term 1 V (x) = (R + gijΓk Γl ) . TS 8 il jk 1 we introduce the notation µ∆ and ∆µ to denote derivation with respect to the ﬁrst or second S + 1 dimensional variable of ∆.

88 3.4. REGULARIZATION SCHEMES CHAPTER 3. PATH INTEGRALS

The other difference between time slicing and other regularization schemes is the presence −1/4 1/4 of a global multiplying factor g(xi) g(x f ) ,explicitly

1/4 D ( ) 1 i j − g x f − gijξ ξ −S K( ) = ( ) 2 2β h int i x f , t f ; xi, ti 2πβ 1/4 e e , g(xi) where the average is performed for a fixed e and then the limit e → 0 is taken. So with the terminology used so far we can say that time-slicing counterterm is the Weyl ordering −D/2 −1/4 1/4 potential VTS, while its global renormalization factor is A = (2πβ) g(xi) g(x f ) ; however we emphasize that in time slicing we don’t need to impose renormalization conditions, or solve any ambiguity, everything is well defined and derives from the very definition of K as a transition element and the application of the midpoint rule with Weyl ordered operators. If we perform phase space calculations in time slicing, i.e. we don’t integrate out momenta in Eq. (3.2.2), but we directly add sources for ps and qs (no ghosts are needed), we can evaluate again the transition element up to order β with discretized phase space propagators: explicit calculations[26] show that even if we have different diagrams, they can be seen as combinations of configuration space diagrams; as a matter of fact nothing changes, −D/2 −1/4 1/4 and again the only “counterterms” are VTS and A = (2πβ) g(xi) g(x f ) .

aaee vlto ftesse hog crdne qaini h unie relativistic quantized the is Hamiltonian equation Schrödinger through system the of evolution parameter A aclt n-opefcieatosfrsaa ed nagaiainlbackground. gravitational a in fields scalar for actions effective classical one-loop calculate a of quantization Gupta-Bleuler metric mass the of from particle a derives describing function action this that showed We opiae h tutr fpoaaosadWc theorem. Wick and Fourier propagators of “tadpole” of the structure space of the presence the complicates the in that see calculations will we explicit However perform regularization). the we to that leads cutoff scheme do a regularization (introducing to any coefficients calcula- order so perturbative In limit, every continuum case results. the a such in same in even that defined show well will is We tion integral. path space phase a as 1 hnw ou ntemi etr fti ok h ramn fteitga kernel integral the of treatment the work: this of feature main the on focus we Then iha ulda fn parametrization. affine Euclidean an with H h rniinelement transition the w aese ntepeiu hpe,w a aeapt nerlfruainof formulation integral path a make can we chapter, previous the in seen have we s g ( ij x ˆ ( , x p ˆ ) = ) nti hpe ewl reydsushwsc oe svr sflee to even useful very is model a such how discuss briefly will we chapter this In . 1 2 g ( x ˆ ) − 1 4 p ˆ i g h ( x x ˆ ) f , 1 2 euaiain npaespace phase in Regularizations t g M f ij | x ( nteFuirmdsi h limit the in modes Fourier the in m x i ˆ , ) t ≥ p i ˆ i j g fteHmloinfnto htdsrbsteaffine the describes that function Hamiltonian the if oigaogageodesic a along moving 0 ( x ˆ ) − 4 1 + V ( x ˆ ) . Mncoo nói,sceo urdió secreto, inmóvil, “Minucioso, 1 ne imos lolaberinto alto su tiempo el en M nacre pc with space curved a in ip → x ˙ ∞ ntefe action free the in og usBorges Luis Jorge ehv mode have we C h a p t e r invisible.” 4 K 91 Part II: CURVED SPACE-TIME

4.1: Worldline methods on curved space

efore we deal with phase space path integrals we make a little digression, and discuss B worldline methods we introduced in Section 2.2 in the presence of gravity. We saw in Sections 2.2.1 and 2.2.2 that if we have a field coupled to an external background field we could write the one-loop effective action as a quantum mechanical path integral. Consider now a real scalar field in a D-dimensional space-time φ coupled to the

metric gµν. The Euclidean quantum ﬁeld action reads

Z √ 1 S[φ, g] = dDx g (gµν∂ φ∂ φ + m2φ2 + ξRφ2) , 2 µ ν

where m is the mass of the scalar particle and ξ is a possible non-minimal coupling to the scalar curvature R. The Euclidean one-loop effective action Γ[g] describes all possible one- loop graphs with the scalar ﬁeld in the loop and any number of gravitons on the external legs. However the ﬂuctuation of the metric, i.e. the graviton, is not quantized. As we saw in Section 2.2.1 we have

Γ[g] = − ln Det−1/2(−∇2 + m2 + ξR) , One-loop Effective Action for a scalar 2 ﬁeld in a where ∇ is the covariant Laplacian; now repeating the same procedure that led to Eq. (2.2.3) gravitational we obtain background by means of Path Z ∞ Z 1 dT −S[x,g] Integral Γ[g] = − [x(t)] e , 2 0 T D PBC

where the action S[x, g] is

Z T 1 µ ν 2 S[x, g] = dt gµν(x)x˙ x˙ + m + ξR(x) ; 0 4

so we see this is exactly the action we have for our relativistic Hamiltonian if we set V(x) = m2 + ξR(x). So the path integral of Eq. (3.3.1)1 is useful to calculate the effective action of a scalar field in a gravitational background by means of a fictitious quantum mechanics. However we would have obtained the same result for Γ[g] first quantizing the scalar particle model not yet gauge-fixed we described in Section 3.1.1: as a matter of fact the action (3.1.5) in Euclidean time and adding the non-minimal coupling to scalar curvature reads Z 1 h i S[x, e] = dt e−1g (x)x˙µx˙ν + e(m2 + ξR(x)) ; 2 µν

we quantized this theory in a Gupta-Bleuler way, alternatively we could have quantized it directly by means of a path integral over the ﬁelds x and e, taking care of the volume of the

1in this case we wrote the conﬁguration space path integral, however we could as well write the phase space path integral.

92 4.2. PHASE SPACE CHAPTER 4. REGULARIZATIONS IN PHASE SPACE gauge group by a Fadeev-Popov gauge fixing procedure, obtaining the same result for Γ[g], namely Z [x] [e] Γ[g] = e−S[x,e] , Vol(Gauge)D D PBC where the division with the volume of the gauge group denotes formally the Fadeev-Popov gauge fixing procedure[14]. We conclude that the “fictitious” quantum mechanics is not so fictitious: it corresponds to the first quantization of the relativistic scalar particle which makes the loop in the effective action.

4.2: Phase Space

n section 3.4.3 we stated that time slicing calculations in phase space led to a fixing of A I and Vph at a regulated level. However we want to show that in fact a regularization is not necessary, because every calculation could be as well performed in the continuum limit. Let’s start from the relativistic Hamiltonian of our model 1 H(p, x) = gij p p + V(x) ; (4.2.1) 2 i j where x and p are the n-dimensional coordinate and conjugate momentum vectors, and V(x) is a scalar function of the coordinates only. We know that the Kernel is given by a phase space path integral in Euclidean time: Z −SeE(p(t),x(t)) K(x f , t f ; xi, ti) ≡ [x(t)] [p(t)]e , (4.2.2) D D x(ti)=xi x(t f )=x f where SeE(p, x) is the Euclidean time action. Since the theory described by such an Hamiltonian as (4.2.1) is super-renormalizable, we have only to fix a global renormalization constant, we will call it as usual A, and a coordinate function Vph(x), correcting the potential V(x), to univocally determine the Kernel at any perturbative order. Since these two terms are fixed at the first order in β = t f − ti , we will 1 evaluate the Kernel at this order and fix A and Vph(x). As we did in the previous chapter we explicit the dependence on the difference t f − ti ≡ 2 β: in order to do so we set t = t f + βτ, and define SbE ≡ βSeE , obtaining :

Z 1 − β SbE(p(t),x(t)) K(x f , t f ; xi, ti) ≡ [x(t(τ))] [p(t(τ))]e , D D x(ti)=xi x(t f )=x f Z 0 1 2 ij 2 SbE = dτ β g pi pj − iβpx˙ + β V(x) + Vph(x) . −1 2

1this is known as “one-loop calculation of the transition amplitude”. 2 we omit the time dependence of x and p in SbE to lighten the notation.

93 Part II: CURVED SPACE-TIME

1 1 i i i i i 2 3 We redeﬁne p → β p and set x = x f − (xi − x f ) τ + q (τ) , ﬁnally obtaining : | {z } ξi

Z 1 − β S(p(τ),q(τ)) K(x f , t f ; xi, ti) ≡ [q(τ)] [p(τ)]e , (4.2.3) D D VBC Z 0 1 ij i i 2 S = dτ g pi pj − iβpiq˙ + ipiξ + β V(x) + Vph(x) . (4.2.4) −1 2

4.2.1: Mode expansion

In order to ﬁnd propagators and use perturbative methods, we split as usual the action S in

an interacting Sint and a free S2 part. So S = S2 + Sint , with:

Z 0 1 ij i i S2 = dτ g (x f )pi pj − iβpiq˙ + ipiξ , (4.2.5) −1 2 Z 0 1 ij ij 2 Sint = dτ g (x) − g (x f ) pi pj + β V(x) + Vph(x) ; (4.2.6) −1 2

ij ij where g (x f ) is the metric calculated at ﬁnal point, and g (x) , V(x) and Vph(x) are intended as functions of the q(τ) path. So deﬁning usual averages over the free action

1 R − S2(p(τ),q(τ)) [q(τ)] [p(τ)] [p(τ), x(τ)] e β VBC D D F h [p(τ), x(τ)] i ≡ 1 , (4.2.7) R − S2(p(τ),q(τ)) F [q(τ)] [p(τ)]e β VBC D D

apart from

Z − 1 S (p(τ),q(τ)) h 1 i ≡ [q(τ)] [p(τ)]e β 2 , (4.2.8) D D VBC

where [p(τ), q(τ)] is an arbitrary functional of coordinate and momentum paths; we can F write K(x f , t f ; xi, ti) as:

−1 K(x f , t f ; xi, ti) = h 1 ih exp{−β Sint} i . (4.2.9)

Since the exponential can be written as a power series, all we have to do is ﬁnd an expres- i in sion for n-point propagators h φ 1 (τ1) ··· φ (τn) i where each φ can be either a p or a q, since is straightforward to see that the average of a sum equals the sum of the averages. Further- more we will see that in that case the 1-point propagator doesn’t vanish, nevertheless every

1this redefinition can be adsorbed in the definition of momenta measure [p(t)] , and affects the path integral only for an overall multiplying constant, i.e. it will be adsorbed in the definitionD of the renormalization constant A. 2the transformation from x to q respects the boundary conditions if q(−1) = q(0) = 0 , i.e. with VBC. Such a transformation does not affect D[x(t(τ))] ≡ D[q(τ)] because path measures are translationally invariant. 3from now on we will omit that the τ-dependence of paths pass through t(τ), and simply write x(τ) and p(τ)

94 4.2. PHASE SPACE CHAPTER 4. REGULARIZATIONS IN PHASE SPACE n-point propagator can be written as a function of 1 and 2-point propagators only, in a mod- ified version of Wick Theorem; effectively all we have to find are 1 and 2-point propagators. Before we delve into the calculation of propagators we find very useful to expand coordinate and momentum paths in Fourier transform, in order to properly define the measure [q(τ)] [p(τ)] and perform the integration over dτ. D We defineD the covariant measure as:

i [q(τ)] [p(τ)] ≡ ∏ ∏ dpi(τ)dq (τ) . D D −1<τ<0 i

Now, since q(τ) vanish at the boundaries, its Fourier expansion will be written in terms of sines, and since Hamilton equations tell us that momenta are proportional to the derivative of coordinates, we will expand p(τ) with cosines1:

∞ pi(τ) = ∑ pim cos(πmτ) , (4.2.10) m=0 ∞ i i sin(πmτ) q (τ) = ∑ qm . (4.2.11) m=1 πm

So the path integral measure reduces to integration over all values of the Fourier coefﬁ- cients2:

N ∞ i i [q(τ)] [p(τ)] ≡ ∏ ∏ dpi(τ)dq (τ) = ∏ ∏ dpi0dpimdqm . D D −1<τ<0 i i=1 m=1

We are therefore able to perform integration over τ on the free action S2, substituting (4.2.10) and (4.2.11) in p(τ) and q(τ). We have to use the following integrals:

Z 0 dτ cos(πmτ) = 0 , −1 Z 0 dτ = 1 , −1 Z 0 δ dτ cos(πmτ) cos(πnτ) = mn ; −1 2 and obtain:

∞ ∞ 1 ij i i 1 ij i S2 = g (x f ) ∑ pim pjm − ∑ pimqm + g (x f )pi0 pj0 + ipi0ξ . (4.2.12) 4 m=1 2 m=1 2

1momentum expansion presents one more term than coordinate one, the zero mode term, which vanish for the qs due to their sines expansion (i.e. due to their vanishing boundary conditions). This additional term requires to be treated apart from the others, and we will see it is responsible for the non-vanishing of the h p i and h pp i propagators. The idea of expanding momenta with cosines was introduced in [36], in the case of ﬂat space-time. 2this procedure bring a normalization factor, that we will adsorb in the deﬁnition of A, so we will omit it here.

95 Part II: CURVED SPACE-TIME

As already stated in the footnotes, we will treat the momentum zero mode apart from the rest n 0 of the free action, so we split S2 = S2 + S2 with:

∞ ∞ n 1 ij i i S2 ≡ g (x f ) ∑ pim pjm − ∑ pimqm , 4 m=1 2 m=1 1 S0 ≡ gij(x )p p + ip ξi . 2 2 f i0 j0 i0

4.2.2: Propagators

We are now ready to calculate the one and two-point propagators from the free action. The easiest way is to deﬁne a generating functional Z[Φ]: Z − 1 (S(φ)−φ·Φ) Z[Φ] = D[φ]e β ,

for any φ-dependent functional action, where φ is a covariant or contravariant vector function and Φ is respectively a contravariant or covariant vector function. Z[Φ] is very useful to calculate propagators, in fact we know very well that the n-point propagator, for an action S(φ), can be calculated as:

1 δ δ i1 in n h φ ··· φ i = β ··· Z[Φ] , Z[Φ] δΦi δΦi n 1 Φ=0

h 1 i = Z[Φ] , Φ=0

δ where δΦ is a functional derivative. Since in our case we already performed the time integral in the action, the action reduces to a function of the momentum and coordinate Fourier 1 s coefﬁcients, and functional derivatives reduce to ordinary derivatives. So we deﬁne S2 ≡ i i i ∑m Pm pim + Qimqm + P0 pi0 , and Z[P, Q] = limM→∞ Zm[P, Q] where

Z N M 1 s i − β (S2−S2) Zm[P, Q] ≡ ∏ ∏ dpi0dpimdqm e i=1 m=1

Z 1 0 s0 Z 1 n sn − β (S2−S2 ) i − β (S2 −S2 ) (4.2.13) = ∏ dpi0 e ∏ ∏ dpimdqm e i i m

≡ Z0[P0]Zm[Pm, Qm] .

The calculation of Zm[P, Q] is quite easy. We deﬁne a 2nm-dimensional vector

µ 1 2 n 1 ζ ≡ ( q1 , q1 ,..., q1 , p11 ..., pn1 , q2 ,..., pn2 ,..., pnm ) ,

i µ so the measure ∏i ∏m dpimdqm ≡ ∏µ ζ , and Zm[Pm, Qm] ≡ Zm[Zµ] , where:

Z 1 µ − β S(ζ,Z) Zm[Zµ] = ∏ dζ e , (4.2.14) µ

1P and Q are called sources.

96 4.2. PHASE SPACE CHAPTER 4. REGULARIZATIONS IN PHASE SPACE with 1 S(ζ,Z) = K ζµζν − Z ζµ , (4.2.15) 2 µν µ

  H 0 ......    0 H 0 . . . K ≡   2nm × 2nm matrix , (4.2.16) µν  ..   0 . . . . 0  0 ...... H

! 0 − i 1 ≡ 2 × H i 1 −1 2n 2n matrix . (4.2.17) − 2 1 2 g (x f )

1 This is a well known Gaussian integral , and the result depends only on K ≡ det Kµν and µν Kµν ≡ K−1 , so since K will enter in our deﬁnition of the renormalization factor A and furthermore cancels out in an average like (4.2.7), we need only to know Kµν to calculate propagators. An elementary calculation yields:   H−1 0 ......  −1   0 H 0 . . .  Kµν =   , (4.2.18)  ..   0 . . . . 0  0 ...... H−1

! 2ig−1(x ) 2i1 H−1 = f . (4.2.19) 2i1 0

So we are now able to perform Gaussian integration and calculate Zm[Zµ]: Z 1 1 1 1 µν µ µ ν µ nm − β K ZµZν Zm[Zµ] = ∏ dζ exp − Kµνζ ζ − Zµζ = β K 2 e 2 ; (4.2.20) µ β 2 restoring proper Pm and Qm dependencies we obtain:

M nm − 1 1 ij i Zm[Pm, Qm] = β K 2 exp ∑ g (x f )QimQjm + 2iQimPm . (4.2.21) β m=1

We remain to consider Z0[P0], but that time we have to proceed more carefully. In that case we have: 1 S0 − S0s = gij(x )p p + ip ξi − Pk p ; 2 2 2 f i0 j0 i0 0 k0 so we are dealing with a Gaussian Integral somewhat similar to (4.2.20), but with an “ef- i i 1 i i j j fective” source term (P0 − iξ )pi0 , that brings a term 2β gij(x f )(P0 − iξ )(P0 − iξ ) in the 1see Appendix C of Part I for a discussion of such integrals.

97 Part II: CURVED SPACE-TIME

exponential after integration1. So we have performed every integration, and the resulting generating functional is:

( M ) 1 ij i i i j j Zm[P, Q] = Ae exp ∑ 2g (x f )QimQjm + 4iQimPm + gij P0 − iξ P0 − iξ ; 2β m=1

nm+1 − 1 1 where Ae = β K 2 g(x f ) 2 . Now with that convention is straightforward to see that n 1 i jo h 1 i = Ae exp − 2β gij(x f )ξ ξ , but in defining path integral measure we omitted some normalization constants, promising to adsorb them later in the definition of the global renormalization constant A. It is now time to remember that promise and define A in a way that2:

1 1 1 i j − − 2β gij(x f )ξ ξ h 1 i ≡ Ag 4 (x f )g 4 (xi) e , (4.2.22)

so the generating functional has to be modiﬁed accordingly and becomes:

1 ( M g(x ) 4 1 [ ] = f ij( ) + i + Zm P, Q A 1 exp ∑ 2g x f QimQjm 4iQimPm g(xi) 4 2β m=1 (4.2.23) ) i i j j +gij(x f ) P0 − iξ P0 − iξ .

We are therefore ready to make derivatives with respect to the sources and calculate propagators. Since we are dealing with exponentials of quadratic forms of sources (i.e. each term is a product of two sources), except for the zero-mode term, which present a linear i j term ( −igijP0ξ ), it is easy to prove that odd-point propagators vanish for pms and qms, and (n > 2)-point propagators are obtained from the 2-point one performing all possible

Wick contractions in a standard way; however odd-point p0 propagators don’t vanish, but it is easy to prove by induction that every propagator depends only on the 1 and 2-point ones The Wick Theorem in the following manner3: has to be modiﬁed in Phase Space Path Integral h pp i = h p ih p i + h pp ic ,

h ppp i = h p ih p ih p i + 3h p ih pp ic ,

h pppp i = h p ih p ih p ih p i + 6h p ih p ih pp ic + 3h pp ich pp ic , h ppppp i = h p ih p ih p ih p ih p i + ··· , ··· .

1 ij since we immediately know the inverse of g (x f ) to be gij(x f ) . 2 1 − 1 the reason to extract the factor g 4 (x f )g 4 (xi) deﬁning A is a matter of convenience, as we will see in a few pages. 3 note that this rule applies even for the pm and qm propagators, noting that their 1-point propagator vanishes.

98 4.2. PHASE SPACE CHAPTER 4. REGULARIZATIONS IN PHASE SPACE

An easy calculation then provides:

j h pi0 i = −igij(x f )ξ , (4.2.24) i j ij h qmqn i = 2βg (x f )δmn , (4.2.25) j j h pimqn i = 2βδi δmn , (4.2.26)

h pim pjn i = 0 , (4.2.27)

h pi0 pj0 ic = βgij(x f ) ; (4.2.28) all other two point propagators vanish. Finally, remembering (4.2.10) and (4.2.11) we obtain

j h pi(τ) i = −igijξ , (4.2.29) h qi(τ)qj(σ) i = −βgij∆(τ, σ) , (4.2.30) j j • h pi(τ)q (σ) i = −iβδi ∆(τ, σ) , (4.2.31)

h pi(τ)pj(σ) ic = βgij ; (4.2.32) where1 ∞ 2 ( ) ≡ − ( ) ( ) ∆ τ, σ ∑ 2 2 sin πmτ sin πmσ , (4.2.33) m=1 π m ∞ 2 • ( ) ≡ − ( ) ( ) ∆ τ, σ ∑ cos πmτ sin πmσ . (4.2.34) Phase Space = πm m 1 Propagators As we saw in the previous chapter, an explicit calculation yields that they are the Fourier series expression respectively of the distributions

∆(τ, σ) = τ(σ + 1)θ(τ − σ) + σ(τ + 1)θ(σ − τ) , •∆(τ, σ) = σ + θ(τ − σ) .

Since in κ([−1, 0])∗ we have that θ(0) = 1/2, we then have the following expressions for distributions at coincident times:

∆(τ, τ) = τ(τ + 1) , 1 •∆(τ, τ) = τ + . 2 If we introduce a cutoff M in the Fourier modes we obtain the mode regulated propagators in phase space:

M 2 ( ) ≡ − ( ) ( ) ∆M τ, σ ∑ 2 2 sin πmτ sin πmσ , m=1 π m M • 2 ∆M(τ, σ) ≡ ∑ − cos(πmτ) sin(πmσ) ; m=1 πm in the limit M → ∞ they become the two distributions ∆ and •∆. 1the symbol •Λ(x, y) is intended to be the derivative with respect to the ﬁrst parameter ( x in this case) of Λ, Λ• the derivative with respect to the second parameter. Henceforth we will always use this notation to mean the i i • derivative of ∆ with respect to τ or σ (so the propagator h q (τ)pj(σ) i = −iβδ j∆ ).

99 Part II: CURVED SPACE-TIME

4.2.3: Perturbative expansion

−1 All we have to do now, to ﬁx A and Vph(x) , is to calculate, up to order β, h exp{−β Sint} i. ij 1 In order to do that we expand g (x) about x f , and obtain Sint = S3 + S4 + ... , with:

Z o h1 ij k k i S3 = dτ ∂kg q pi pj − τξ pi pj , (4.2.35) −1 2 Z 0 h1 ij l k l k 2 l k 2 i S4 = dτ ∂l∂kg q q pi pj − 2τξ q pi pj + τ ξ ξ pi pj + β V + Vph ; (4.2.36) −1 4

ij −1 −1 where g , V and Vph are calculated at ﬁnal point x f . Since exp{−β Sint} = 1 − β Sint + 3 1 −2 2 2 2 2 β Sint + . . . and S3 + S4 = O(β ) + O(β ), to order β we need to calculate:

1 h exp{−β−1S } i = 1 − β−1h S i − β−1h S i + β−2h S2 i + O(β2) int 3 4 2 3 1 ≡ exp{−β−1(h S i + h S i) + β−2h S2 i } + O(β2) , 3 4 2 3 c

2 2 2 where h S3 ic is the average containing only connected diagrams, i.e. h S3 ic ≡ h S3 i − 2 2 h S3 i . We write down explicitly S3 :

Z 0 Z 0 2 h1 ij mn k l k l S3 = dτ dσ ∂kg ∂l g q q pi pj pm pn − τξ p pi pj pm pn −1 −1 4 (4.2.37) k l k l i −σq ξ pi pj pm pn + τσξ ξ pi pj pm pn .

Now we can calculate h S3 i , h S4 i and h S3 ic . After applying contraction rules for n-point propagators described in the previous section, we need only to perform some integrations over τ and σ involving ∆ functions. As a matter of fact such integrals are perfectly well- deﬁned in the continuum limit, since we don’t have nor divergent terms, nor ambiguous products of distributions; all the integrals are here reported2:

Z 0 = • = I1 dτ ∆ τ 0 −1 Z 0 1 I2 = dτ τ = − −1 2 Z 0 = • = I3 dτ ∆ τ 0 −1

1Subscripts and in the expansion stands for the square of the order in β of these parts of the action. In 3 4 p fact note that each p , q and ξ weigh as β . 2see Appendix E for examples of explicit calculations.

100 4.2. PHASE SPACE CHAPTER 4. REGULARIZATIONS IN PHASE SPACE

Z 0 1 = = − I4 dτ ∆ τ −1 6 Z 0 1 = • 2 = I5 dτ ∆ τ −1 12 Z 0 1 = • = I6 dτ τ ∆ τ −1 12 Z 0 2 1 I7 = dτ τ = −1 3 Z 0 Z 0 1 I8 = dτ dσ ∆ = − −1 −1 12 Z 0 Z 0 1 = • • = I9 dτ dσ ∆ τ ∆ −1 −1 12 Z 0 Z 0 • • 1 I10 = dτ dσ ∆ ∆ = − −1 −1 12 Z 0 Z 0 = • • = I11 dτ dσ ∆ τ ∆ σ 0 −1 −1 Z 0 Z 0 • 1 I12 = dτ dσ τ ∆ = −1 −1 12 Z 0 Z 0 = • = I13 dτ dσ τ ∆ σ 0 −1 −1 Z 0 Z 0 1 I14 = dτ dσ τσ = −1 −1 4

= where τ , σ means that the ∆ function is calculated in τ σ and so reduces to a single- variable function, and every non-simmetric integral with respect to τ and σ has its inverted counterpart which brings the same result. We write explicitly h S3 i and h S4 i in dependence of such integrals, for the sake of completeness1:

Z 0 1 ij k k k h S3 i = dτ ∂kg h q pi ih pj i + h q pj ih pi i − τξ h pi pj i −1 2 1 = − ∂ gij 2βδk g ξlI − ξkI βg − ξlξmg g , 2 k i jl 1 2 ij il jm Z 0 1 ij l k l k l k h S4 i = dτ ∂l∂kg h q q ih pi pj i + h q pi ih q pj i + h q pj ih q pi i −1 4 l k k 2 l k 2 − 2τξ h q pi ih pj i + h q pj ih pi i + τ ξ ξ h pi pj i + β V + Vph 1 = − ∂ ∂ gij βglkI βg − ξmξng g + 2β2δl δk I − 4ξlξmδk g I 4 l k 3 ij im jn i j 4 i jm 5 l k m n 2 − ξ ξ I6 βgij − ξ ξ gimgjn + β V + Vph .

1 2 h S3 ic takes an analogous form, but since there a lot of different contractions with 4-point propagators, we omit it here.

101 Part II: CURVED SPACE-TIME

Finally, after some boring calculations1, we obtain:

1 1 1 − h S i = − ξkξiξ j∂ g + ξ jΓi , (4.2.38) β 3 4β k ij 2 ij 1 1 2 − h S i = ∂ ∂ g βgijgkl − gikgjl − ξkξl gij − ξiξ jgkl + 2ξkξ jgil − ξiξ jξkξl β 4 24 k l ij β 1 − β V + V + ξkξl gij∂ ∂ g , (4.2.39) ph 8 k l ij 1 1 h S2 i = ∂ g ∂ g β gijgkl gmn − 2gijgknglm − 2gikgjl gmn + 4gikgjmgln 2β 3 c 96 k ij l mn + 8gil gjmgkn − 4gimgjngkl + 4ξiξm gjngkl − gjl gkn + 2ξkξl gimgjn + 2ξiξ j 2gkmgln − gkl gmn + 4ξiξk gjl gmn − 2gjmgln 1 + ξiξ jξmξngkl − 4ξiξkξmξngjl + 4ξiξkξlξmgjn β 1 − ξkξl gimgjn∂ g ∂ g ; (4.2.40) 8 k ij l mn

i where Γ ij is the connection coefficient calculated at x f . Now it will be very useful to isolate the cyan terms we have explicited from the rest of the average, so from now on when we 1 1 1 2 write − β h S3 i, − β h S4 i and 2β h S3 ic we will consider the ones above without cyan n 1 j i 1 i j k o terms; the latter will be written apart as exp 2 ξ Γ ij + 4 ξ ξ ∂iΓ kj since from the definition k 1 ij im jn of connection coefficients ∂iΓ kj ≡ 2 g ∂k∂l gij − g g ∂kgij∂l gmn .

4.2.4: Fixing A and Vph

We have calculated all the averages we needed, so now we are able to fix A and Vph once 2 and for all. In order to do that we compare our Kph(x f , t f ; xi, ti) at order β with the configuration space, mode regulated one that we already figured out previously. Since the kernel does not depend on the regularization chosen, nor whether we perform our calculations in configuaration or phase space, it has to be satisfied the following condition:

MR MR Kph (x f , t f ; xi, ti) ≡ Kcon f (x f , t f ; xi, ti) .

1remembering that since

ia ia ∂k g gaj = ∂k∂l g gaj = 0 ,

then we have:

ij im jn ∂k g = −g g ∂k gmn , ij im jn im jq np im jq np ∂k∂l g = −g g ∂k∂l gmn + g g g ∂l gmn∂k gpq + g g g ∂k gmn∂l gpq .

−1 Furthermore all −β h S4 i terms derived from last two terms of this last expression are included in −1 2 (2β) h S3 ic, and modify accordingly the result to the one reported. 2 ph is added to distinguish between differently calculated kernels.

102 4.2. PHASE SPACE CHAPTER 4. REGULARIZATIONS IN PHASE SPACE

Now our phase space kernel up to order β can be written as:

1 1 i j g 4 (x ) 1 j i 1 i j k 1 1 2 MR − 2β gijξ ξ f 2 ξ Γ ij+ 4 ξ ξ ∂iΓ kj − β h S3 i+h S4 i + 2β h S3 ic 2 Kph (x f , t f ; xi, ti) = Ae 1 e e + O(β ) g 4 (xi) (4.2.41) where metric and connection coefficients are calculated at the final point if not otherwise specified. The last thing to point out to make the comparison straightforward is that the following relation is satisfied:

1 1 1 j i 1 i j k 2 ξ Γ ij+ 4 ξ ξ ∂iΓ kj 2 g 4 (xi) = g 4 (x f )e + O(β ) .

1 1 Proof. Expanding g 4 (xi) around x f up to second order in ξ ≡ xi − x f we obtain:

1 1 i 1 1 i j 1 2 g 4 (x ) = g 4 (x ) + ξ ∂ g 4 (x ) + ξ ξ ∂ ∂ g 4 (x ) + O(β ) i f i f 2 i j f 1 1 1 ∂ig 4 (x f ) 1 ∂i∂jg 4 (x f ) 4 i i j 2 = g (x f ) 1 + ξ 1 + ξ ξ 1 + O(β ) g 4 (x f ) 2 g 4 (x f ) 1 1 1 1 ∂ig 4 (x f ) 1 ∂jg 4 (x f ) 1 ∂ig(x f ) ∂jg 4 (x f ) 4 i i j i j 2 = g (x f ) 1+ ξ 1 + ξ ξ ∂i 1 + ξ ξ 1 +O(β ) g 4 (x f ) 2 g 4 (x f ) 8 g(x f ) g 4 (x f ) 1 i 1 1 i j 1 = g 4 (x ) 1 + ξ ∂ ln g 4 (x ) + ξ ξ ∂ ∂ ln g 4 (x ) f i f 2 i j f 1 i j 1 1 2 + ξ ξ ∂ ln g 4 (x ) ∂ ln g 4 (x ) + O(β ) 8 i f j f

−1 but since ∂i ln det A = tr A ∂iA , 1 1 i jk 1 i j jk 1 i j jk lm 2 = g 4 (x ) 1+ ξ g ∂ g + ξ ξ ∂ g ∂ g + ξ ξ g ∂ g g ∂ g +O(β ) f 4 i jk 8 i j jk 32 i jk j lm 1 1 i j 1 i j k 1 i j k l 2 = g 4 (x ) 1 + ξ Γ + ξ ξ ∂ Γ + ξ ξ Γ Γ + O(β ) f 2 ji 4 i kj 8 ki lj

n 1 j i 1 i j k o now changing dummy index accordingly, and noting that exp 2 ξ Γ ij + 4 ξ ξ ∂iΓ kj ≡ 1 + 1 i j 1 i j k 1 i j k l 2 2 ξ Γ jk + 4 ξ ξ ∂iΓ kj + 8 ξ ξ Γ kiΓ lj + O(β ), we obtain:

1 1 1 j i 1 i j k 2 ξ Γ ij+ 4 ξ ξ ∂iΓ kj 2 g 4 (xi) = g 4 (x f )e + O(β ) .

So Eq.(4.2.41) reduces to:

1 i j 1 1 2 MR − 2β gijξ ξ − β h S3 i+h S4 i + 2β h S3 ic 2 Kph (x f , t f ; xi, ti) = Ae e + O(β ) ; (4.2.42) p 1since each ξ counts as β in a power expansion.

103 Part II: CURVED SPACE-TIME

MR so a straightforward comparison with Kcon f (x f , t f ; xi, ti) ﬁxes A and Vph as:

− n A = (2πβ) 2 , (4.2.43) R(x) 1 V (x) = − + gij(x)Γk (x)Γl (x) . (4.2.44) ph 8 8 il jk We note that mode-regulated, phase space counterterm is identical to the same one in time- slicing, as we expected.

4.2.5: Regularization is not needed for perturbative calculations

We have just proved that introducing a regularization scheme is not very useful if one cal- culates perturbative corrections in phase space. In fact we were able to solve all integrals without introducing a cutoff M in the Fourier modes, i.e. without recurring to mode regularization. Furthermore we obtained the same result as performing calculations in time

slicing. As a matter of fact phase space path integral counterterm Vph is just the term due to the Weyl ordering of quantum Hamiltonian, necessary to evaluate transition elements at intermediate times; furthermore A can be ﬁxed univocally imposing a suitable consistency

condition: we imposed the equality of Kph with the one calculated in configuration space mode regularization at order β, since it is the most general consistency condition we could require. We note a peculiar fact: while in time slicing counterterms in phase and configuration space are exactly the same, this is not true in mode regularization; we already pointed out that integrating out the momenta in the continuum limit was a dangerous and not well defined manipulation: it leads to ambiguities and divergencies, to the introduction of ghosts and finally to a modification of the counterterms even if we treat the same regularization scheme. Thus one could try to derive the mode regularization configuration space path integral from the phase space one integrating out the momenta at the regulated level, and try to understand where the modifications to the counterterm arise.

104 Appendices

105

otetasto element transition the so expressions symmetrized terms such of call expression we symmetrized the general in contain will ic hsi refrany for true is this since Proof. with substituted O now where O where D.0.1. Theorem W ( M M M noperator An x wn opoetefollowing the prove to want e W m = = ≡ ≡ ( p x r Z Z Z Z ) , S W p d d d d ) = D D D D sa prtri elodrdfr,then form, ordered Weyl in operator an is ( 2 1 z p p p p fw en as M deﬁne we If 2 1 ( m + h h 2 z 1 z z m O + ∑ | | y z p p W ) ∑ + i | y 2 /2 O l O ) nWy reigcnb rte nTyo eisof series Taylor in written be can ordering Weyl in = y m l , W W M n h prtrpwt h aibep. variable the with p operator the and p = ( and 0 x m m = 0 , 1 2 stefnto orsodn oO to corresponding function the is p m p ( r h ) r z h m z ih l z h hoe sproved. is theorem the | + m ( | l p p x x | y ih m y m ) z i p , p − m r p | ( − l ) y x p S , l i r m y | h x y l p p p l i r ; | r ) a ewitnas: written be can y h S ; i z h olwn oml holds: formula following the , | p ih . p | y i x W m elOrdering Weyl p ( x r , iharbitrary with p ) hnteoeao is x operator the when x

and A p p e n d i x m D p and hsit thus , r If . 107 Part II: CURVED SPACE-TIME

As it concerns the Trotter-like approximation

Z Z 3 D ipk·(xk−xk−1) −eH(x¯k−1/2,pk) D ipk·(xk−xk−1) −eHW (x¯k−1/2,pk) d pk e e = d pk e e + O(e 2 ) , W

it is justiﬁed since the difference between (exp(−eH))W and exp(−eHW ) consists of terms without a p, surely of higher order in e, and terms with at least one p, however if we split the

Hamiltonian in a free and interacting part and calculate phase space propagators h pk,i pk,j i j and h pk,ix¯l−1/2 i they are both of order unity, the formula holds at ﬁrst order in e.

108 nyin only eev oetikn is thinking some deserve where becomes have we limit continuum the in since that h euae ee ednthv uhapolm nfc vni h limit the in even fact In problem. of a series such Fourier have the don’t we level regulated the and T RR τ ) ( 2 1 6 = sa xml eepiil calculate explicitly we example an As τθ rbescudb rsn ttebudre u ooclaigfcoswt nnt rqec,btsrl at surely but frequency, inﬁnite with factors oscillating to due boundaries the at present be could problems integration the clearly • ∆ iuu n ut ayt ov fw eebrtecniumlmtepeso of expression limit continuum the remember we if solve to easy quite and biguous itiuinlintegrals distributional he ∆ I I • ∆ nasto ulmaueteitga vanishes integral the measure null of set a in 0 ( 10 10 • RR τ τ ( ( τ τ h eodfco ftels xrsincmsfo h terms the from comes expression last the of factor second The . − = = = , , tnsfritgainfrom integration for stands σ σ σ ZZ Z Z = ) = ) itiuinlitgaini hs space phase in integration Distributional )) − − 0 0 1 1 htaesmercadtu ietesm result same the give thus and symmetric are that τσ d d σ τ τ τ θ ( + + + Z Z t − − − 0 0 θ θ 2 1 1 s ( ( Z d d ) σ R τ egv nScin134(nti case this (in 1.3.4 Section in gave we σ σ d − − τ • ( σθ ∆∆ ( RR σ σ τ τ ( ) ) + τ • + θ − θ ( 1 , , ( σ , τ I + ) τ ) 1 − – ie h factor the gives − I 14 ZZ σ σ ) ))( htw aet efr nScin423aentam- not are 4.2.3 Section in perform to have we that θ − ( θ σ o0bt in both 0 to 1 ( τ τ − + − τ θ σ ) ( oee ic h function the since however , τ σ ) θ + − ( σ htmultiplies that 1 τ − )) β 2 τ τ = tagtowr aclto the calculation straightforward A . ) and )yed zero. yields 1) , M σ , 1 → o h nyitga that integral only the Now . R τ ∞ nerto from integration nepii aclto using calculation explicit an RR ( σθ θ ( (

τ A p p e n d i x σ − − σ E τ − ) )) θ o0 to 1 ( and σ − ∆ 109 Part II: CURVED SPACE-TIME

leads to:

Z 0 Z 0 • • 1 I10 = dτ dσ ∆∆ = − . −1 −1 12 The other integrals are even simpler than this one.

110 Conclusions

e have proved that the phase space path integral describing the affine evolution dic- W tated by a relativistic Hamiltonian, i.e. the quantum motion of a particle in curved space, is independent of the regularization scheme used to define the functional measure, in contrast with the configuration space path integral that is explicitly dependent on the regularization scheme chosen, as far as we perform perturbative calculations. In particular, if we treat this path integral as a one dimensional QFT, we have a super- renormalizable theory that needs the introduction of ghosts, in configuration space, in order to manage divergencies; and a regularization scheme as well as a suitable counterterm in order to solve ambiguities and obtain the same result in any case for the integral kernel of affine evolution at any order. Since in the continuum limit we can obtain the configuration space functional integral from the phase space one formally integrating out the momenta, we can say that such formal manipulation is quite problematic: it leads to divergent and ambiguous diagrams and thus to the necessity to introduce ghosts and counterterms depending on the regularization scheme. Phase space path integral instead is well defined in the continuum limit, i.e. it gives the same perturbative results for any regularization scheme chosen to define precisely the functional measure of momentum and coordinate paths: in fact apart a global factor that we could adsorb in the definition of such functional measure, the only modification we need to introduce in the relativistic action is an additional factor that arises from the particular ordering (the Weyl ordering) we need to give to the quantum Hamiltonian operator to evaluate its transition element between a position and a momentum eigenstate. The only subtlety in the phase space path integral formulation is that since there is a non-vanishing momentum zero-mode linear term in the free action, the structure of Wick theorem is different from the usual one, since odd-points correlation functions don’t vanish; however it is possible to see that any correlation function can be written by means of the one and two-point propagators. The integration of momenta at a regulated level in time slicing does not modify the structure of the counterterm: in fact in both configuration and phase space path integral time-slicing has the same counterterm. Instead mode regularization counterterm in configuration space is different from the one in phase space; it would be worth to analyze the way such modifications to phase space counterterm arise when we integrate out the momenta at the regulated level: in order to derive unambiguously mode regulated configuration path integral from the phase space one.

111

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114 Thanks

Un sentito ringraziamento a Fiorenzo Bastianelli, per l’aiuto, la pazienza, la disponibilità e la cordialità grazie a cui il sottoscritto ha potuto portare a termine questo lavoro; e a Roberto Bonezzi per il prezioso confronto dialettico. Un pensiero anche a Jorge Luis Borges, Samuel Taylor Coleridge e Eikichi Onizuka per aver illuminato le notti più buie della mia mente.

“Di questa poesia mi resta quel nulla d’inesauribile segreto”