MATHEMATICAL INTRODUCTION TO QUANTUM ELECTRODYNAMICS

JAN DEREZINSKI´

Slides of the minicourse at IHP 21-25.06.2010

Contents

3 0 Introduction 5

1 General scattering theory 29

2 Axioms of quantum field theory 65

3 Neutral scalar bosons 89

4 Massive photons 137 5 Massless photons 175

6 Linearly perturbed quadratic Hamiltonians 209

7 Charged scalar bosons 221

8 Dirac fermions 267 Chapter 0

Introduction 5 Quantum Electrodynamics (QED) is an extremely successful theory. It depends on very few adjustable parameters – the fine structure constant and the masses of charged particles – and it can be used to compute with great precision many physical quantities, explaining a wide range of phenomena. It has a rich and complex formal structure, even though it can be developed from a very small number of postulates without any arbitrariness, The agreement of the QED computations with the experimental data is astounding. Its most spectacular successes are the computation of the anomalous magnetic moment of the electron and of the energy levels of light atoms (the Lamb shift). Its range of applicability is however much wider: one can argue that a large part of modern physics can be derived from QED. QED has serious problems. It has been noticed already by F. Dyson that the perturbation expansion computed in QED is probably divergent. Even worse: as noted by L. D. Landau, if we try to sum up this series we encounter a singularity, the so-called Landau pole. The common wisdom says that QED does not exist as a consistent satisfactory nonperturbative theory. QED is an example of a quantum field theory (QFT). It is a relatively sophisticated example of a QFT because of the gauge invariance. It is natural to ask whether QED, or more generally, other QFT’s, fit into the framework of usual quantum mechanics and of some common-sense postulates: whether they can be described on a Hilbert space, with a bounded from below Hamiltonian, satisfy the Poincar´e covariance, and the Einstein causality. In the literature two basic sets of such postulates can be found: the Wightman axioms and the Haag-Kastler axioms. They are viewed by many physicists as an irrelevant pedantry of narrow-minded mathematicians. In my opinion, this is a wrong assessment – in reality, most of these postulates express basic intuitions coming from quantum mechanics. One can view these axioms not as an unshakable foundation, but as a set guiding principles. Unfortunately, so far the only known nontrivial models of Wightman or Haag-Kastler axioms in 1+3 dimensions are free theories. Interacting quantum field theories, such as eg. the λ:φ4: theory, satisfy these axioms at best in the sense of formal power series. There is probably no hope that a model of interacting QFT will be rigorously and non-perturbatively constructed in a near future. In particular, it is commonly believed that it is impossible to construct the λ:φ4: theory satisfying the Wightman axioms (because of the Landau pole). The situation with QED is, to my understanding, even worse, since it does not even satisfy Wightman axioms on the formal level, because of the gauge invariance. According to the Wightman axioms, the full informations about a QFT can be encoded in the so-called Wightman functions. Actually, in usual physics textbooks, QFT is studied following a somewhat different philosophy. According to them, the objective of QFT is computation of various quantities related to the scattering operator (of course, as formal, probably divergent power series). These quantities are easier to describe in massive theories. In theories containing massless particles, the situation is messed-up by the so-called the infra-red problem. Basic quantities computed in QFT

massive QFT QFT possessing massless particles

N-point time-ordered N-point time-ordered Green’s functions Green’s functions (with a physical normalization); (without a physical normalization);

scattering amplitudes; ————-

scattering cross-sections; inclusive scattering cross-sections;

(complex) energy levels; (complex) energy levels. In standard modern textbooks, N-point time-ordered Green’s functions are treated as the ba- sic objects of a given QFT. Other quatities, such as the energy levels and scattering amplitudes and cross-sections can be in principle derived from Green’s functions. Scattering amplitudes (that is matrix elements of the scattering operator) do not, however, exist in the massless case (even as formal power series). Physically measurable are inclusive cross-sections and energy levels. In QED (both massive and massless), there is a problem with Green’s functions, because they depend on the gauge (both in the massless and massive case). Neither the (inclusive) scattering amplitudes nor energy levels depend on the gauge. One can name 4 major issues in QED that need a special attention:

1. the ultra-violet problem;

2. the infra-red problem;

3. the gauge-invariance;

4. bound states of many particle systems.

The ultra-violet problem is common to all kinds of interacting relativistic quantum field theories. It is the main reason why we seem to be limited to a perturbative treatment and why the renormalization is necessary. Historically, it posed the biggest obstacle in the development of QED, and was mastered only in the late 40’s, with major contributions of Schwinger, Feynman, Tomonaga and Dyson. The infra-red problem is common to all QFT’s possessing massless particles. In QED it is due to the presence of massless photons and is absent in the massive QED. It appears already in non-relativistic quantum mechanics – in scattering theory with Coulomb forces. These forces are long-range, which makes the usual definition of scattering operators impossible. Its another manifestation is the appearance of inequivalent representations of canonical commutation re- lations when we consider scattering of photons against a classical current that has a different direction in the past and in the future. Thus, even in these toy non-relativistic situations the usual scattering operator is ill-defined. Therefore, it is not surprising that (much bigger) problems are present in the full relativistic case. One can cope with the infra-red problem by approximating massless QED with the massive one and restricting computations only to inclusive cross-sections justified by an imperfect resolution of the measuring device. The expression gauge invariance has in the context of QED several meanings.

1. The most common, discussed in classical electrodynamics, is the fact that if we add to a 4-potential solving the Maxwell equation a total derivative, then it still solves the Maxwell equations. Of course, this no longer holds for the Proca equations – the massive gener- alization of the Maxwell equations. Therefore, it is often stressed that gauge-invariance implies that the photons are massless.

2. There exists another meaning of the gauge-invariance: we can multiply charged fields by a space-time dependent phase factor and compensate it by changing the external potentials. 1. and 2. go together in the full Lagrangian of QED, which is invariant with respect to these two gauge transformations applied simultaneously. 3. In QED one often uses the term “gauge invariance” in yet another meaning: we have the freedom of the choice of the (free) photon propagator. This meaning applies both to the massive and massless QED. Some of these propagators are distinguished. In particular, the Feynman propagator plays a special role in the renormalization in the ultraviolet regime.

There are several distinct approaches to treat the problem of the gauge invariance QED, all of them equivalent what concerns the physical quantities. These approaches are quite different in the massive and massless case. Nevertheless, all physical quantities depend continuously on the mass, down to zero. Bound states energies are defined as the positions of singularities of Green’s functions. They usually have a nonzero imaginary part. The understanding of bound states seems to be one of the major problems of QFT. It is in fact dificult to study bound states by purely perturbative methods, since the basic quantities that we compute involve elementary particles as asymptotic states. The same problem would appear if we wanted to compute the scattering matrix of an N-body Schr¨odingeroperator in a purely perturbative way. QED is a quantized theory that describes charged particles interacting with photons. Charged particles are either fermions described by the Dirac equation or bosons described by the Klein- Gordon equation. From the point of view of applications, the case of charged fermions is much more important, nevertheless, from the theoretical point of view, charged bosons can be considered on the same footing as charged fermions.

Photons are massless vector bosons, described by the Maxwell equation. The zero mass of photons leads to the infra-red divergencies, which are difficult from the theoretical point of view. The usual way to cope with them is to assume that photons are massive, described by the Proca equation. Then we can treat the (physical) massless QED as the limit of the (non- physical) massive one. Therefore, it is useful to discuss in parallel the massless and massive QED. In full QED both photons and charged particles are quantized. It is however useful to consider the formalism, where only one kind of particles is quantized. In fact, the following three theories can be viewed as the main pillars of the full QED:

1. charged scalar bosons interacting with an external potential;

2. Dirac fermions interacting with an external potential;

3. (massless or massive) photons interacting with an external current.

All the above three theories are (or at least can be) well understood in the non-perturbative sense. They have a well-dynamics on a Hilbert space, or at least in an appropriate C∗-algebra. They can be used to compute quite a number of physically relevant quantities and to explain tricky theoretical questions. One can distinguish three, increasingly complex, basic varieties of the above theories: (i) theory without external potentials, resp. currents; (ii) theory with external potentials, resp. currents in the vacuum representation; (iii) theory with external potentials, resp. currents in the interaction (Furry) representation. Only the first theory is Poincar´einvariant. All of them are in some sense very simple-minded. Their Hamiltonians are quadratic, and hence their equations of motions are linear. After solving the classical equations, we can say everything about their quantizations. Full QED is much more difficult. It involves quantizing simultaneously charged particles and photons. One way to formulate full QED goes as follows. First we formulate the theory of charged particles in an external potential and the theory of photons in an external current as perturbative theories for many-point Green’s functions and scattering amplitudes. An elegant and convenient way to do this a representations of terms in the perturbation expansions as diagrams. Rules to evaluate them are often called the Feynman rules. Then we put the Feynman rules for charged particles and photons together obtaining the Feynman rules for the full QED. These rules (in a somewhat different language) were known already in the 30’s. If one applies them to the so-called tree diagram, one obtains physically meaningful and well defined answers (such as the Klein-Nishina formula, Bhabha formula, etc.). When one tries to evaluate diagrams involving loops, one usually encounters divergencies. Only at the end of 40’s physicists found methods that enabled them to make sense of these diagrams. As a result, the quantities of interest can be expressed in terms of formal power series. These power series, though probably divergent, involve a very small coupling constant. Therefore, for practical purposes, if we take a few first terms, they seem to behave as convergent series. To my experience, in most textbooks, QED is not presented as a consistent theory. Essen- tially all commonly available textbooks on QED present it as a collection of “cookbook recipes”, somewhat chaotic and not quite honest. The reader is supposed to excuse the inconsistencies and use his or her “physical intuition” to cope with its problems. To my understanding, in reality, properly interpreted, QED is a rigorous theory with unam- biguous rules. There are several methods of renormalizing Feynman diagrams. All of them, to my understanding, are equivalent and lead to a uniquely defined formal power series for various quantities of interest depending on very few parameters. The parameters of QED do not appear explicitly in the inital Feynman rules. They enter the theory in the so-called normalization conditions, usually imposed on the values of certain Green’s functions at some chosen points in the momentum space. There is a considerable freedom in choosing these conditions. For instance, we can demand that the 2-point Green’s functions have an appropriate singularity at the value of the physical mass (the on-shell normalization condition). However, other conditions can often be more convenient. The value of the electric charge (and hence of the fine structure constant) can be fixed by demanding that the appropriate Ward(-Takahashi) identity holds. One can distinguish at least 3 basic varieties of the full QED: (i) with no external potentials; (ii) with weak external potentials; (iii) in the interaction (Furry) picture. QED with no external potentials is fully Poincar´ecovariant. It can be viewed as the basic theory. The normalization conditions for the masses and the charge are formulated within its framework. QED with external potentials can in principle be derived as an effective approximation of the QED with no external potentials. We need to assume that some infinitely heavy charged particles generate a potential, which can be treated classically (in addition to quantized photons already present in the theory). One possibility is to keep the vacuum representation and expand the external potential in the perturbation series. Another possibility is to use the interaction (Furry) picture, where we assume that the external potential (or at least its major part) is time-invariant and its effect can be included in the propagator of charged particles. (Of course, there can be an additional time-dependent external potential treated perturbatively). QED in an external potential is very important from the point of view of practical compu- tations. The definition of the anomalous magnetic moment involves QED in a weak external potential. A natural setup for the Lamb shift is the QED in the Furry picture. To sum up, one can view QED as a family of closely related theories. Beside the full QED, this family contains the theories of one kind of particles interacting with external potentials or currents. These partial theories can be understood non-perturbatively. Even though they are in some sense “trivial”, they are actually quite interesting, both physically and mathe- matically. A number of nontrivial and often pathological problems can be partly understood within their setup, such as the infra-red problem, the infinite renormalization of the charge, non-implementability of the dynamics, gauge invariance. The full QED is much less trivial. It seems to be necessarily perturbative.

Chapter 1

General scattering theory 29

Heisenberg picture Let H be a Hilbert space with a self-adjoint operator (called the Hamiltonian) H. Let B be an operator and t ∈ R. The operator B in the Heisenberg picture at time t is defined as

itH −itH BH(t) := e Be . Wightman functions Let Φ be a normalized vector satisfying HΦ = EΦ.

Given operators BN ,...,B1 and times tn, . . . , t1 (not necessarily ordered), the corresponding Wightman function is defined as

W (BN , tN ,...,B1, t1)

:= (Φ|BN,H(tN ) ··· B1,H(t1)Φ)   i(−tN +tN−1)(H−E) i(−t2+t1)(H−E) = Φ|BN e ··· e B1Φ . Dyson’s time ordered product

Let tn . . . , t1 ∈ R be pairwise distinct. We define Dyson’s time-ordered product of Bn(tn),...,

Bi(t1) by

T(Bn(tn) ··· B1(t1)) := Bin (tin ) ··· Bi1 (ti1 ),

where i1, . . . , in is a permutation such that tin ≥ · · · ≥ ti1 . Green’s functions

The (time-ordered) Green’s functions are defined for pairwise distinct tN , . . . , t1 as

G (BN , tN ,...,B1, t1)

:= (Φ|T(BN,H(tN ) ··· B1,H(t1)) Φ) .

Clearly, we can pass from the Wightman to Green’s functions:

G(BN , tN ,...,B1, t1) X

= θ tσ(N) − tσ(N−1) ··· θ tσ(2) − tσ(1) W (Bσ(N), tσ(N),...,Bσ(1), tσ(1)).

σ∈SN Analytic continuation of Wightman functions Assume in addition, that Φ is a ground state of H, that is H ≥ E. Then it is easy to see that Wightman functions can be extended to a function defined for Im(−zN + zN−1) ≥ 0,

Im(−z2 + z1) ≥ 0:

W (BN , zN ,...,B1, z1)   i(−zN +zN−1)(H−E) i(−z2+z1)(H−E) = Φ|BN e ··· e B1Φ . The Wightman function is holomorphic in the interior of its domain and continuous up to the boundary. For any real tN > ··· > t1 we have

G(BN , tN ,...,B1, t1) = W (BN , tN ,...,B1, t1). (1.1)

This allows us in principle to determine the Wightman functions from Green’s functions. This follows from the Edge of the Wedge Theorem, which says that in certain situations a holomor- phic function is determined by its values on the boundary of its domain. Note that the holomorphic extension of thw Wightman function at its “euclidean points” is often called the Schwinger function:

S(BN , sN ,...,B1, s1) := W (BN , isN ,...,B1, is1). Standard Møller and scattering operators

Assume now that H0 and V are self-adjoint operators and H := H0 + λV . The standard Møller and scattering operators (if they exist) equal

S± := s− lim eitH e−itH0 , t→±∞ S := S+∗S−.

± Theorem 1.1 1. If the standard Møller operators exist, they are isometric and satisfy S H0 = HS±.

2. If the standard scattering operator exists, it satisfies H0S = SH0. Problem with eigenvalues It is easy to show the following fact:

Theorem 1.2 If the standard Møller operators exist and H0Ψ = EΨ, then HΨ = EΨ.

This is the reason why in practice the standard formalism of scattering theory is usually applied in a suituations where the unperturbed Hamiltonian H0 has only absolutely continuous spectrum.

In quantum field theory, even when one can define (even formally) H0 and H, typically both have ground states, and these ground states are different. Thus standard scattering theory is not applicable. Instead one can sometimes try other approaches to scattering theory. Time-ordered exponentials

Consider a family of operators [t−, t+] 3 t 7→ B(t). We define the time-ordered exponential

∞ Z t+  X Z Z Texp B(t)dt := ··· B(tn) ··· B(t1)dtn ··· dt1 t− n=0 t+≥tn≥···≥t1≥t− ∞ X Z t+ Z t+ 1 = ··· T(B(t ) ··· B(t )) dt ··· dt , t ≥ t ; n! n 1 n 1 + 1 n=0 t− t− −1 Z t+   Z t−  Texp B(t)dt := Texp B(t)dt t− t+ ∞ Z Z X n = ··· (−1) B(t1) ··· B(tn)dt1 ··· dtn, t+ ≤ t−. n=0 t+≤t1≤···≤tn≤t− For shortness, let us write

Z t+  U(t+, t−) := Texp B(t)dt . t− Note that d U(t+, t−) = B(t+)U(t+, t−), dt+ d U(t+, t−) = U(t+, t−)B(t−), dt− U(t, t) = 1l,

U(t2, t1)U(t1, t0) = U(t2, t0).

(t −t )B If B(t) = B, then U(t+, t−) = e + − . Time dependent Hamiltonian Let R 3 t 7→ H(t) be a family of self-adjoint operators. H(t) is called the Hamiltonian at time t. We introduce the dynamics (evolution)

 Z t+  U(t+, t−) := Texp −i H(s)ds . t−

The dynamics U(t+, t−) is unitary. Heisenberg picture for time-dependent Hamiltonians If R 3 t 7→ A(t) is an operator-valued function, the operator A(t) in the Heisenberg picture is defined as

AH(t) := U(0, t)A(t)U(t, 0).

Note that if A(t) does not depend on t, then AH(t) is the solution of d A (t) = i [H (t),A (t)] , dt H H H AH(0) = A. Interaction picture

Let H0 be a self-adjoint operator. Let R 3 t 7→ V (t) be a family of self-adjoint operators. Set H(t) := H0 + λV (t). Let R 3 t 7→ A(t) be an operator-valued function. The operator A(t) in the interaction picture is defined as

itH0 −itH0 AI(t) := e A(t)e .

The evolution in the interaction picture is defined as

it+H0 −it−H0 UInt(t+, t−) := e U(t+, t−)e .

The interaction Hamiltonian is defined as

HInt := VI(t).

Note that

 Z t+  UInt(t+, t−) = Texp −iλ HInt(t)dt . t− Scattering operator for time-dependent perturbations

Quite often, the dynamics in the interaction picture has a limit t− → −∞ or t+ → ∞. This is in particular the case if V (t) decays in time sufficiently fast.

Definition 1.3 The Møller or wave operators, resp. the scattering operator (if they exist) are defined as

± S := s− lim UInt(0, t), t→±∞ S := S+∗S−. Theorem 1.4 1. If S± exist, then they are isometric.

2. If both S+ and S− exist, then so does S and

S = w− lim UInt(t+, t−). t+,−t−→∞

3. If RanS+ = RanS−, then S is unitary.

± Note that all the operators UInt(t+, t−), resp. S can be viewed as special cases of the scattering operator, if we multiply V (t) by 1l[t−,t+](t), resp. 1l[0,+∞[(±t). Wick’s time-ordered product For further applications, it is convenient to modify the definition of Dyson’s time-order product. Assume that I is a unitary involution on H. We call an operator B even, resp. odd, if B = ±IBI. Let t 7→ Bk(t),...,B1(t) be time dependent operators of pure parity.

Definition 1.5 Let tn . . . , t1 ∈ [t+, t−] be pairwise distinct. We define Wick’s time-ordered product of Bn(tn),..., Bi(t1) by

T(Bn(tn) ··· B1(t1)) := sgnaσBσn (tσn ) ··· Bσ1 (tσ1 ),

where σ1, . . . , σn is a permutation such that tσn ≥ · · · ≥ tσ1 and sgnaσ is the sign of the permutation restricted to the odd elements among Bi. Note that Dyson’s time-ordered product is a special case of Wick’s time-ordered product for I = 1l. Therefore, we will keep the same symbol for Dyson’s and Wick’s time-ordered product. Note also that if B(t) is even for all t, then all the formulas of (1) remain true for Wick’s time-ordered product. Green’s functions

Assume that H0 and V (t) are even. Let Φ0 be a fixed vector with H0Φ0 = 0. Suppose that there exist ± Φ := lim UInt(0, t)Φ0 = lim U(0, t)Φ0. (1.2) t→±∞ t→±∞ The Green’s function for the dynamics U is defined as

G(Ak, tk,...,A1, t1) +
−
:= Φ |T Ak,H(tk) ··· A1,H(t1) Φ . (1.3)

Let G0(... ) denote Green’s functions for the free dynamics. We can express interacting Green’s functions by the free ones:

G(Ak, tk,...,A1, t1) ∞ X (−iλ)n Z ∞ Z ∞ = ds ··· ds G (V, s , ··· , V, s ,A , t , ··· ,A , t ) . n! n 1 0 n 1 k k 1 1 n=0 −∞ −∞ In fact, it suffices to assume that tk ≥ · · · ≥ t1. Then the left hand side of (1.3) equals  −it+H0 lim lim U(0, t+)e Φ0|U(0, tk)AkU(tk, 0) ··· t+→∞ t−→−∞  −it−H0 ×U(t2, t1)A1U(t1, 0)U(0, t−)e Φ0   = Φ0|UInt(∞, tk)Ak,I(tk)UInt(tk, tk−1) ··· UInt(t2, t1)A1,I(t1)UInt(t1, −∞)Φ0 , which equals the right hand side of (1.3). Abstract LSZ approach

Setting some of ti = ±∞ in Green’s functions (1.3) has the obvious meaning – we take the corresponding limit limti→±∞. Green’s functions can be used to compute the scattering operator. In fact, if B1,...,Bm,Ak,...,A1 are some operators, then

(Bm ··· B1Φ0|SAk ··· A1Φ0) ∗ ∗
= G B1, ∞, ··· ,Bm, ∞,Ak, −∞, ··· A1, −∞ . (1.4) Adiabatic dynamics. −|t| Let V be a self-adjoint operator and  > 0. We define V(t) := e V . Then we will write ± H(t), U(t+, t−), S(t+, t−), S , S for the corresponding time-dependent Hamiltonian, etc. Proposition 1.6 We have  H(t+)U(t+, t−) − U(t+, t−)H(t−) 0 ≥ t+ ≥ t−; iλ∂λU(t+, t−) = −H(t+)U(t+, t−) + U(t+, t−)H(t−) t+ ≥ t− ≥ 0;

 HInt(t+)S(t+, t−) − S(t+, t−)HInt(t−) 0 ≥ t+ ≥ t−; iλ∂λUInt(t+, t−) = −HInt(t+)S(t+, t−) + S(t+, t−)HInt(t−) t+ ≥ t− ≥ 0;

± ± ± iλ∂λS = ±HS ∓ S H0,

iλ∂λS = −H0S + SH0. Proof. Display the dependence on λ by writing U,λ(t+, t−). For t+ ≥ t− ≥ 0 we have

U,λ(t+, t−) = U,λe−θ (t+ + θ, t− + θ).

Hence, d 0 = U,λe−θ (t+ + θ, t− + θ) dθ θ=0 d d = −λ∂λU,λ(t+, t−) + U,λ(t+, t−) + U,λ(t+, t−) dt+ dt− = −λ∂λU,λ(t+, t−) − iH,λ(t+)U,λ(t+, t−) + iU,λ(t+, t−)H,λ(t−).

This proves the first identity, from which the remaing follow. 2 Assume that Φ0 is an eigenvector of H0 with H0Φ0 = E0Φ0. Set ± ± S Φ0 Φ := ± , (Φ0|S Φ0) ± ± ± (Φ0|HS Φ0) (Φ0|HΦ ) E := ± = ± . (Φ0|S Φ0) (Φ0|Φ ) Proposition 1.7

± (H − E0 ∓ iλ∂λ)S Φ0 = 0, ± ± ±iλ∂λ log(Φ0|S Φ0) = E − E0, ±
± H − E ∓ iλ∂λ Φ = 0. The Gell-Mann and Low formula for eigenvectors.

Theorem 1.8 Assume that there exist

± ± ± ΦGL := lim |(Φ0|S Φ0)|Φ . &0 and ± lim λ∂λΦ = 0. &0 Then there exist ± ± EGL := lim E &0 and ± ± ± HΦGL = EGLΦGL. Suppose that E0 is a discrete nondegenerate eigenvalue of H0. Then for small enough λ0 > 0 and |λ| < λ0, H has a unique nondegenerate eigenvalue close to E0, which we denote E. If ± + − + EGL depends continuously on λ, then we see that EGL = EGL = E and ΦGL is proportional − to ΦGL. Suppose now that the we have time reversal invariance. More precisely, assume that T is −1 −1 an antiunitary operator such that TH0T = H0, TVT = V , T Φ0 = Φ0. Then, obviously, −1 + −1 − + − + − TU(t+, t−)T = U(−t+, −t−), TS T = S . Therefore, T ΦGL = ΦGL, EGL = EGL and

+ − (Φ0|ΦGL) = (Φ0|ΦGL).

If we assume both the discreteness of E0 and the time reversal invariance, then we can + − conclude the ΦGL is equal ΦGL. + − We will assume in the following that ΦGL = ΦGL =: Φ.

Lemma 1.9 Let B be an operator. Then (S+Φ |BS−Φ ) (Φ|BΦ) = lim  0  0 . (1.5) &0 (Φ0|SΦ0) Proof. The right hand side of (1.5) equals

+ − + − + − (S Φ0|BS Φ0) (Φ |BΦ ) (ΦGL|BΦGL) lim + − = lim + − = + − . &0 (S Φ0|S Φ0) &0 (Φ |Φ ) (ΦGL|ΦGL) ± But ΦGL = Φ and (Φ|Φ) = 1. 2 The Sucher formula

Theorem 1.10 iλ E − E0 = lim ∂λ log(Φ0|SΦ0). (1.6) &0 2 Proof.

+∗ − iλ∂λS = H0S + SH0 − 2S HS ,

We sandwich it with Φ0 and divide with (Φ0|SΦ0) obtaining

+ − (S Φ0|HS Φ0) iλ∂λ log(Φ0SΦ0) = 2E0 − 2 + − . (S Φ0|S Φ0) The last term, by Lemma 1.9, converges to 2E. 2 The Gell-Mann and Low formula for the N-point function

Theorem 1.11

G (Ak, tk, ··· ,A1, t1) (1.7) ∞ n Z ∞ Z ∞ −1 X (iλ) = lim Z dsn ··· ds1 &0 n! n=0 −∞ −∞ ×G0 (V, sn, ··· V, s1,Ak, tk, ··· A1, t1) , where ∞ X (iλ)n Z ∞ Z ∞ Z = ds ··· ds  n! n 1 n=0 −∞ −∞ ×G0 (V, sn, ··· ,V, s1) . Proof. The left-hand side of (1.7) equals

lim (Φ|Ak,H(tk) ··· A1,H(t1)Φ) &0 (S+Φ |A (t ) ··· A (t )S−Φ ) = lim  0 k,H k 1,H 1  0 &0 (Φ0|SΦ0)

But Z = (Φ0|SΦ0). 2 Adiabatic Møller wave operators

Theorem 1.12 1. Assume that there exists the adiabatic or Gell-Mann–Low wave operator

± ± |(Φ0|S Φ0)| ± SGL := w− lim ± S , (1.8) &0 (Φ0|S Φ0) and ± |(Φ0|S Φ0)| ± w− lim λ∂λ ± S = 0. (1.9) &0 (Φ0|S Φ0) Then

± ± SGL(H0 − E0) = (H − E)SGL, (1.10) ± ± SGLΦ0 = ΦGL. (1.11) 2. Define the wave function renormalization operators

± ±∗ ± Z := SGLSGL.

Clearly Z± ≥ 0,

± ± Z H0 = H0Z , ± Z Φ0 = Φ0.

3. Assume that KerZ± = {0}.Define the renormalized Møller operators

± ± ± −1/2 Sren := SGL(Z ) .

± Then Sren are isometric and

± ± Sren(H0 − E0) = (H − E)SGL., (1.12) ± ± SrenΦ0 = ΦGL. (1.13) ± ± 4. Assume that (1.8) is true in the sense of strong limits. Then SGL are isometric and Z = 1l, so that there is no need to renormalize Gell-Mann – Low wave operators. q |f| f |f| |f| i∂λf Proof of 1. Using f = f , we obtain i∂λ f = − f Re f . Therefore, setting f := ± (Φ0|S Φ0) we compute ± ± ± ± |(Φ0|S Φ0)| |(Φ0|S Φ0)| (Φ0|(HS − S H0)Φ0) ±iλ∂λ ± = − ± Re ± (Φ0|S Φ0) (Φ0|S Φ0) (Φ0|S Φ0) ± |(Φ0|S Φ0)| ±
= − ± Re E − E0 . (Φ0|S Φ0) Therefore,

± ± |(Φ0|S Φ0)| ± |(Φ0|S Φ0)| ± ± ±
±iλ∂λ ± S = − ± HS − S H0 + Re(E − E0) . (Φ0|S Φ0) (Φ0|S Φ0) ± ± Using ReEGL = EGL, we obtain (1.10). (1.11) follows by definition. 2 Adiabatic scattering operator ± Assume that SGL exist. Define the adiabatic or Gell-Mann–Low scattering operator

+∗ − SGL = SGLSGL. (1.14)

Set

+ −1/2 − 1/2 Sren := (Z ) SGL(Z ) +∗ − = SrenSren.

+ − Assume that ΦGL = ΦGL. Theorem 1.13 1. We have

H0SGL = SGLH0,

SGLΦ0 = Φ0.

2. We have

H0Sren = SrenH0,

SrenΦ0 = Φ0.

+ − 3. If RanSren = RanSren, then Sren is unitary.

± 4. If SGL exist as strong limits, then

S SGL := w− lim . &0 (Φ0|SΦ0)

Chapter 2

Axioms of quantum field theory 65

Consider the Minkowski space R1,3 with variables xµ, µ = 0, 1,..., 3. By definition, it is the vector space R4 of signature (− + ++). In particular, it is equipped with the pseudo-Euclidean quadratic form for x = (xµ) ∈ R1,3 equal to

3 X −(x0)2 + (xi)2. i=1 Writing R3 we will mean the spatial part of the Minkowski space obtained by setting x0 = 0. If x ∈ R1,3, then ~x will denote the projection of x onto R3. Latin letters i, j, k will sometimes denote the spatial indices of a vector. On R1,3 we have the standard Lebesgue measure denoted dx. The notation d~x will be used for the Lebesgue measure on R3 ⊂ R1,3. A nonzero vector x ∈ R1,3 is called

timelike if x2 < 0, causal if x2 ≤ 0, lightlike if x2 = 0, spacelike if x2 > 0.

A causal vector x is called

future oriented if x0 > 0, past oriented if x0 < 0. The set of future/past oriented causal vectors is denoted J ±. We set J := J + ∪ J −. If O ⊂ R1,3, its causal shadow is defined as J(O) := O + J. We also define its future/past shadow J ±(O) := O + J ±. 1,3 Let Oi ⊂ R . We will write O1 × O2 iff J(O1) ∩ O2 = ∅, or equivalently, O1 ∩ J(O2) = ∅.

We then say that O1 and O2 are causally separated. A function on R1,3 is called space-compact iff there exists a compact K ⊂ R1,3 such that suppf ⊂ J(K). It is called future/past space-compact iff there exists a compact K ⊂ R1,3 such that suppf ⊂ J ±(K). ∞ 1,3 The set of space-compact smooth functions will be denoted Csc (R ). The set of fu- ∞ 1,3 ture/past space-compact smooth functions will be denoted C±sc(R ). Lorentz and Poincar´egroup The pseudo-Euclidean group O(1, 3) called the Lorentz group. Its connected component of unity is denoted SO↑(1, 3). The affine extension of the Lorentz group R1,3 o O(1, 3) is called the Poincar´egroup. These groups have double coverings P in(1, 3), Spin↑(1, 3) and R1,3 o P in(1, 3). Note that Spin↑(1, 3) happens to be isomorphic to Sl(2, C). An element of R1,3 o P in(1, 3). will be often written as (a, Λ)˜ and then the corresponding element of R1,3 o O(1, 3) will be written as (a, Λ). Representations of Spin↑(1, 3) can be divided into two categories. The integer spin repre- sentations induce a representation of SO↑(1, 3) and the half-integer representations do not. Basic rules of relativistic quantum mechanics Physical states are described by a Hilbert space H equipped with a strongly continuous unitary representation of the double cover of the Poincar´egroup (or one of its subgroups)

1,3 ↑ R o Spin (1, 3) 3 (x, Λ) 7→ U(x, Λ) ∈ U(H).

The self-adjoint generator of space-time translations is denoted P = (H, P~ ). H is called the Hamiltonian and P~ – the momentum. Thus

U((t, ~x), 1) = e−itH+i~xP~ . It is natural to impose the following conditions:

1. Existence of a Poincar´ecovariant vacuum: There exists a (normalized) vector Ω invariant with respect to R1,3 × Spin↑(1, 3). 2. Spectral condition: sp(P ) ⊂ J +.

3. Uniqueness of the vacuum: The vector Ω is unique up to a phase factor.

Note that (2) implies H ≥ 0. Conversely, the Poincar´einvariance and H ≥ C implies (2). (2) implies also that Ω is the ground state of H. (3) implies that this groundstate is nondegenerate. In the mathematical literature one can find two basic sets of axioms for a quantum theory, which try to express the concept of causality: Haag-Kastler and Wightman axioms. Even though the Wightman axioms were formulated earlier, it is in my opinion more natural to start with the Haag-Kastler axioms. Note that we keep the “basic rules of relativistic quantum mechanics” as a part of both sets of axioms. Haag-Kastler axioms To each open bounded set O ⊂ R1,3 we associate a distinguished von Neumann algebra A(O) ⊂ B(H). The family A(O), O ⊂ R1,3, satisfies the following conditions

1. Isotony: O1 ⊂ O2 imples A(O1) ⊂ A(O1).

2. Covariance: for (a, Λ)˜ ∈ R1,3×Spin↑(1, 3) we have U(a, Λ)˜ A(O)U(a, Λ)˜ ∗ = A ((a, Λ)O)).

3. Einstein causality: O1 × O2 and Ai ∈ A(Oi), i = 1, 2 imply A1A2 = A2A1. This means that measurements in spatially separated regions are independent. S 0 4. Irreducibility: ( O A(O)) = C1l. Local algebras The algebras A(O) describe algebras of observables in O. This means that in principle an observer contained in O can perturb the dynamics by a self-adjoint operator from A(O), and only from A(O). The least obvious axiom is that of irreduciblity. Quantum fields In practical computations of quatum field theory the information is encoded in a family of 1,3 functions R 3 x 7→ φa(x), where a = 1, . . . , n is a finite index set enumerating the “internal degrees of freedom”, typically, species of particles and the value of their spin projected on a distinguished axis. These functions can be viewed as “operator valued distributions”, which become (possibly unbounded) self-adjoint operators when smeared out with test functions in ∞ 1,3 Cc (R ). Some of the particles are bosons, some are fermions. They commute or anticom- mute for causally separated points, which is expressed with the commutation/anticommutation relations 2 [φa(x), φb(y)]± = 0, (x − y) > 0. Smeared quantum fields We can organize the internal degrees of freedom into a finite dimensional vector space n ∞ 1,3 n V = R . Thus for any f = (fa) ∈ Cc (R , R ) we obtain an operator X Z φ[f] := fa(x)φa(x)dx. a From the mathematical point of view it is natural to treat φ[f] as the basic objects. This leads to the following set of axioms. Wightman axioms

We assume that V = Vs ⊕ Va is a finite dimensional real vector space. Vs, resp. Va is called the space of bosonic, resp. fermionic particles. We set  := 1l ⊕ (−1l) on V. We will write |s| = 0, |a| = 1. The space V is equipped with a representation

Spin↑(1, 3) 3 Λ˜ 7→ S(Λ)˜ , (2.1) which preserves the decomposition V = Vs ⊕ Va.

We assume that H = Hs ⊕ Ha. We set I := 1l ⊕ (−1l) on H. Hs, resp. Ha is called even fermionic, resp. odd fermionic. We suppose that D is a dense subspace of H containing Ω. We have a map

∞ 1,3 Cc (R , V) 3 f 7→ φ[f] (2.2) into linear operators on D satisfying the following conditions:

1. Continuity: For any Φ, Ψ ∈ D,

∞ 1,3 Cc (R , V) 3 f 7→ (Φ|φ[f]Ψ) (2.3)

is continuous. h i 2. Covariance: for (x, Λ)˜ ∈ R1,3×Spin↑(1, 3) we have U(x, Λ)˜ φ[f]U(x, Λ)˜ ∗ = φ S(Λ)˜ f ◦ (x, Λ)−1 . ∞ |j1||j2| 3. Einstein causality: O1×O2 and fi ∈ Cc (Oi, Vji ), i = 1, 2, imply φ[f1]φ[f2] = (−1) φ[f2]φ[f1]. nS o 4. Irreducibility: Span f φ[f]Ω is dense in H. 5. Self-adjointness: φ[f]∗ = φ[f].

6. Fermionic fields change the fermionic parity: φ[f] = Iφ[f]I

One can show the theorem about the connection of spin and statistics saying that the representation (2.1) has integer spin in Vs and half-integer in Va. Wightman axioms are satisfied by free fields. Neutral vs charged fields In practice, only the so-called neutral fields are assumed to be self-adjoint. One has operator 1,3 ∗ valued distributions R 3 x 7→ ψa(x), ψa(x), a = 1, . . . , m. They are the so-called charged fields. After smearing with complex test functions we obtain operators X Z ψ[g] := ga(x)ψa(x)dx, a Z ∗ X ∗ ψ [g] := ga(x)ψa(x)dx, a such that ψ[g]∗ = ψ∗[g]. Clearly, by setting

1 ∗ φa,R(x) := √ (ψa(x) + ψ (x)), 2 a 1 ∗ φa,I(x) := √ (ψa(x) − ψ (x)), i 2 a for any charged field we obtain a pair of neutral fields. Thus one can organize the space describing the species of particles into two finite dimensional spaces: a real space Vn describing neutral fields and a complex space Vc describing charged fields. Relationship between Haag-Kastler and Wightman axioms ”Morally”, Wightman axioms are stronger than the Haag-Kastler axioms. In fact, let Falg(O) be the ∗-algebra in L(D) (linear operators on D) generated by φ[f] with suppf ⊂ O. It is equipped with an involution α given by α(φ[f]) = φ[f], so that Falg(O) as a vector space is alg alg alg a direct sum of its even and odd part, F0 (O) and F1 (O). Then the family F0 (O) almost alg satisfies the Haag-Kastler axioms, except that elements of F0 (O) are defined only on D and not on the whole H, and often do not extend to bounded operators on H. We know that the fields φ[f] are hermitian on D. Suppose they are essentially self-adjoint. Then we could consider the von-Neumann algebra F(O) generated by bounded functions of

φ[f], suppf ⊂ O. Then the family F0(O) satisfies the the Haag-Kastler axioms.

In practice, observable algebras A(O) can be even smaller than F0(O). This is so in particular if we have a group acting G on the space V responsible for a (global) symmetry of the fields. For example, in the case of charged fields the group equals U(1) and this symmetry is obtained by multiplying fields with a phase factor. The action of G on fields induces an action of G on

F0(O). One can argue that the subalgebra of fixed points of this action should be taken as the observable algebra. (Note that what we described is the case of a global symmetry and not of a local gauge invariance). N-point Wightman functions Under the Wightman axioms we obtain a multilinear map

1,3 1,3 Cc(R , V) × · · · × Cc(R , V)

3 (fN , . . . , f1) 7→ (Ω|φ[fN ] ··· φ[f1]Ω) ∈ C, (2.4) which is separately continuous in its arguments. By the Schwartz Kernel Theorem, it can be extended to a linear map Z 1,3 N ⊗N
Cc (R ) , V 3 F 7→ W (xN , . . . , x1)F (xN , . . . , x1)dxN ··· dx1,

(1,3)N (1,3)N where R 3 (xN , . . . , x1) 7→ W (xN , . . . , x1) is a distribution on R with values in the space dual to V⊗N , called the N-point Wightman function, so that (2.4) equals Z W (xN , . . . , x1)fN (x1) ··· f1(x1)dxN ... dx1. From the point of view of the Wightman axioms, the collection of Wightman functions WN , N = 0, 1,... , contains all the information about a given quantum field theory. In particular,

(φ[fN ] ··· φ[f1]Ω|φ[gM ] ··· φ[g1]Ω) Z = W (y1, . . . , yN , xM , . . . , x1)

×f1(x1) ··· fN (xN )gM (yM ) ··· g1(y1)dx1 ··· dxN dyM ··· dy1. N-point Green’s functions In practical computations, the functions that play an important role are not Wightman functions but the so-called (time-orderded) Green’s functions. Their formal definition is as follows:

G(xN , . . . , x1) (2.5) X  0 0   0 0  := sgn(σ)θ xσ(N) − xσ(N−1) ··· θ xσ(2) − xσ(1) W (xσ(N), . . . , xσ(1)),

σ∈SN where sgna(σ) is the sign of the permutation of the fermionic elements among N,..., 1. Note that in (2.5), we multiply a distribution with a discontinuous function, which strictly speaking is illegal. Disregarding this problem, the Green’s function is a covariant, due to the (anti-)commutativity of fields at spacelike separations. Chapter 3

Neutral scalar bosons 89

In this chapter we consider the Klein-Gordon equation with an external source

(−2 + m2)φ(x) = j(x). (3.1)

We will quantize the space of its real solutions. Special solutions of the homogeneous equation Every function ζ that solves the Klein-Gordon equation

(−2 + m2)ζ(x) = 0. (3.2) can be written as 1 Z ζ(x) = eikxg(k)δ(k2 + m2)dk (2π)3 Z ~ q √ 1 X dk 0 ~ 2 2 ~ = g(± ~k2 + m2,~k)e±ix k +m ∓i~xk, (2π)3 p 2 2 ± ~k + m where g is a function on the hyperboloid (mass shell) k2 + m2 = 0. A special role is played by the following 3 solutions of the homogeneous Klein-Gordon equa- tion:

1. The standard positive, resp. negative frequency solution. i Z D(±)(x) = ∓ eikxθ(±k0)δ(k2 + m2)dk (2π)3 Z ~ √ i dk 0 ~ 2 2 ~ = ∓ e±ix k +m ∓i~xk 3 p (2π) 2 ~k2 + m2 2 2 0 √ 1 0 2 mθ(−x ) ±sgnx p 2 miθ(x ) 2 = sgnx δ(x ) ∓ √ H (m −x ) ∓ √ K1(m x ). 4π 8π −x2 1 4π2 x2 ± where H1 are the Hankel functions and K1 is the MacDonald function of the 1st order. 2. The Pauli-Jordan function (or the commutator function) i Z D(x) = e−ikxsgn(k0)δ(k2 + m2)dk (2π)3   1 Z d~k ~ q = ei~xk sin x0 ~k2 + m2 3 p (2π) ~k2 + m2 2 1 0 2 mθ(−x ) p 2 = sgnx δ(x ) − √ J1(m −x ). 2π 4π −x2 D(x) is the unique solution of the homogeneous Klein-Gordon equation satisfying

D(0, ~x) = 0, D˙ (0, ~x) = δ(~x).

We have, suppD ⊂ J. Green’s functions Solutions of (−2 + m2)ζ(x) = δ(x), (3.3) are called Green’s functions or fundamental solutions of the Klein-Gordon equation. In partic- ular, let us introduce the following important Green’s functions:

1. advanced, resp. retarded Green’s function. 1 Z e−ikx D±(x) = dk (2π)4 k2 + m2 ∓ i0sgnk0 2 0 1 0 2 mθ(−x )θ(±x ) p 2 = θ(±x )δ(x ) − √ J1(m −x ). 2π 4π −x2 We have suppD± ⊂ J ±. In the literature, D+(x) is usually denoted Dret(x) and D−(x) is usually denoted Dadv(x). 2. causal or Feynman(-Stueckelberg) Green’s function.

1 Z e−ikx Dc(x) = dk (2π)4 k2 + m2 − i0 2 2 √ 1 2 mθ(−x ) − p 2 miθ(x ) 2 = δ(x ) − √ H (m −x ) + √ K1(m x ). 4π 8π −x2 1 4π x2 Relationships between special solutions and Green’s functions

D(x) = −D(−x) = D(+)(x) + D(−)(x) = D+(x) − D−(x), D(+)(x) = −D(−)(−x), D+(x) = D−(−x) = θ(x0)D(x) D−(x) = θ(−x0)D(x) Dc(x) = Dc(−x) = θ(x0)D(−)(x) − θ(−x0)D(+)(x). Let us prove the last identity. 1 Z e−ikx Dc(x) = dk (2π)4 k2 + m2 − i0 1 Z e−ikx = 4  dk (2π) p~ 2 2 p~ 2 2 2 k + m k + m − |k0| − i0 1 Z e−ikx + 4  dk (2π) p~ 2 2 p~ 2 2 2 k + m k + m + |k0| + i0 ~ 1 Z eik0x0−ik~x = 4  dk (2π) p~ 2 2 p~ 2 2 2 k + m k + m − k0 − i0 ~ 1 Z eik0x0+ik~x + 4  dk (2π) p~ 2 2 p~ 2 2 2 k + m k + m + k0 + i0 ~ 1 Z eik0x0−ik~x = 4  dk (2π) p~ 2 2 p~ 2 2 2 k + m k + m − k0 − i0 ~ 1 Z eik0x0+ik~x + 4  dk (2π) p~ 2 2 p~ 2 2 2 k + m k + m + k0 + i0 This equals

~ ~ i Z eik0x0−ik~x i Z e−ik0x0+ik~x = θ(x ) d~k + θ(x ) d~k 3 0 p 3 0 p (2π) 2 ~k2 + m2 (2π) 2 ~k2 + m2 (−) (−) = θ(x0)D (x) + θ(−x0)D (−x), where we used the identity Z eits ds = 2πie−itaθ(t). a − s − i0 Retarded and advanced solutions of the inhomogeneous problem Among solutions of the inhomogeneous Klein-Gordon equation one can distinguish the ad- vanced and the retarded solution. They can be obtained by the convolution with the ad- vanced/retarded Green’s function.

∞ 1,3 ± ∞ 1,3 Theorem 3.1 For any f ∈ Cc (R ) there exist unique functions ζ ∈ C±sc(R ), solutions of (−2 + m2)ζ± = f, suppζ± ⊂ J ±(suppf).

Moreover, ζ±(x) = R D±(x − y)f(y)dy.

Note that by duality D± can be applied to distributions of compact support. Therefore, in the above theorem we can assume that f is a distribution of a compact support, and then ζ± is a distribution. The Cauchy problem Solutions of the Cauchy problem are uniquely parametrized by their Cauchy data (the value and the normal derivative on a Cauchy surphace). They can be expressed by the Cauchy data with help of the Jordan-Pauli function.

∞ 3 ∞ 1,3 Theorem 3.2 Let ς, ϑ ∈ Cc (R ). Then there exists a unique ζ ∈ Csc (R ) that solves

(−2 + m2)ζ = 0 (3.4) with initial conditions ζ(0, ~x) = ς(~x), ζ˙(0, ~x) = ϑ(~x). It satisfies suppζ ⊂ J(suppς ∪ suppϑ) and is given by Z Z ζ(t, ~x) = − D˙ (t, ~x − ~y)ς(~y)d~y + D(t, ~x − ~y)ϑ(~y)d~y. R3 R3 Conserved current

Let YKG,R, resp. YKG denote the space of real, resp. complex, space-compact solutions of the Klein Gordon equation. Clearly, YKG = CYKG,R. ∞ 1,3 Let ζ1, ζ2 ∈ C (R , R). We define

µ µ µ µ j (x) = j (ζ1, ζ2, x) := ∂ ζ1(x)ζ2(x) − ζ1(x)∂ ζ2(x).

We easily check that

µ 2 2 ∂µj (x) = −(−2 + m )ζ1(x)ζ2(x) + ζ1(x)(−2 + m )ζ2(x),

Therefore, if ζ1, ζ2 ∈ YKG, then µ ∂µj (x) = 0. One says that jµ(x) is a conserved current. Symplectic form The flux of jµ across a space-like subspace S of codimension 1 does not depend on its choice. It defines a symplectic form on YKG,R Z µ ζ1ωζ2 = j (ζ1, ζ2, x)dsµ(x) S Z  ˙ ˙  = ζ1(t, ~x)ζ2(t, ~x) − ζ1(t, ~x)ζ2(t, ~x) d~x. (3.5)

Clearly, the form (3.5) is well defined if only ζ2 ∈ YKG,R, and ζ1 is a distributional solution of the Klein-Gordon equation. Poincar´ecovariance

The Poincar´egroup acts on YKG,R by

−1
r(a,Λ)ζ(x) := ζ (a, Λ) x . r(a,Λ) are symplectic for Λ in the orthochronous Poincar´egroup, otherwise they are antisym- plectic. 1,3 We will write for shortness ra for r(a,1l), where a ∈ R . Solutions parametrized by space-time functions The Jordan-Pauli function D can be used to construct solutions of the Klein-Gordon equa- tion, which are expecially useful in the axiomatic formulation of QFT. We will write Z Df(x) := D(x − y)f(y)dy.

∞ 1,3 Theorem 3.3 1. For any f ∈ Cc (R , R), Df ∈ YKG,R.

2. Every element of YKG,R is of this form. R 3. (Df1)ωDf2 = f1(x)D(x − y)f2(y)dxdy.

4. If suppf1 × suppf2, then

(Df1)ωDf2 = 0. Classical fields

We will also consider the space dual to YKG,R. More precisely, we can endow the space ∞ 3 ∞ 3 YKG,R with the standard topology of Cc (R ) ⊕ Cc (R ) given by the initial conditions. The space of real continuous functionals on Y . will be denoted by Y # , complex valued by KG,R KG,R Y # . The action of T ∈ Y # on ζ ∈ Y will be denoted by hT |ζi. C KG,R C KG,R KG,R 1,3 For x ∈ R , φ(x), π(x) will denote the functionals on YKG,R given by

hφ(x)|ζi := ζ(x), hπ(x)|ζi := ζ˙(x).

They are called classical fields. Clearly, for any ζ ∈ YKG,R we have

(−2 + m2)hφ(x)|ζi = 0.

Thus the equation (−2 + m2)φ(x) = 0 is a tautology. On Y # we have the action of the Poincar´egroup (a, Λ) 7→ r#−1 . Note that KG,R (a,Λ)

#−1 r(a,Λ)φ(x) = φ(Λx + a). Clearly, φ˙(x) = π(x) and Z Z φ(t, ~x) = D˙ (t, ~x − ~y)φ(0, ~y)d~y + D(t, ~x − ~y)π(0, ~y)d~y.

The symplectic form can be written as Z ζ1ωζ2 = (hπ(t, ~x)|ζ1ihφ(t, ~x)|ζ2i − hφ(t, ~x)|ζ1ihπ(t, ~x)|ζ2i) d~x, or more simply, Z ω = π(t, ~x) ∧ φ(t, ~x)d~x. Spatially smeared fields We can also consider smeared out fields. One way to smear them out is to use the symplectic form to pair Y # and Y . KG,R KG,R

More precisely, for ζ ∈ YKG,R we introduce the functional on YKG,R given by

hφ((ζ))|ρi := ζωρ, ρ ∈ YKG,R.

Clearly, Z   φ((ζ)) = ζ˙(t, ~x)φ(t, ~x) − ζ(t, ~x)π(t, ~x) d~x. (3.6)

Note that

{φ((ζ1)), φ((ζ2))} = ζ1ωζ2. (3.7) Space-time smeared fields ∞ 1,3 We can also smear fields with space-time functions f ∈ Cc (R ): Z φ[f] := f(x)φ(x)dx.

Clearly, φ[f] = φ((Df)), ZZ {φ[f1], φ[f2]} = f1(x)D(x − y)f2(y)dxdy. (3.8) Poisson structure

The symplectic structure on the space YKG,R leads to a Poisson bracket on functions on

YKG,R:

{φ(t, ~x), φ(t, ~y)} = {π(t, ~x), π(t, ~y)} = 0, {φ(t, ~x), π(t, ~y)} = δ(~x − ~y) (3.9)

The relations (3.9) can be viewed as mnemotechnic identities that yield the correct Poisson bracket for more regular functions, eg. the smeared out fields in (3.7) or (3.8). Note that formally φ(t, ~x) and π(t, ~x) generate all functions on YKG,R. Using (3.6) we obtain {φ(x), φ(y)} = D(x − y). Therefore, the Jordan-Pauli solution is often called the commutator function. The Hamiltonian density and momentum density are quadratic functionals on YKG,R: 1   H(x) = π(x)2 + ∇~ φ(x)2 + m2φ(x)2 , 2 P~ (x) = π(x)∇~ xφ(x).

They give the total Hamiltonian and momentum Z Z H = H(0, ~x)d~x, P~ = P~ (0, ~x)d~x, the generators of the time and space translations. The Poisson bracket of H and P~ vanishes. Plane waves We would like to diagonalize simultaneously the Hamiltonian, the momentum and the sym- p plectic form. To this end, for any k = (ω(~k),~k) with ω(~k) = ~k2 + m2, we introduce the corresponding plane wave 1 ixk Φ(k) = q e . (3.10) (2π)3/2 2ω(~k) Note that

iΦ(k)ωΦ(k0) = 0, iΦ(k)ωΦ(k0) = δ(~k − ~k0). Plane waves in a box Sometimes we use the box normalization:

1 ixk ΦV (k) = q e . V 2ω(~k)

Then

0 iΦV (k)ωΦV (k ) = 0, 0 iΦV (k)ωΦV (k ) = δ~k,~k0 . Plane wave functionals

Define the following functionals on YKG,R:

a(k) = φ((iΦ(k))) s  Z ~ − 3 ω(k) −i~x~k i −i~k~x = (2π) 2  e φ(0, ~x) − e π(0, ~x) d~x,  2 q  2ω(~k) a(k) = φ(( − iΦ(k))) s  Z ~ − 3 ω(k) i~x~k i i~k~x = (2π) 2  e φ(0, ~x) + e π(0, ~x) d~x.  2 q  2ω(~k) Diagonalization Note that

{a(k), a(k0)} = {a(k), a(k0)} = 0, {a(k), a(k0)} = iδ(~k − ~k0), Z H = d~kω(~k)a(k)a(k), Z P~ = d~k~ka(k)a(k).

If ζ1, ζ2 ∈ YKG,R, then Z ~ iζ1ωζ2 = (ha(k)|ζ1iha(k)|ζ2i − ha(k)|ζ1iha(k)|ζ2i) dk. The fields can be written as Z ~ − 3 dk ikx −ikx
φ(x) = (2π) 2 q e a(k) + e a(k) 2ω(~k) Z   = Φ(k)a(k) + Φ(k)a(k) d~k.

Thus, every ζ ∈ YKG,R can be written as Z ~ − 3 dk ikx −ikx
ζ(x) = (2π) 2 q e ha(k)|ζi + e ha(k)|ζi . 2ω(~k) Positive/negative frequency solutions (±) YKG will denote the subspace of YKG consisting of positive, resp. negative frequency solu- tions, that is,

(+) ~ ~ YKG := {ζ ∈ YKG : ha(k)ζi = 0, k = (ω(k), k)}, (−) ~ ~ YKG := {ζ ∈ YKG : ha(k)ζi = 0, k = (ω(k), k)}. Hilbert space of positive energy solutions (+) Every ζ ∈ YKG can be written as Z ~ − 3 dk ikx ζ(x) = (2π) 2 q e ha(k)|ζi 2ω(~k)

(+) For ζ1, ζ2 ∈ YKG we define the scalar product Z ~ (ζ1|ζ2) := iζ1ωζ2 = ha(k)|ζ1iha(k)|ζ2idk

(+) (+) We set ZKG to be the completion of YKG in this scalar product. The ortochronous Poincar´e (+) group leaves ZKG invariant. Quantization of free fields The Klein-Gordon equation defines a symplectic space equipped with a linear symplectic dynamics generated by a strictly positive Hamiltonian. For such systems, we can apply the prcedure of the positive energy quantization. Let us describe the quantization of the Klein-Gordon equation. We will use the “hat” to denote the quantized objects. We want to construct a Hilbert space H, a positive self-adjoint operator Hˆ called the Hamiltonian, a normalized vector Ω, which is a ground state of Hˆ and a distribution

1,3 R 3 x 7→ φˆ(x), (3.11)

∞ 1,3 which smeared with Cc (R , R) functions has values in self-adjoint operators satisfying 1. (−2 + m2)φˆ(x) = 0, [φˆ(x), φˆ(y)] = iD(x − y).

2. eitHˆ φˆ(x0, ~x)e−itHˆ = φˆ(x0 + t, ~x).

3. Ω belongs to the domain of polynomials in smeared out φˆ(x) and is cyclic for these operators. There exists an alternative equivalent formulation of the quantization program, which uses the smeared fields instead of point fields. Instead of (3.11) we look for a linear function

ˆ YKG,R 3 ζ 7→ φ((ζ)) with values in self-adjoint operators such that

1.’ ˆ ˆ [φ((ζ1)), φ((ζ2))] = iζ1ωζ2. (3.12)

2.’ ˆ itHˆ ˆ −itHˆ φ((r(t,~0)ζ)) = e φ((ζ))e .

3.’ Ω belongs to the domain of polynomials in φˆ((ζ)) and is cyclic for these operators. One can pass between these two versions of the quantization problem by Z   φˆ((ζ)) = ζ˙(t, ~x)φˆ(t, ~x) − ζ(t, ~x)ˆπ(t, ~x) d~x, (3.13)

˙ where πˆ(x) := φˆ(x). The above problem has a solution, which is unique up to a unitary equivalence. We set H = (+) ∗ ~ ~ Γs(ZKG ). The creation/annihilation operators will be denoted by aˆ and aˆ. For k = (ω(k), k) we set

aˆ∗(k) :=a ˆ∗(−iΦ(k)).

Ω will be the Fock vacuum. We set

3 Z d~k   ˆ − 2 ikx ~ −ikx ∗ ~ φ(x) = (2π) q e aˆ(k) + e aˆ (k) . 2ω(~k)

The Hamiltonian and the momentum are Z Hˆ = aˆ∗(~k)ˆa(~k)ω(~k)d~k, ~ Z Pˆ = aˆ∗(~k)ˆa(~k)~kd~k. Note that the whole orthochronous Poincar´egroup is unitarily implemented on H by U(a, Λ) :=   Γ r This is true even though in our construction we only required that time trans- (a,Λ) (+) ZKG lations are implemented. We have

U(a, Λ)φˆ(x)U(a, Λ)∗ = φˆ(a, Λ)x
.

Note the identities

(Ω|φˆ(x)φˆ(y)Ω) = −iD(+)(x − y), (Ω|T(φˆ(x)φˆ(y))Ω) = −iDc(x − y). In fact,

0 ZZ d~kd~k 0 (Ω|φˆ(x)φˆ(y)Ω) = (2π)−3 √ √ eikx−ik y(Ω|aˆ(~k)ˆa∗(~k0)Ω) 2ω 2ω0 Z d~k = (2π)−3 eik(x−y) 2ω = −iD(−)(x − y); (Ω|T(φˆ(x)φˆ(y))Ω) = θ(x0 − y0)(Ω|φˆ(x)φˆ(y)Ω) + θ(y0 − x0)(Ω|φˆ(y)φˆ(x)Ω) = −iθ(x0 − y0)D(−)(x − y) − iθ(y0 − x0)D(−)(y − x) = −iDc(x − y). Axioms ∞ 1,3 ˆ R ˆ The family Cc (R , R) 3 f 7→ φ[f] := φ(x)f(x)dx satisfies the Wightman axioms with fin (+) D := Γs (ZKG ). For an open set O ⊂ Rd we set

ˆ ∞ 00 A(O) := {exp(iφ[f]) : f ∈ Cc (O, R)} .

The algebras A(O) satisfy the Haag-Kastler axioms. External source We go back to the classical theory. In the presence of an external source R1,3 3 x 7→ j(x) ∈ R, the Klein-Gordon equation becomes

(−2 + m2)ζ(x) = −j(x). (3.14)

Introducing the fields R1,3 3 x 7→ φ(x) we can rewrite (3.14) as

(−2 + m2)φ(x) = −j(x). (3.15) Lagrangian formalism (3.15) can be obtained from the Lagrangian formalism. The Lagrangian density can be taken as

1 µ 1 2 2 L(x) = −2∂µφ(x)∂ φ(x) − 2m φ(x) − j(x)φ(x).

The Euler-Lagrange equations ∂L ∂φL − ∂µ = 0 (3.16) ∂(∂µφ) yield (3.15). Hamiltonian formalism The variable conjugate to φ(x) is ∂L π(x) := = ∂0φ(x). ∂0φ(x) We have the equal-time Poisson-brackets (3.9). The Hamiltonian density equals 1 1 m2 H(x) = π2(x) + ∂ φ(x)∂ φ(x) + φ2(x) + j(x)φ(x). 2 2 i i 2 The Hamiltonian Z H(t) = d~xH(t, ~x) can be used to generate the dynamics

φ˙(t, ~x) = {H(t), φ(t, ~x)}, π˙ (t, ~x) = {H(t), π(t, ~x)}.

If j does not depend on time, then the Hamiltonian does not depend on time as well and we can write H instead of H(t). We can also introduce the stress-energy tensor density 1 T (x) := ∂ φ(x)∂ φ(x) − g L(x), µν 2 µ ν µν which is conserved

∂µTµν(x) = 0. We have

H(x) = T00(x), Pi(x) = T0i(x). Quantization in the presence of external source We are looking for quantum fields satisfying

(−2 + m2)φˆ(x) = −j(x) (3.17) coinciding with the free fields for t = 0. Such fields are given by  Z 0   Z t  φˆ(t, ~x) := Texp −i Hˆ (s)ds φˆ(0, ~x)Texp −i Hˆ (s)ds . t 0 where the Hamiltonian Hˆ (t), and the corresponding Hamiltonian in the interaction picture equal Z 1 1 m2  Hˆ (t) = d~x : πˆ2(~x) + ∂ φˆ(~x)∂ φˆ(~x) + φˆ2(~x) + j(t, ~x)φˆ(~x) :, 2 2 i i 2 Z ˆ HˆInt(t) = j(t, ~x)φI(t, ~x)d~x.

In Hˆ (t), the fields are at time zero: φˆ(~x) = φˆ(0, ~x), πˆ(~x) =π ˆ(0, ~x). ˆ In HˆInt(t), the fields φI(x) are free, that is defined with H0. N-point Green’s functions

For xN , . . . , x1, the N-point Green’s function are defined as follows:

ˆ ˆ G(φ, xN ,..., φ, x1)  + ˆ ˆ  − := Ω |T φ(xN ) ······ φ(x1) Ω , where  Z t  Ω± := lim Texp −i Hˆ (s)ds Ω. t→±∞ 0 Note that the fields φ(t, ~x) according to the notation from first chapter would be denoted

φH(t, ~x). Feynman rules in momentum space for Green’s functions. We have 1 kind of lines and 1 kind of vertices. At each vertex just one line ends.

1. In the nth order we draw all possible topologically distinct Feynman diagrams with n vertices and N external lines. This means, we put N dots (for external legs) and n dots (for vertices). Then we connect them in all possible ways.

2. To each line we associate a propagator −i . k2 + m2 − i0

d4k For internal lines we integrate over the variables with the measure (2π)4 . 3. To each vertex we associate the factor −ij(k). Scattering operator

 Z ∞  S = Texp −i HˆInt(t)dt . −∞ A special role is played by matrix elements of the scattering operator between plane waves, called scattering amplitudes

+ + − −
Φ(k1 ) ··· Φ(kN + )| S Φ(kN − ) ··· Φ(k1 ) .

For simplicity, we assume that the momenta are distinct. Feynman rules in momentum space for scattering amplitudes. To compute scattering amplitudes with N − incoming and N + outgoing particles we draw the same diagrams as for N − + N +-point Green’s functions, where we adopt the convention that the incoming lines are drawn on the right and outgoing lines on the left. The rules are changed only concerning the external lines.

1. With each incoming external line we associate √ 1 . V 2ω(~k) 2. With each outgoing external line we associate √ 1 . V 2ω(~k) Chapter 4

Massive photons 137

Let Aν(x) be a vector field. Set

Fµν := ∂µAν − ∂νAµ. (4.1)

In this chapter we discuss the quantization of the so-called Proca equation

µν 2 ν −∂µF (x) + m A (x) = 0. (4.2) Let J ν(x) be a given vector function called current. We will assume that the current is conserved, that is ν ∂νJ (x) = 0. (4.3) We will also consider the Proca equation in the presence of the current J µ:

µν 2 ν ν −∂µF (x) + m A (x) = J (x). (4.4) Space of solutions

Let YPr,R, resp. YPr denote the set of real, resp. complex smooth space-compact solutions of µ 2 −∂ (∂µζν − ∂νζµ) + m ζν(x) = 0, (4.5) or what is equivalent

(−2 + m2)ζµ(x) = 0, (4.6) ν ∂νζ (x) = 0. (4.7)

~ Only ζ is dynamical and satisfies the obvious analog of Theorem 3.2. ζ0 can be calculated by

2 −1 ~˙ ζ0 = (−∆ + m ) divζ. Symplectic form Clearly, the following expression defines a conserved current:

µ µ ν µ ν j (ζ1, ζ2, x) := ∂ ζ1,ν(x)ζ2 (x) − ζ1,ν(x)∂ ζ2 (x).

YPr,R is a symplectic space with the symplectic form Z µ ζ1ωζ2 = j (ζ1, ζ2, x)dsµ(x) S Z  ˙ ν ˙ν  = ζ1,ν(t, ~x)ζ2 (t, ~x) − ζ1,ν(t, ~x)ζ2 (t, ~x) d~x. Alternatively, we could have used another conserved current to define the symplectic form:

µ j0 (ζ1, ζ2, x) µ µ ν µ ν ν µ := (∂ ζ1,ν(x) − ∂νζ1 (x)) ζ2 (x) − ζ1,ν(x)(∂ ζ2 (x) − ∂ ζ2 (x)) .

Using the integration by parts and the Lorentz condition we see that it leads to the same symplectic form on YPr,R. Classical potentials

We introduce the functionals Aµ(x)

hAµ(x)|ζi := ζµ(x).

Moreover, we introduce the functionals Ei(x)

˙ hEi(x)|ζi := ζi(x) − ∇iζ0(x). ˙ Clearly, E = A~ − ∇A0. Poisson structure

The symplectic structure on the space YPr,R leads to a Poisson bracket on functions on

YPr,R:

{Ai(t, ~x),Aj(t, ~y)} = {Ei(t, ~x),Ej(t, ~y)} = 0,

{Ai(t, ~x),Ej(t, ~y)} = δijδ(~x − ~y). (4.8)

Using 2 −1 ˙ A0 = (−∆ + m ) divA,~ we deduce  ∂ ∂  {A (x),A (y)} = g − µ ν D(x − y). µ ν µν m2 Spatially smeared fields

We can use the symplectic form to pair distributions and solutions. For ζ ∈ YPr,R we introduce also the functional on YPr,R given by

hA((ζ))|ρi := ρωζ.

Note that

{A((ζ1)),A((ζ2))} = ζ1ωζ2. Z  ˙ µ µ  A((ζ)) = ζµ(t, ~x)A (t, ~x) − ζµ(t, ~x)E (t, ~x) d~x. (4.9) Space-time smeared fields ∞ 1,3 1,3 We can also smear the potentials with space-time vector valued functions f ∈ Cc (R , R ): Z µ A[f] := fµ(x)A (x)dx.

Note that A[f] = A((ζ)), where ∂ ∂ν ζ = Df − µ Df . µ µ m2 ν Let ijk be the standard fully antisymmetric symbol. We introduce 1   H(x) = E~ 2(x) + m−2(divE~ )2(x) + (∇~ A~)2(x) − (divA~)2(x) + m2A~2(x) 2 P~ (x) = Ei(x)∇~ Ai(x),

S(x) = Ei(x)ijk∇kAj(x).

They give the total Hamiltonian, momentum and polarization Z Z Z H = H(0, ~x)d~x, P~ = P~ (0, ~x)d~x, S = S(0, ~x)d~x.

The Poisson bracket of any pair from H, P,S~ is zero. Plane waves We would like to diagonalize simultaneously the Hamiltonian H, the momentum P~ , the polarization S and the symplectic form ω. 1,3 ~ ~ ~ p~ 2 2 ~ ~ Let k ∈ R with k = (ω(k), k), ω(k) = k + m . We fix two spatial vectors e1(k), e2(k) that form an oriented orthonormal basis of the plane orthogonal to ~k. Define ! |~k| ω(~k)~k u(k, 0) := , , (4.10) m m|~k|  1  u(k, ±1) := 0, √ (e1 ± ie2) . (4.11) 2 Note that

µ uµ(k, σ)k = 0, µ 0 uµ(k, σ)u (k, σ ) = δσ,σ0 , X kµkν u (k, σ)u (k, σ) = g + . µ ν µν m2 σ=0,±1 A plane wave is defined as

1 ikx Φ(k, σ) = q uµ(k, σ)e . (4.12) (2π)3/2 2ω(~k)

0 0 ~ ~ 0 Note that iΦ(k, σ)ωΦ(k , σ ) = δ(k − k )δσ,σ0 . Plane wave functionals

Define the following functionals on YPr,R:

a(k, σ) = A((iΦ(k, σ))) s Z ~ − 3  ω(k) −i~x~k µ = (2π) 2 e u (k, σ)A (0, ~x) 2 µ i −i~k~x µ  −q e uµ(k, σ)E (0, ~x) d~x. 2ω(~k) Diagonalization

{a(k, σ), a(k0, σ0)} = {a(k, σ), a(k0, σ0)} = 0, 0 0 ~ ~ 0 {a(k, σ), a(k , σ )} = iδ(k − k )δσ,σ0 , Z X H = d~k ω(~k)a(k, σ)a(k, σ), σ=0,±1 Z X P~ = d~k ~ka(k, σ)a(k, σ), σ=0,±1 Z X S = d~k σ|~k|a(k, σ)a(k, σ). σ=0,±1

If ζ1, ζ2 ∈ YPr,R, then Z X ~ iζ1ωζ2 = (ha(k, σ)|ζ1iha(k, σ)|ζ2i − ha(k, σ)|ζ1iha(k, σ)|ζ2i) dk. σ=0,±1 The fieds can be written as Z ~ − 3 X dk  ikx −ikx  Aµ(x) = (2π) 2 q uµ(x, σ)e a(k, σ) + uµ(x, σ)e a(k, σ) σ=0,±1 2ω(~k) X Z   = Φ(k, σ)a(k, σ) + Φ(k, σ)a(k, σ) d~k. σ=0,±1 Positive/negative frequency solutions (±) YPr will denote the subspace of YPr consisting of positive, resp. negative frequency solu- (+) tions. Every ζ ∈ YPr can be written as Z ~ − 3 X dk ikx ζµ(x) = (2π) 2 q e uµ(k, σ)ha(k, σ)|ζi σ=0,±1 2ω(~k)

(+) For ζ1, ζ2 ∈ YPr we define the scalar product Z X ~ (ζ1|ζ2) := iζ1ωζ2 = ha(k, σ)|ζ1iha(k, σ)|ζ2idk σ=0,±1

(+) (+) We set ZPr to be the completion of YPr in this scalar product. The ortochronous Poincar´e (+) group leaves ZPr invariant. Quantization of free fields We want to construct a Hilbert space H, a positive self-adjoint operator Hˆ , a normalized vector Ω, which is a ground state for Hˆ and a self-adjoint operator-valued distribution R1,3 3 x 7→ Aˆµ(x) satisfying

1.

µ 2 −∂ (∂µAˆν − ∂νAˆµ) + m Aˆν(x) = 0,  ∂ ∂  [Aˆ (x), Aˆ (y)] = i g − µ ν D(x − y). µ ν µν m2

2. itHˆ 0 −itHˆ 0 e Aˆµ(x , ~x)e = Aˆµ(x + t, ~x).

3. Ω belongs to the domain of polynomials in smeared out Aˆµ(x) and is cyclic for these operators. An alternative set of conditions: ˆ We look for a linear function YPr,R 3 ζ 7→ A((ζ)) satisfying 1.’ h i Aˆ((ζ1)), Aˆ((ζ2)) = iζ1ωζ2.

2.’ ˆ itHˆ ˆ −itHˆ A((r(t,~0)ζ)) = e A((ζ))e .

3.’ Ω belongs to the domain of polynomials in Aˆ((ζ)) and is cyclic for these operators. (+) The above problem has a solution unique up to a unitary equivalence. We set H = Γs(ZPr ). The creation/annihilation operators will be denoted by aˆ∗ and aˆ. In particular, we set

aˆ∗(k, σ) :=a ˆ∗(−iΦ(k, σ)).

Ω will be the Fock vacuum. We set Z ~ 3 dk X   ˆ − 2 −ikx ikx ∗ Aµ(x) = (2π) q uµ(k, σ)e aˆ(k, σ) + uµ(k, σ)e aˆ (k, σ) , 2ω(~k) σ=0,±1 The quantum Hamiltonian, momentum and polarization are X Z Hˆ = aˆ∗(k, σ)ˆa(k, σ)ω(~k)d~k, σ=0,±1 ~ X Z Pˆ = aˆ∗(k, σ)ˆa(k, σ)~kd~k, σ=0,±1 Z X Sˆ = d~k σ|~k|aˆ∗(k, σ)ˆa(k, σ). σ=0,±1 The whole orthochronous Poincar´egroup is unitarily implemented on H by U(a, Λ) :=   Γ r We have (a,Λ) (+) ZPr ˆ ∗ ν ˆ
U(a, Λ)Aµ(x)U(a, Λ) = ΛµAν (a, Λ)x .

Note the identities  ∂ ∂  (Ω|Aˆ (x)Aˆ (y)Ω) = −i g − µ ν D(+)(x − y), µ µ µν m2  ∂ ∂  (Ω|T(Aˆ (x)Aˆ (y))Ω) = −i g − µ ν Dc(x − y). µ ν µν m2 Axioms ∞ 1,3 1,3 For f ∈ Cc (R , R ) set Z µ Aˆ[f] := Aˆµ(x)f (x)dx.

fin We obtain a family that satisfies the Wightman axioms with D := Γs (ZPr). For an open set O ⊂ Rd we set

n ˆ ∞ 1,3 1,3 o A(O) := exp(iA[f]) : f ∈ Cc (R , R ) .

The algebras A(O) satisfy the Haag-Kastler axioms. External current We return to the classical Proca equation. Recall that in the presence of the external current the Proca equations take the form

ν ∂νJ (x) = 0, (4.13) 2 ν ν −∂µ(∂µAν − ∂νAµ) + m A (x) = J (x). (4.14)

Note that (4.14) and (4.13) imply

ν ∂νA (x) = 0. (4.15)

We have therefore (−2 + m2)Aµ(x) = J µ(x). (4.16) The temporal component of (4.14) has no time derivative:

∆A0 − ∂0divA~ − m2A0 = J 0. (4.17)

Therefore, we can compute A0 in terms of A~ at the same time:

A0 = (∆ − m2)−1(∂0divA~ + J 0). (4.18)

The only dynamical variables are the spatial components, satisfying the equation

2 2 ~ ~ (−∂0 + ∆ − m )A = J. (4.19) The Lagrangian The Lagrangian density equals 1 m2 L := − F F µν − A Aµ + J Aµ 4 µν 2 µ µ 1 1 m2 = − ∂ A ∂µAν + ∂ A ∂νAµ − A Aµ + J Aµ 2 µ ν 2 µ ν 2 µ µ 1 1 1 1 = − ∂ A ∂ A + ∂ A ∂ A + ∂ A ∂ A + ∂ A ∂ A − ∂ A ∂ A 2 i j i j 2 i 0 i 0 2 0 i 0 i 2 i j j i 0 i i 0 m2 m2 + A2 − A2 + J A − J A 2 0 2 i i i 0 0 2 2 1 1 1 ˙ 2 ˙ m m = − (rotA~)2 + (∇~ A )2 + A~
− A~∇~ A + A2 − A~2 + J~A~ − J A . 4 2 0 2 0 2 0 2 0 0 As noted before, only spatial components Ai are dynamical. We have the conjugate variables ∂L Ei = = F0i ∂A˙ i ˙ or E~ = A~ − ∇~ A0. In terms of E~ , we have 1 A = (J − divE~ ). (4.20) 0 m2 0 The Hamiltonian The Hamiltonian density

H = E~ · A~ − L 1 1 m2 m2 = E~ 2 + E~ ∇~ A + (rotA~)2 − A2 + A~2 − J~A~ + J A 2 0 4 2 0 2 0 0 1 1 1 1 m2 = E~ 2 + (J − divE~ )2 + (∇~ A~)2 − (divA~)2 + A~2 2 2m2 0 2 2 2 −1 −J~A~ + J0(−∆) J0 + total derivative.

~ ~ ~ 1 ~ 2 1 ~ ~ 2 1 ~ 2 In the last step we replaced E∇A0 with −divEA0 and 4(rotA) with 2(∇A) − 2(divA) , the difference being a total derivative in spatial variables. We also inserted A0 as in (4.20). Quantization in the presence of external source 1,3 ˆ We are looking for quantum potentials R 3 x 7→ Aµ(x) satisfying

µ ν ν µ 2 ν ν −∂µ(∂ Aˆ − ∂ Aˆ ) + m Aˆ (x) = J (x).

They are given by  Z 0   Z t  Aˆµ(t, ~x) := Texp −i Hˆ (s)ds Aˆµ(~x)Texp −i Hˆ (s)ds , t 0 where the Hamiltonian Hˆ (t), and the corresponding Hamiltonian in the interaction picture equal

Z 1 ~ 1 ~ 1 ~ 1 ~ m2 ~ Hˆ (t) = d~x : Eˆ2 + (J − divEˆ)2 + (∇~ Aˆ)2 − (divAˆ)2 + Aˆ2 2 2m2 0 2 2 2 ~ −1  −J~Aˆ + J0(−∆) J0 : Z  1 ~ ~ 1  Hˆ (t) = d~x : − divEˆ J − J~Aˆ + J 2 + J (−∆)−1J :. Int m2 I 0 I 2m2 0 0 0 N-point Green’s functions

For xN , . . . , x1, the N-point Green’s function are defined as follows:

ˆ ˆ G(AµN , xN ,..., Aµ1 , x1)  + ˆ ˆ − := Ω |T(AµN (xN ),..., Aµ1 (x1))Ω , where  Z t  Ω± := lim Texp −i Hˆ (s)ds Ω. t→±∞ 0 Feynman rules for Green’s functions. We have 1 kind of lines and 1 kind of vertices. At each vertex just one leg ends.

1. In the nth order we draw all possible topologically distinct Feynman diagrams with n vertices and external lines.

−i  kµkν  2. To each line we associate a propagator m2+k2−i0 gµν − m2 . For internal lines we inte- d4k grate over the variables with the measure (2π)4 . 3. To each vertex we associate the factor −iJ µ(k). Scattering operator

 Z ∞  S = Texp −i HˆInt(t)dt . −∞ As usual, matrix elements of the scattering operator between plane waves are called scattering amplitudes + + + + − − − −
Φ(k1 , σ1 ) ··· Φ(kN + , σN + )| S Φ(kN − , σN − ) ··· Φ(k1 , σ1 ) . For simplicity, we assume that the momenta are distinct. Feynman rules for scattering amplitudes. To compute scattering amplitudes with N − incoming and N + outgoing particles we draw the same diagrams as for N − + N +-point Green’s functions, where we adopt the convention that the incoming lines are drawn on the right and outgoing lines on the left. The rules are changed only concerning the external lines.

1. With each incoming external line we associate √ 1 u(k, σ). V 2ω(~k) 2. With each outgoing external line we associate √ 1 u(k, σ). V 2ω(~k) Propagators for massive photons The causal propagator used to compute Green’s functions and scattering amplitudes that follows directly from the interaction Hamiltonian is 1  k k  Dc = g + µ ν . (4.21) µν m2 + k2 − i0 µν m2 If we compute scattering amplitudes, we can pass from this propagator to another by adding kµfν(k) + fµ(k)kν for an arbitrary function fµ(k). To see this note that after adding kµfν(k) + fµ(k)kν the contribution of each line changes by µ ν J (k)(kµfν(k) + fµ(k)kν) J (k),

µ which is zero, because kµJ (k) = 0. For scattering amplitudes the external line does not involve the propagator. Therefore, scattering amplitudes do not change. However, Green’s functions change, because of external lines. We will list a number of useful propagators. We will drop the superscript c (for causal). For any α ∈ R, we can pass to the following propagators 1  k k  Dα,I = g − (α − 1) µ ν , µν m2 + k2 − i0 µν m2 1  k k  Dα,II = g − (α − 1) µ ν . µν m2 + k2 − i0 µν k2 The above propagators coincide for α = 1, and will be called the propagator in the Feynman gauge, that is 1 DFeyn(k) = . µν m2 + k2 − i0 We can introduce the propagator in the Yukawa gauge:   Yuk 1 Yuk Yuk 1 kikj D00 = − ,D0j = 0,Dij = δij − , m2 + ~k2 m2 + k2 − i0 m2 + ~k2

Yuk Feyn Yuk Yuk We have Dµν = Dµν + kµfν (k) + fµ (k)kν, where

Yuk k0 Yuk ki f0 (k) = , fi (k) = − . (k2 − i0)2(m2 + ~k2) (k2 − i0)2(m2 + ~k2) The propagator in the Yukawa gauge is the massive analog of the propagator in the Coulomb gauge. The propagator in the temporal gauge equals   tem tem tem 1 kikj D00 = 0,D0j = 0,Dij = 2 δij − 2 . k − i0 k0

tem Feyn tem tem We have Dµν = Dµν + kµfν (k) + fµ (k)kν, where

tem 1 tem ki f0 (k) = 2 2 , fi (k) = − 2 2 2 . (m + k − i0)2k0 (m + k − i0)2k0 Chapter 5

Massless photons 175

For a vector potential Aν(x) we define

Fµν := ∂µAν − ∂νAµ,

as in (4.1). In this chapter we discuss the quantization of the Maxwell equation

µν −∂µF (x) = 0. (5.1) We will also consider an external conserved current, that is a given vector function J ν(x) satisfying ν ∂νJ (x) = 0. (5.2) The Maxwell equation in the presence of the current J reads

µν ν −∂µF (x) = J (x), (5.3) Gauge invariance It is well known that the Maxwell equation

µ ν ν µ −∂µ (∂ ζ (x) − ∂ ζ (x)) = 0 (5.4) is invariant wrt the replacement of ζµ with ζµ + ∂µχ, where χ is an arbitrary smooth function on the space-time. In particular, there is no uniqueness of the Cauchy problem for ζµ. This poses problems both for the classical and quantum theory. One could avoid the problem of the gauge invariance by considering fields and not potentials as basic objects. However, when one quantizes the Maxwell equation including the currents, it is more convenient to use potentials. Therefore, we will stick to potentials. Space of solutions of the Maxwell equation There exist several ways to cope with gauge invariance. In the first approach, we start with the space of real/complex smooth space-compact solutions of the Maxwell equation (5.4), ˜ ˜ which we denote by YMax,R/YMax We check that the following expression defines a conserved current

µ ˜j (ζ1, ζ2, x) µ µ ν µ ν ν µ := (∂ ζ1,ν(x) − ∂νζ1 (x)) ζ2 (x) − ζ1,ν(x)(∂ ζ2 (x) − ∂ ζ2 (x)) .

It defines in the usual way the antisymmetric form Z 0 ζ1ωζ˜ 2 = ˜j (ζ1, ζ2, (t, ~x))d~x. (5.5)

(5.5) does not depend on the gauge. The ortochronous Poincare group acts on Y˜Max by transformations preserving ω˜:

µ µ −1
r˜(a,Λ)ζ (x) := Λν ζµ (a, Λ) x . The Coulomb gauge

We say that a solution ζ of the Maxwell equation is in the Coulomb gauge if ζ0 = 0 and divζ~ = 0.

Note that every ζ ∈ Y˜Max is gauge-equivalent to a unique solution of the Maxwell equation in the Coulomb gauge, which however does not have to be space-compact. If for ζ ∈ Y˜Max, we denote by ζCoul the corresponding Coulomb gauge solution, then

Coul Coul ζ1ωζ˜ 2 = ζ1 ωζ˜ 2 . Symplectic reduction ˜ The space YMax,R is presymplectic. The kernel of the symplectic form coincides with po- tentials that are gauge equivalent to zero. We define YMax,R to be the symplectic reduction ˜ ˜ of YMax,R. In other words, YMax,R is YMax,R divided by the gauge equivalence. YMax,R is equipped with a natural symplectic form ω. The ortochronous Poincare group acts on YMax,R by symplectic transformations.

Analogously we define the space YMax of gauge classes of complex smooth space-compact solutions of (5.4). Coul We introduce the functionals Aµ (x) on YMax,R, called the classical potentials in the Coulomb gauge, Coul Coul hAµ (x)|ζi := ζµ (x), where ζCoul on the right hand side is the representative of the class ζ in the Coulomb gauge. Clearly, Coul ~Coul A0 (x) = 0, divA (x) = 0.

Moreover, we introduce the functionals Fµν(x) called the fields:

hFµν(x)|ζi := ∂µζν(x) − ∂νζµ(x).

We will write Ei(x) = F0i(x). Clearly,

Coul Coul E~ = ∂tA~ , divA~ (x) = 0. Poisson structure

The symplectic structure on the space YMax,R leads to a Poisson bracket on the level of functions on YMax,R:

Coul Coul {Ai (t, ~x),Aj (t, ~y)} = {Ei(t, ~x),Ej(t, ~y)} = 0,  ∂ ∂  {ACoul(t, ~x),E (t, ~y)} = δ − i j δ(~x − ~y) i j ij ∆ From the above relations we deduce  ∂ ∂  {ACoul(x),ACoul(y)} = δ − i j D(x − y). i j ij ∆ Spatially smeared potentials

Let us study various constructions related to YMax. In what follows we drop the superscript Coul Coul on Aµ (x).

We can use the symplectic form to pair distributions and solutions. For ζ ∈ YMax,R we introduce also the functional on YMax,R given by

hA((ζ))|ρi := ρωζ.

Note that

{A((ζ1)),A((ζ2))} = ζ1ωζ2. Z  ˙ µ µ  A((ζ)) = ζµ(t, ~x)A (t, ~x) − ζµ(t, ~x)E (t, ~x) d~x. (5.6)

Let us stress that A((ζ)) depends on ζ only modulo gauge transformations. Space-time smeared potentials ∞ 1,3 1,3 We can also smear the potentials with space-time vector valued functions f ∈ Cc (R , R ): Z µ A[f] := fµ(x)A (x)dx. (5.7)

Note that A[f] = A((ζ)), where

 ∂ ∂j  ζ = D f − i f , ζ = 0. i i ∆ j 0 As for massive photons, we introduce 1   H(x) = E~ (x)2 + ∇~ A~(x)2 , 2 P~ (x) = Ei(x)∇~ Ai(x),

S(x) = Ei(x)ijk∇kAj(x).

They give the total Hamiltonian, momentum and polarization Z Z Z H = H(0, ~x)d~x, P~ = P~ (0, ~x)d~x, S = S(0, ~x)d~x.

The Poisson bracket of any pair from H, P,S~ is zero. Plane waves p Let k ∈ R1,3 with k = (ω,~k), ω(~k) := ~k2. The vectors u(k, ±1) are defined as in (4.11). u(k, 0) are not defined at all. Note that

~ ~u(k, σ) · k = 0, u0(k, σ) = 0, µ 0 uµ(k, σ)u (k, σ ) = δσ,σ0 , X kikj ui(k, σ)uj(k, σ) = δij − . ~ 2 σ=±1 k A plane wave is defined as

1 ikx Φ(k, σ) = q uµ(k, σ)e . (5.8) (2π)3/2 2ω(~k)

0 0 ~ ~ 0 Note that iΦ(k, σ)ωΦ(k , σ ) = δ(k − k )δσ,σ0 . Plane wave functionals

Define the following functionals on YMax,R:

a(k, σ) = A((iΦ(k, σ))) s Z ~ − 3  ω(k) −i~x~k µ = (2π) 2 e u (k, σ)A (0, ~x) 2 µ i −i~k~x µ  −q e uµ(k, σ)E (0, ~x) d~x. 2ω(~k) Diagonalization

{a(k, σ), a(k0, σ0)} = {a(k, σ), a(k0, σ0)} = 0, 0 0 ~ ~ 0 {a(k, σ), a(k , σ )} = iδ(k − k )δσ,σ0 , Z X H = d~k ω(~k)a(k, σ)a(k, σ), σ=±1 Z X P~ = d~k ~ka(k, σ)a(k, σ), σ=±1 Z X S = d~k σ|~k|a(k, σ)a(k, σ). σ=±1

If ζ1, ζ2 ∈ YMax,R, then Z X ~ iζ1ωζ2 = (ha(k, σ)|ζ1iha(k, σ)|ζ2i − ha(k, σ)|ζ1iha(k, σ)|ζ2i) dk. σ=±1 The potentials can be written as Z ~ − 3 X dk  ikx −ikx  Aµ(x) = (2π) 2 q uµ(x, σ)e a(k, σ) + uµ(x, σ)e a(k, σ) σ=±1 2ω(~k) X Z   = Φ(k, σ)a(k, σ) + Φ(k, σ)a(k, σ) d~k. σ=±1 Positive/negative frequency solutions (±) YMax will denote the subspace of YMax consisting of positive, resp. negative frequency solutions. (+) Every ζ ∈ YMax can be written as Z ~ − 3 X dk ikx ζ(x) = (2π) 2 q e u(k, σ)ha(k, σ)|ζi σ=±1 2ω(~k)

(+) For ζ1, ζ2 ∈ YMax we define the scalar product Z X ~ (ζ1|ζ2) := iζ1ωζ2 = ha(k, σ)|ζ1iha(k, σ)|ζ2idk. σ=±1

(+) (+) We set ZMax to be the completion of YMax in this scalar product. The ortochronous Poincar´e (+) group leaves ZMax invariant. Spin averaging For a given k ∈ R1,3 with k2 = 0, let M,N be vectors with

µ ν M kµ = N kν = 0.

Then X µ ν µ M uµ(k, σ)uν(k, σ)N = M Nν. (5.9) σ=±1 In fact, ~ ~ X kµkν u (k, σ)u (k, σ) = g + δ δ − . µ ν µν µ0 ν0 ω2 σ=±1 Therefore, the left hand side of (5.9) equals

(M~ ~k)(N~ ~k) M µg N ν + M 0N 0 − . µν ω2 But ~kM~ ~kN~ M 0 = ,N 0 = . ω ω Quantization of free fields There are several approaches to the quantization of Maxwell equation. In particular, one can either reduce first and then quantize, or first quantize and then reduce. We will treat the former choice as the basic one. This means, we will use the space YMax,R as the input for quantization. The other possibilities will be discussed later.

The quantization of YMax,R is very similar to the quantization of YPr,R. The main difference is that Condition 1. is replaced with

2 (−2 + m )Aˆi(x) = 0,

∂iAˆi(x) = 0,

Aˆ0(x) = 0  ∂ ∂  [Aˆ (x), Aˆ (y)] = i δ − i j D(x − y). j i ij ∆ (+) The above problem has a solution unique up to a unitary equivalence. We set H = Γs(ZMax). The creation/annihilation operators will be denoted by aˆ∗ and aˆ. In particular, we set

aˆ∗(k, σ) :=a ˆ∗(−iΦ(k, σ)).

Ω will be the Fock vacuum. We set Z ~ − 3 dk X  −ikx ikx ∗  Aˆi(x) = (2π) 2 √ ui(k, σ)e aˆ(k, σ) + ui(k, σ)e aˆ (k, σ) , 2ω σ=±1 The Hamiltonian, the momentum and the polarization are X Z Hˆ = aˆ∗(k, σ)a(k, σ)ω(~k)d~k, σ=±1 ~ X Z Pˆ = aˆ∗(k, σ)ˆa(k, σ)~kd~k, σ=±1 X Z Sˆ = d~kσ|~k|aˆ∗(k, σ)ˆa(k, σ). σ=±1 The whole orthochronous Poincar´egroup is unitarily implemented on H by U(a, Λ) :=   Γ r We have (a,Λ) (+) ZPr ˆ ∗ µ0 ν0 ˆ
U(a, Λ)Fµ,ν(x)U(a, Λ) = Λµ Λν Fµ0,ν0 (a, Λ)x .

Note the identities  ∂ ∂  (Ω|Aˆ (x)Aˆ (y)Ω) = −i δ − i j D(+)(x − y), i j ij ∆  ∂ ∂  (Ω|T(Aˆ (x)Aˆ (y))Ω) = −i δ − i j Dc(x − y). i j ij ∆ Axioms ∞ 1,3 3 ˆ R ˆ fin The family Cc (R , R ) 3 f 7→ A[f] := Ai(x)fi(x)dx with D := Γs (ZMax) does not satisfy the Wightman axioms because of two problems: the nonlocality of the commutator and the absence of the Lorentz covariance.

If we replace the fields Aˆµ with Fˆµν, we restore the Poincare covariance. For an open set O ⊂ R1,3 we set

ˆ ∞ 1,3 3 A(O) := {exp(iF [f]) : f ∈ Cc (R , R )}.

The algebras A(O) satisfy the Haag-Kastler axioms. External current

We return to the classical Maxwell equation. We consider an external conserved current Jµ. The Maxwell equation with an external current reads

µ ν ν µ ν −∂µ∂ ζ + ∂ ∂µζ = J . (5.10)

Given an arbitrary solution ζµ, by setting

Coul ζµ := ζµ + ∂µχ, ˙ where χ(t, ~x) := −∆−1divA~(t, ~x) we obtain a soluton satisfying

~Coul Coul −1 0 divζ = 0, ζ0 = −∆ J .

We call ζCoul the representative of ζ in the Coulomb gauge. Consider the Maxwell equation written in terms of Aµ:

µ ν ν µ ν −∂µ∂ A + ∂ ∂µA = J . (5.11)

We write separately the temporal and spatial equations: ˙ −∆A0 + divA~ = J 0, 2
0 (∂0) − ∆ A~ − ∇~ A˙ + ∇~ divA~ = J.~ We can compute A0 in terms of A~ at the same time:

A0 = ∆−1(∂0divA~ − J 0). (5.12)

We can insert this into spatial equations, using J˙0 = divJ~, obtaining

2 ~ ~ (−∂0 + ∆)Atr = Jtr. (5.13) where

−1 A~tr := A~ − ∇~ ∆ divA,~ −1 Jtr := J~ − ∇~ ∆ divJ.~

Thus the only dynamical variables are the transversal spatial fields. divA~ =: Θ is an arbitrary space-time function. The Lagrangian and Hamiltonian The Lagrangian density 1 L := − F F µν + J Aµ 4 µν µ 1 1 1 ˙ 2 ˙ = − (rotA~)2 + (∇~ A )2 + A~
− A~∇~ A + J~A~ − J A 4 2 0 2 0 0 0 1 1 ˙ 1 = − (∇~ A~ )2 + (A~ )2 + J~ A~ + J 0∆−1J 0 − ∂0(J 0∆−1divA~) 2 tr 2 tr tr tr 2 +total spatial derivative. ˙ The conjugate variables are E~ = A~ − ∇A0, where clearly divE~ = J0. The Hamiltonian density is

H = E~tr · A~tr − L 1 1 = E~ 2 + (∇~ A~ )2 − J~ A~ + J (−∆)−1J + total derivative. 2 tr 2 tr tr tr 0 0 Quantization in the presence of external source 1,3 ˆ We are looking for quantum potentials R 3 x 7→ Aµ(x) satisfying

µ ν ν µ 2 ν ν −∂µ(∂ Aˆ − ∂ Aˆ ) + m Aˆ (x) = J (x), (5.14) ~ divAˆ(x) = 0, (5.15) ~ Aˆ0 = ∆−1(∂0divAˆ − J 0). (5.16) To construct them note that it is enough to impose the condition (5.15) at t = 0. Then this condition propagates for all times. (5.16) can be used as a definiton of Aˆ0(x). One can use the Hamiltonian Hˆ (t) and the corresponding Hamiltonian in the interaction picture equal

Z 1 ~ 1 ~ ~  Hˆ (t) = d~x : Eˆ2 + (∇~ Aˆ)2 − J~Aˆ + J (−∆)−1J : 2 2 0 0 Z  ~ −1  HˆInt(t) = d~x −J~AˆI + J0(−∆) J0 . Propagators for massless photons The causal propagator used to compute Green’s functions and scattering amplitudes that follows directly from the interaction Hamiltonian is the propagator in the Coulomb gauge   Coul 1 Coul Coul 1 kikj D00 = − ,D0j = 0,Dij = δij − , ~k2 k2 − i0 ~k2 If we compute scattering amplitudes, we can pass from this propagator to another by adding kµfν(k) + fµ(k)kν for an arbitrary function fµ(k). Let us list a number of useful propagators in other gauges. In particular, we distinguish the family of propagators 1  k k  g + (α − 1) µ ν . k2 − i0 µν k2 They have the following names

Lan 1  kµkν  Dµν := k2−i0 gµν − k2 Landau or Lorentz gauge, Feyn 1 Dµν := k2−i0gµν Feynman gauge, FY 1  kµkν  Dµν := k2−i0 gµν + 2 k2 Fried and Yennie gauge. Coul Feyn Coul Coul We have Dµν = Dµν + kµfν (k) + fµ (k)kν, where

Coul k0 Coul ki f0 (k) = , fi (k) = − . (k2 − i0)2~k2 (k2 − i0)2~k2 The propagator in the temporal gauge   tem tem tem 1 kikj D00 = 0,D0j = 0,Dij = 2 δij − 2 . k − i0 k0

tem Feyn tem tem We have Dµν = Dµν + kµfν (k) + fµ (k)kν, where

tem 1 tem ki f0 (k) = 2 , fi (k) = − 2 2 . (k − i0)2k0 (k − i0)2k0

Chapter 6

Linearly perturbed quadratic Hamiltonians 209

Baker-Campbell-Hausdorff formula The BCH formula says that if [A, [A, B]] = [B, [A, B]] = 0, then

A+B A B − 1 [A,B] e = e e e 2 . Time-dependent Baker-Campbell-Hausdorff formula Let us describe a time-dependent version of the BCH formula.

Let [t−, t+] 3 t 7→ A(t),B(t) be operator valed functions such that

[A(s),A(s0)] = [B(s],B(s0)] = 0,

[[A(s),B(s0)],A(s00)] = [[A(s),B(s0)],B(s00)] = 0. Then Z t+ ! Texp ds(A(s) + B(s)) t− Z t+ ! Z t+ ! = exp dsA(s) exp dsB(s) t− t− Z t+ Z t+ ! × exp ds1 ds2θ(s1 − s2)[A(s2),B(s1)] t− t− + ! + ! + + ! Z t Z t 1 Z t Z t = exp dsA(s) exp dsB(s) exp ds1 ds2G(s2, s1) , t− t− 2 t− t− where

G(s2, s1) = θ(s1 − s2)[A(s2),B(s1)] + θ(s2 − s1)[A(s1),B(s2)]. Time-dependent Van Hove Hamiltonians 2 2 Cosider a Hilbert space L (Ξ, dξ) and the corresponding Fock space Γs(L (Ξ, dξ)). Consider the self-adjoint operator Z ∗ H0 = ω(ξ)a (ξ)a(ξ)dξ We perturb it by a time-dependent operator. Z Z V (t) = v(t, ξ)a∗(ξ)dξ + v(t, ξ)a(ξ)dξ.

Clearly, the Hamiltonian in the interaction picture equals Z Z itω(ξ) ∗ −itω(ξ) HInt(t) = e v(t, ξ)a (ξ)dξ + e v(t, ξ)a(ξ)dξ. Scattering for van Hove Hamiltonians The scattering operator is then given by  Z   Z  S = exp i v(ω(ξ), ξ)a∗(ξ)dξ exp i v(ω(ξ), ξ)a(ξ)dξ ! i Z v(τ, ξ)v(τ, ξ)ω(ξ) × exp dτdξ , 2π ω(ξ)2 − τ 2 − i0 where v(τ, ξ) = R v(t, ξ)e−itτ dt. Time-independent Van Hove Hamiltonians Consider now a time-independent operator Z Z V = v(ξ)a∗(ξ)dξ + v(ξ)a(ξ)dξ.

Clearly, the Fock vacuum Φ0 = Ω is the eigenstate of H0 vacuum with the eigenvalue E0 = 0.

The eigenstate and eigenvalue of H := H0 + V are Z |v(ξ)|2 E = − dξ, ω(ξ) ! Z v(ξ) Z v(ξ) Φ = exp a∗(ξ)dξ − a(ξ)dξ Ω. ω(ξ) ω(ξ) Adiabatic scattering theory for van Hove Hamiltonians + To compute the Gell-Mann and Low wave operators for the pair H0, H, consider v := −|t| + −iv(ξ) θ(t)e v(ξ). Then v (τ, ξ) = τ−i . Thus ! Z v(ξ)  Z v(ξ) S+ = exp a∗(ξ)dξ exp − a(ξ)dξ  ω(ξ) − i ω(ξ) + i  i Z |v(ξ)|2ω(ξ)dτdξ  × exp . 2π (τ 2 + 2)(ω(ξ)2 − τ 2 − i0) Using the residue calculus we see that i Z ω(ξ)dτdξ i = (6.1) 2π (τ 2 + 2)(ω(ξ)2 − τ 2 − i0) 2(ω(ξ) − i) iω(ξ) 1 = − . (6.2) 2(ω(ξ)2 + 2) 2(ω(ξ)2 + 2)

1 Now the real part of (6.2) equals −2(ω(ξ)2+2). Therefore

+ Z  Z ! |(Ω|S Ω)| + v(ξ) ∗ v(ξ) + S = exp a (ξ)dξ exp − a(ξ)dξ (Ω|S Ω) ω(ξ) − i ω(ξ) + i  1 Z |v(ξ)|2  × exp − dξ . 2 ω(ξ)2 + 2 R |v(ξ)|2 Therefore, if ω(ξ)2 dξ < ∞, then Z  Z ! + + v(ξ) ∗ v(ξ) SGL = s− lim S = exp a (ξ)dξ exp − a(ξ)dξ &0 ω(ξ) ω(ξ)  Z |v(ξ)|2  × exp − dξ . ω(ξ)2

R |v(ξ)|2 + If ω(ξ)2 dξ = ∞, then SGL does not exist. − + An analogous computation yields SGL = SGL. Therefore,

+∗ − SGL := SGLSGL = 1l.

−|t| 2 One can also set v(t, ξ) := e v(ξ). Then v(τ, ξ) = τ 2+2 v(ξ). Therefore,  Z  Z ! S v(ξ)2 ∗ v(ξ)2 = exp i 2 2 a (ξ)dξ exp −i 2 2 a(ξ)dξ . (Ω|SΩ)  + τ  + τ

This converges strongly to 1l. Infrared problem A classical particle travels with trajectory t 7→ ~y(t), and a constant profil f(~x). Then its current equals ~y(t) J 0(t, ~x) = f(~x − ~y(t)),J i(t, ~x) = f(~x − ~y(t)) . dt Assume that ~y(t) = t~v± for ±t > 0. Then

Z ~ J µ(k) = J µ(t, ~x)e−ik~x+ik0tdxdt  i(1,~v )µ i(1,~v )µ  = − + + − f(~k) ~ ~ k~v+ − k0 − i0 k~v− − k0 + i0  ipµ ipµ  = − + + − f(~k), kp+ − i0 kp− + i0 where p = √ m (1,~v±). 1−(~v±)2 Consider photons of mass m ≥ 0 coupled to the current J µ. Then the scattering operator is well defined if Z J (k)J µ(k) µ dk (6.3) ~ 2 2 2 k + m − k0 − i0 is finite. This is the case if m > 0. If m = 0, then (6.3) is infinite. If we want to define the scattering operator in this case, we need to consider different representations of the CCR for the incoming and outgoing photons. Chapter 7

Charged scalar bosons 221

In this chapter we consider again the Klein-Gordon equation

(−2 + m2)ψ(x) = 0.

This time we will quantize the space of its complex solutions. The analysis of this equation is essentially a trivial generalization of its real case. Instead of smooth space-compact real solutions we consider smooth space-compact complex solutions. However, the formalism used to quantize in the complex case is different from the real case, therefore we devote to it a separate chapter. Another difference between the real and complex case is the possibility to include an external 4-potential Aµ(x) and to consider the equation

µ µ 2
− (∂µ − iAµ(x)) (∂ − iA (x)) + m ψ(x) = 0. Space of solutions

Recall from the previous chapter that YKG denotes the space of smooth space-compact complex solutions of the Klein-Gordon equation

(−2 + m2)ζ = 0 (7.1)

Clearly, the space YKG is equipped with the complex conjugation ζ 7→ ζ and a U(1) symmetry ζ 7→ eiθζ. Complex classical fields # 1,3 We will also consider the space dual to YKG, denoted YKG. In particular, for x ∈ R , we introduce the functionals on YKG given by 1 1 hψ(x)|ζi := √ ζ(x), hψ(x)|ζi := √ ζ(x), 2 2 1 1 hη(x)|ζi := √ ζ˙(x), hη(x)|ζi := √ ζ˙(x). 2 2 Clearly, Z Z ψ(t, ~x) = D˙ (t, ~x − ~y)ψ(0, ~y)d~y + D(t, ~x − ~y)η(0, ~y)d~y. Hermitian form Just as for the real Klein-Gordon equation, we can define the symplectic (bilinear) form

ζ1ωζ2. As the basic structure it is however more natural to choose the hermitian form iζ1ωζ2.

In fact, on YKG we have a conserved current

µ µ  µ µ  J (x) = ij (ζ1, ζ2, x) := i ∂ ζ1(x)ζ2(x) − ζ1(x)∂ ζ2(x) .

The flux of J µ across an arbitrary space-like subspace S of codimension 1 defines a hermitian form on YKG: Z µ iζ1ωζ2 = J (ζ1, ζ2, x)dsµ(x) (7.2) S Z  ˙ ˙  = i ζ1(t, ~x)ζ2(t, ~x) − ζ1(t, ~x)ζ2(t, ~x) d~x. Poisson structure The imaginary part of the hermitian form (7.2) is a symplectic form and leads to a Poisson bracket on functions on YKG. To describe it, it is sufficient to restrict oneself to operators for equal times, since they (in some sense) generate all functions on YKG. The only non-vanishing equal-time Poisson brackets are

{ψ(t, ~x), η(t, ~y)} = {ψ(t, ~x), η(t, ~y)} = δ(~x − ~y). (7.3)

Using (7.3) we obtain

{ψ(x), ψ(y)} = {ψ(x), ψ(y)} = 0, {ψ(x), ψ(y)} = D(x − y). Spatially smeared fields

We can use the charged symplectic form to pair distributions and solutions. For ζ ∈ YKG we introduce also the functional on YKG given by

hψ((ζ))|ρi := −iζωρ,

hψ((ζ))|ρi := iζωρ, ρ ∈ YKG.

Note that

{ψ((ζ1)), ψ((ζ2))} = {ψ((ζ1)), ψ((ζ2))} = 0,

{ψ((ζ1)), ψ((ζ2))} = ζ1ωζ2. Space-time smeared fields ∞ 1,3 We can also smear fields with space-time functions f ∈ Cc (R , C): Z ψ[f] := f(x)ψ(x)dx, Z ψ[f] := f(x)ψ(x)dx.

Clearly, ψ[f] = ψ((Df)), ψ[f] = ψ((Df)),

{ψ[f1], ψ[f2]} = {ψ[f1], ψ[f2]} = 0, ZZ {ψ[f1], ψ[f2]} = f1(x)D(x − y)f2(y)dxdy Charge and current

We can introduce the current of a single ζ ∈ YKG:

J µ(ζ, x) := J µ(ζ, ζ, x) = i∂µζ(x)ζ(x) − iζ(x)∂µζ(x).

(Note that the imaginary unit makes it real). It can be written as

J µ(x) = i∂µψ(x)ψ(x) − iψ(x)∂µψ(x).

It satisfies µ ∂µJ (x) = 0. The total charge is Z Q = J 0(0, ~x)d~x. We have the Hamiltonian and momentum density

H(x) = η(x)η(x) + ∇~ ψ(x) · ∇~ ψ(x) + m2ψ(x)ψ(x), P~ (x) = η(x)∇~ ψ(x) + ∇~ ψ(x)η(x).

Acting on ζ ∈ YKG, the Hamiltonian density equals

H(ζ, x) = |ζ˙(x)|2 + |∇~ ζ(x)|2 + m2|ζ(x)|2.

The total Hamiltonian and momentum are Z Z H = H(0, ~x)d~x, P~ = P~ (0, ~x)d~x.

The Poisson brackets between the Hamiltonian, momentum and charge vanish. Plane waves We would like to diagonalize simultaneously of the Hamiltonian, the momentum, the charge, and the symplectic form. It will be convenient to use different letters for the generic notation of the energy-momentum in the neutral and charged case. In the charged case, the momentum will be denoted generically by p and the energy by E = E(~p) = p~p2 + m2 (which were denoted k and ω(~k) in the neutral case.) 1,3 2 2 For p ∈ R satisfying p + m = 0 (which is equivalent to |p0| = E(~p)), we define the corresponding plane waves 1 Φ(p) = eixp. (7.4) (2π)3/2p2E(~p) Note that

iΦ(p)ωΦ(p0) = 0, 0 0 0 iΦ(p)ωΦ(p ) = sgnp0δ(~p − ~p ), sgnp0p0 > 0, 0 0 iΦ(p)ωΦ(p ) = 0, sgnp0p0 < 0.

For plane waves with negative frequency, we will also use the alternative notation

Ξ(p) := Φ(−p). Plane wave functionals

Define the following functionals on YKG:

a(p) = ψ(( − iΦ(p))) Z r ! − 3 E(~p) −i~x~p i −i~p~x = (2π) 2 e ψ(0, ~x) − e η(0, ~x) d~x, 2 p2E(~p) a(p) = ψ((iΦ(p))) Z r ! − 3 E(~p) i~x~p i i~p~x = (2π) 2 e ψ(0, ~x) + e η(0, ~x) d~x. 2 p2E(~p) Diagonalization

{a(p), a(p0)} = {a(p), a(p0)} = 0, 0 0 0 {a(p), a(p )} = isgnp0δ(~p − ~p ), sgnp0p0 > 0, 0 0 {a(p), a(p )} = 0, sgnp0p0 < 0, Z X H = d~p E(~p)a(p)a(p),

p0=±E(~p) Z X P~ = d~p ±~pa(p)a(p),

p0=±E(~p) Z X Q = d~p ±a(p)a(p).

p0=±E(~p)

Z X iζ1ωζ2 = d~p ±ha(p)|ζ1iha(p)|ζ2i.

p0=±E(~p) The fields can be written as Z − 3 X d~p ipx ψ(x) = (2π) 2 e a(p) p2E(~p) p0=±E(~p) X Z = Φ(p)a(p)d~p

p0=±E(~p) Z   = Φ(p)a(p) + Ξ(p)a(−p) d~p.

Thus, every ζ ∈ YKG can be written as Z − 3 X d~p ipx ζ(x) = (2π) 2 e ha(p)|ζi. p2E(~p) p0=±E(~p) Positive/negative frequency solutions (±) YKG were defined in Section 3. Every ζ ∈ YKG can be uniquely decomposed as ζ = (+) (−) (±) (±) ζ + ζ with ζ ∈ YKG . For ζ1, ζ2 ∈ YKG we define the scalar product

(+) (+) (−) (−) (ζ1|ζ2) := iζ1 ωζ2 − iζ1 ωζ2 Z Z = ha(p)|ζ1iha(p)|ζ2id~p + ha(−p)|ζ1iha(−p)|ζ2id~p,

(+) (−) where p = (E(~p), ~p). We set ZKG = ZKG ⊕ ZKG to be the completion of YKG in this scalar (+) (−) product. The ortochronous Poincar´egroup leaves ZKG and ZKG invariant. Quantization of free fields In this section we describe the quantization of the complex Klein-Gordon equation. As usual, we will use the “hat” to denote the quantized objects. In principle, we could quantize it as a pair of real Klein-Gordon fields. However, we will use a different formalism, adapted to the complex (charged) case. We want to construct a Hilbert space H, a positive self-adjoint operator H called the Hamiltonian, a normalized vector Ω, which is a ground state of Hˆ and a distribution

1,3 R 3 x 7→ ψˆ(x), (7.5)

∞ 1,3 which smeared with Cc (R , C) functions has values in closed operators satisfying 1. (−2 + m2)ψˆ(x) = 0, [ψˆ(x), ψˆ∗(y)] = iD(x − y).

[ψˆ(x), ψˆ(y)] = 0.

2. eitHˆ ψˆ(x0, ~x)e−itHˆ = ψˆ(x0 + t, ~x).

3. Ω belongs to the domain of polynomials in smeared out ψˆ(x), ψˆ∗(x) and is cyclic for these operators. An alternative equivalent formulation of the quantization program uses the smeared fields instead of point fields. Instead of (3.11) we look for an antilinear function

ˆ YKG 3 ζ 7→ ψ((ζ)) with values in closed operators such that

1.’

ˆ ˆ∗ [ψ((ζ1)), ψ ((ζ2))] = iζ1ωζ2. (7.6) ˆ ˆ [ψ((ζ1)), ψ((ζ2))] = 0.

ˆ itHˆ ˆ −itHˆ 2.’ ψ((r(t,~0)ζ)) = e ψ((ζ))e . 3.’ Ω belongs to the domain of polynomials in ψ((ζ)), ψ∗((ζ)) and is cyclic for these operators. One can pass between these two versions of the quantization problem by Z   ψˆ((ζ)) = ζ˙(t, ~x)ψˆ(t, ~x) − ζ(t, ~x)ˆη(t, ~x) d~x, (7.7)

˙ where ηˆ(x) := ψˆ(x). (+) The above problem has a solution unique up to a unitary equivalence. We set H = Γs(ZKG ⊕ (−) (+) ∗ ZKG ). Creation/annihilation operators on ZKG will be denoted aˆ /aˆ; creation/annihilation (−) ˆ∗ ˆ operators on ZKG will be denoted b /b. For plane waves we use the notation

∗ ∗ aˆ (p) =a ˆ (−iΦ(p)), p0 = E(~p); ˆ∗ ˆ∗ ˆ∗ b (p) = b (iΦ(−p)) = b (iΞ(p)), p0 = −E(~p).

Ω will be the Fock vacuum. We set Z − 3 d~p  ipx −ipx ∗  ψˆ(x) = (2π) 2 e aˆ(p) + e ˆb (−p) p2E(~p) Z   = Φ(p)a(p) + Ξ(p)b∗(p) d~p.

The Hamiltonian, the momentum and the total charge are Z   Hˆ := aˆ∗(p)ˆa(p) + ˆb∗(p)ˆb(p) E(~p)d~p, ~ Z   Pˆ := aˆ∗(p)ˆa(p) + ˆb∗(p)ˆb(p) ~pd~p, Z   Qˆ := aˆ∗(p)ˆa(p) − ˆb∗(p)ˆb(p) d~p. Note that the whole orthochronous Poincar´egroup acts unitarily on H by U(a, Λ) :=   Γ r(a,Λ) , with ZKG U(a, Λ)ψˆ(x)U(a, Λ)∗ = ψˆ(a, Λ)x
.

Note the identities

(Ω|ψˆ(x)ψˆ∗(y)Ω) = −iD(+)(x − y), (Ω|T(ψˆ(x)ψˆ∗(y))Ω) = −iDc(x − y). Quantized current The quantized current density is given by   Jˆµ(x) = : i∂µψˆ∗(x)ψˆ(x) − iψˆ∗(x)∂µψˆ(x) :.

It satisfies

µ µ ∂µJˆ (x) = 0, (Ω|Jˆ (x)Ω) = 0, [Jˆµ(t, ~x), ψˆ(t, ~y)] = −ψˆ(t, ~y)δ(~x − ~y), [Jˆµ(t, ~x), ψˆ∗(t, ~y)] = ψˆ∗(t, ~y)δ(~x − ~y), [Jˆµ(t, ~x), Jˆν(t, ~y)] = 0

The total charge equals Z Qˆ = Jˆ0(0, ~x)d~x. The current at point x can be defined without the use of the Wick ordering, by the so-called point-splitting method. ∞ 3 R If g ∈ Cc (R ) with g(~x)d~x = 1, then Z   Jˆµ(x) = lim g(~y/) i∂µψˆ∗(x + ~y)ψˆ(x) − iψˆ∗(x + ~y)∂µψˆ(x) d~y − −3. &0 Axioms ∞ 1,3 For f ∈ Cc (R , C) we set Z ψˆ[f] := ψˆ(x)f(x)dx, Z ψˆ∗[f] := ψˆ∗(x)f(x)dx.

We obtain an operator valued distribution satisfying the Wightman axioms with D := fin (+) (−) Γs (ZKG ⊕ ZKG ). For an open set O ⊂ Rd we set 00 n  ˆ∗ ˆ  ∞ o F(O) := exp iψ [f] + iψ[f] : f ∈ Cc (O, C) .

We define A(O) to be the subalgebra of F(O) fixed by the automorphism eiθQˆ · e−iθQˆ. The algebras A(O) satisfy the Haag-Kastler axioms. External potential Let us go back to the classical theory. Let R1,3 3 x 7→ A(x) ∈ R1,3 be a given function. The (complex) Klein-Gordon in an external potential A is

µ µ 2
−(∂µ − ieAµ(x))(∂ − ieA (x)) + m ζ(x) = 0. (7.8)

If ζ satisfies (7.8) and R1,3 3 x 7→ χ(x) ∈ R is an arbitrary function, then eiχζ satisfies (7.8) with A replaced with A + ∇χ. If we introduce the fields R1,3 3 x 7→ ψ(x), ψ(x), then we can rewrite (7.8) as µ µ 2
−(∂µ − ieAµ(x))(∂ − ieA (x)) + m ψ(x) = 0. (7.9) The Lagrangian (7.9) can be obtained as the Euler-Lagrange of a variational problem. The Lagrangian density can be taken as

µ µ 2 L(x) = −(∂µ − ieAµ(x)) ψ(x)(∂ − ieA (x)) ψ(x) − m ψ(x)ψ(x),

The Euler-Lagrange equations ∂L ∂ψL − ∂µ = 0 (7.10) ∂(∂µψ) yield (7.9). The Hamiltonian Let us introduce the variable conjugate to ψ(x): ∂L η(x) := = ∂0ψ(x) − ieA0(x)ψ(x). ∂0ψ(x) We have the equal-time Poisson-brackets (7.3). The Hamiltonian density   H(x) = η(x)η(x) − ieA0(x) ψ(x)η(x) − η(x)ψ(x) 2 +(∂i − ieAi(x))ψ(x)(∂i − ieAi(x))ψ(x) + m ψ(x)ψ(x). The Hamiltonian Z H(t) = d~xH(t, ~x) can be used to generate the dynamics

ψ˙(t, ~x) = {H(t), ψ(t, ~x)}, η˙(t, ~x) = {H(t), η(t, ~x)}. We can also introduce the stress-energy tensor density

Tµν(x) := ∂µψ(x)∂νψ(x) − gµνL(x), which is conserved

∂µTµν(x) = 0. We have

H(x) = T00(x), Pi(x) = T0i(x). Quantization in the presence of external potentials We are looking for quantum fields satisfying

µ µ 2
ˆ −(∂µ − ieAµ(x))(∂ − ieA (x)) + m ψ(x) = 0. (7.11) coinciding with the free fields for t = 0. Such fields are given by  Z 0   Z t  ψˆ(t, ~x) := Texp −i Hˆ (s)ds ψˆ(0, ~x)Texp −i Hˆ (s)ds , t 0 where the Hamiltonian Hˆ (t), and the corresponding Hamiltonian in the interaction picture equal Z  ∗ ˆ∗ ∗ ˆ
Hˆ (t) = d~x: ηˆ (~x)ˆη(~x) − ieA0(t, ~x) ψ (~x)ˆη(~x) − ηˆ (~x)ψ(~x) ˆ∗ ˆ +(∂i − ieAi(t, ~x))ψ (~x)(∂i − ieAi(t, ~x))ψ(~x)  +m2ψˆ∗(~x)ψˆ(~x) :, (7.12) Z ˆ  ˆµ 2 ~ 2 ˆ∗ ˆ  HInt(t) = d~x: ieAµ(t, ~x)JInt(t, ~x) + e A(t, ~x) ψInt(t, ~x)ψInt(t, ~x) : Z  ˆµ 2 2 ˆ∗ ˆ = d~x: ieAµ(t, ~x)JInt(t, ~x) + e A(t, ~x) ψInt(t, ~x)ψInt(t, ~x) 2 2 ˆ∗ ˆ  +e A0(t, ~x) ψInt(t, ~x)ψInt(t, ~x) :. (7.13)

Note that the above Hamiltonians should be understood as formal expressions. They need not correspond to (time-dependent) self-adjoint operators, and if they do, it is not guaranteed that they generate a unitary dynamics. Some of the terms in the following perturbation expansions may turn out to be infinite – however, they lead to well defined scattering cross-sections. 2N-point Green’s functions.

For yN , . . . y1, xN , . . . , x1, the 2N point Green’s function are defined as follows:

 ˆ∗ ˆ∗ ˆ ˆ  G ψ , y1, ··· ψ , yN , ψ, xN , ··· ψ, x1  +  ˆ∗ ˆ∗ ˆ ˆ  − := Ω |T ψ (y1) ··· ψ (yN )ψ(xN ) ··· ψ(x1) Ω . Feynman rules for Green’s functions. We have 1 kind of lines and 2 kind of vertices. Each line has an arrow. At each vertex two lines meet, one with an arrow pointing towards, one with an arrow pointing away from hthe vertex. The 1-photon vertex is denoted by an attached wavy line, the 2-photon vertex, by attached two wavy lines. The names of vertices are motivated by the full QED.

1. In the nth order we draw all possible topologically distinct Feynman diagrams with n vertices and 2N external lines.

2. To each line we associate a propagator −i Dc(p) = . p2 + m2 − i0

d4p For internal lines we integrate over the variables with the measure (2π)4 . 3. (i) To each 1-photon vertex we associate a factor

+ ˆν + − ˆν + − − e(pν A (p − p ) + A (p − p )pν ).

(ii) To each 2-photon vertex we associate a factor

2 ν + − −2ie (A Aν)(p − p ). The derivation of the Feynman rules for charged scalar bosons is more complicated than for neutral bosons, since it involves not just expectation values of “configuration space fields”, but also the “momentum space fields”:

ˆ ˆ∗ c (Ω|T(ψI(x)ψI (y))Ω) = −iD (x − y), ˆ∗ c (Ω|T(ˆηI(x)ψI (y))Ω) = −i∂x0 D (x − y), ˆ ∗ c (Ω|T(ψI(x)ˆηI (y))Ω) = −i∂y0 D (x − y), ∗ c (Ω|T(ˆηI(x)ˆηI (y))Ω) = −i∂x0 ∂y0 D (x − y) − iδ(x − y). Scattering operator

 Z ∞  S = Texp −i HˆInt(t)dt . −∞ Scattering amplitudes: matrix elements of the scattering operator between plane waves

 + + +0 +0 −0 −0 − −  Φ(p1 ) ··· Φ(pN + )Ξ(p1 ) ··· Ξ(pN +0 )| S Ξ(pN −0 ) ··· Ξ(p1 )Φ(pN − ) ··· Φ(p1 ) .

For simplicity, we assume that the momenta are distinct. Feynman rules for scattering amplitudes. To compute scattering amplitudes with N − incoming and N + outgoing particles we draw the same diagrams as for N − + N +-point Green’s functions, where we adopt the convention that the incoming lines are drawn on the right and outgoing lines on the left. The rules are changed only concerning the external lines.

1. With each incoming external line we associate (i) charged boson: √ 1 . V 2E(~p) (ii) charged anti-boson: √ 1 . V 2E(~p0) 2. With each outgoing external line we associate (i) charged boson: √ 1 . V 2E(~p) (ii) charged anti-boson: √ 1 . V 2E(~p0) Abstract gauge covariance Let us adopt for a moment an abstract setting. Let R 3 t 7→ H(t) be a time-dependent Hamiltonian generating the dynamics

 Z t+  U(t+, t−) := Texp −i H(s)ds . t− Let t 7→ R(t) be a time dependent family of self-adjoint operators. Introduce the time- dependent gauge transformation  Z t  W (t) := Texp −i R(s)ds . −∞ Assume that W (t) converges to identity as t → −∞. Then  Z t  ∗ W (t+)U(t+, t−)W (t−) = Texp −i HR(s)ds . −∞ where the gauge-transformed Hamiltonian equals

∗ HR(t) := W (t)H(t)W (t) + R(t). (7.14) Gauge covariance Let us go back to the setting of quantized charged scalar fields. Let R1,3 3 x 7→ χ(x) ∈ R be a smooth function that decays in all directions. We define the gauge transformation  Z  W (χ, t) := exp −i d~xχ(t, ~x)Jˆ0(~x) .

 Z t Z  = exp −i ds d~x χ˙(s, ~x)Jˆ0(~x) (7.15) −∞  Z t Z  = Texp −i ds d~x χ˙(s, ~x)Jˆ0(~x) . −∞ To see the second above identity it is enough to note that [Jˆ0(~x), Jˆ0(~y)] = 0, hence we can replace Texp with exp in (7.15). In the interaction picture we have

itHˆ0 −itHˆ0 WInt(χ, t) := e W (χ, t)e  Z  ˆ0 = exp −i d~xχ(t, ~x)JI (t, ~x) . (7.16) Let Hˆ (A, t), resp. HˆInt(A, t) denote (7.12), resp. (7.13). Let U(A, t+, t−), UInt(A, t+, t−) and S(A) be the corresponding dynamics, the dynamics in the interaction picture and the scattering matrix. We have the following identities, which express the gauge covariance:

∗ W (χ, t+)U(A, t+, t−)W (χ, t−) = U(A + ∇χ, t+, t−), (7.17) ∗ WInt(χ, t+)UInt(A, t+, t−)WInt(χ, t−) = UInt(A + ∇χ, t+, t−). (7.18)

To see (7.17), we use (7.14) and Z W (χ, t)Hˆ (A, t)W ∗(χ, t) + d~xχ˙(t, ~x)Jˆ0(~x)

= Hˆ (A + ∇χ, t).

The second identity follows from the first. Ward(-Takahashi) identities for the scattering operator

Using that WInt(χ, ±∞) = 1l, we see that (7.18) implies

S(A) = S(A + ∇χ).

Differentiating this identity wrt χ and setting χ = 0 we obtain one of versions of the identities for the scattering operator ∂ ∂yµ S(A) = 0. ∂Aµ(y) In the momentum representation these identities read ∂ pµ S(A) = 0. ∂Aµ(p) Ward(-Takahashi) identities for Green’s functions   ˆ∗ 0 ˆ We will write G A; ψ , x1, ··· , ψ, x1 to express the dependence of Green’s functions on the external potential A. The following property follows from (7.17):

 ˆ∗ 0 ˆ  G A; ψ , x1, ··· , ψ, x1

 0  iχ(x1) ˆ∗ 0 −iχ(x1) ˆ = G A + ∇χ; e ψ , x1, ··· , e ψ, x1 . By differentiating with respect to χ(y) and setting χ = 0 we obtain the Ward identities for Green’s functions in the position representation:

∂  ˆ∗ 0 ˆ  ∂yµ G A; ψ , x1, ··· , ψ, x1 ∂Aµ(y) N N ! X 0 X  ˆ∗ 0 ˆ  = i δ(y − xj) − i δ(y − xj) G A; ψ , x1, ··· , ψ, x1 . j=1 j=1 In the momentum representation these identities read

∂  ˆ∗ 0 ˆ  kµ G A; ψ , p1,, ··· , ψ, p1 ∂Aµ(k)  ˆ∗ 0 ˆ   ˆ∗ 0 ˆ  = iG A; ψ , p1 − k, ··· , ψ, p1 + · · · − iG A; ψ , p1, ··· , ψ, p1 + k . Chapter 8

Dirac fermions 267

Let γµ be matrices satisfying

µ ν µ,ν [γ , γ ]+ = −2g , µ −1 µ∗ γ0γ γ0 = γ . In this chapter we study the Dirac equation

µ (−iγ ∂µ + m)ψ(x) = 0 and its quantization. For further reference, let us note the Dirac equation in the momentum representation: µ (γ pµ + m)ψ(p) = 0. We will also consider the Dirac equation in the presence of an external potential Aµ(x):

µ (−iγ (∂µ − ieAµ(x)) + m) ψ(x) = 0. Representations of Dirac matrices To find a representation of Dirac matrices, that is to find γµ, µ = 0,..., 3, such that

(γ0)2 = 1l, (γi)2 = −1l, i = 1, 2, 3; γµγν + γνγµ = 0, 0 ≤ µ < ν ≤ 3; (γ0)∗ = γ0, (γi)∗ = −γi, i = 1, 2, 3. we need the space C4. One of the most common representations is the so-called Dirac representation " # " # 1 0 0 ~σ γ0 = , ~γ = . 0 −1 −~σ 0

Here is the Majorana representation: " # " # " # " # 0 −1 0 σ −1 0 0 σ γ0 = i , γ1 = i 1 , γ2 = i , γ3 = i 3 , 1 0 σ1 0 0 1 σ3 0 and the spinor representation: " # " # 0 1 0 −~σ γ0 = , ~γ = . 1 0 ~σ 0 We also introduce the spin operator 1 σµν := [γµ, γν]. 2 In the Dirac representation " # 0 σi σ0i = , σi 0 " # 1 ij σk 0 σ ijk = . 2 0 σk Special solutions and Green’s functions Note the identity

−(−iγ∂ + m)(iγ∂ − m) = −2 + m2. Therefore, if (−2 + m2)ζ(x) = 0,

µ then (iγ ∂µ + m)ζ(x) is a solution of the homogeneous Dirac equation:

µ µ (−iγ ∂µ + m)(iγ ∂µ + m)ζ(x) = 0.

In particular, we have special solutions of the homogeneous Dirac equation

S(±)(x) = −(iγ∂ + m)D(±)(x), S(x) = −(iγ∂ + m)D(x).

We have suppS ⊂ J. If (−2 + m2)ζ(x) = δ(x),

µ then −(iγ ∂µ − m)ζ(x) is a Green’s function of the Dirac equation, that is

µ (−iγ∂ + m)(iγ ∂µ + m)ζ(x) = δ(x).

In particular, a special role is played by the Green functions

S±(x) = −(iγ∂ + m)D±(x), Sc(x) = −(iγ∂ + m)Dc(x).

We have suppS± ⊂ J ±. Note the identities

S(x) = −S(−x) = S(+)(x) + S(−)(x) = S+(x) − S−(x), S(+)(x) = S(−)(−x), S+(x) = S−(−x) = θ(x0)S(x) S−(x) = θ(−x0)S(x) Sc(x) = Sc(−x) = θ(x0)S(−)(x) − θ(−x0)S(+)(x). Solution of the inhomogeneous equation

∞ 1,3 4 ± ∞ 1,d 4 Theorem 8.1 For any f ∈ Cc (R , C ) there exist unique functions ζ ∈ C±sc(R , C ) solutions of (−iγ∂ + m)ζ± = f. (8.1)

They are given by Z ζ±(x) = (S±f)(x) = S±(x, y)f(y)dy. R1,3 The Cauchy problem We set αi = iγ0γi, i = 1,..., 3, and β := γ0, obtaining

2 2 β = 1l, (αi) = 1l, i = 1,..., 3;

βαi + αiβ = 0, αiαj + αjαi = 0, 1 ≤ i < j ≤ 3; ∗ ∗ β = β, αi = αi, i = 1,..., 3.

In the Dirac representation we have " # " # 1 0 0 ~σ β = , ~α = . 0 −1 ~σ 0

We can rewrite the Dirac equation in the hamiltonian form

i i∂tζ(t, ~x) = Dζ, D := α pi + mβ. ∞ 3 4 ∞ 1,3 Theorem 8.2 Let ϑ ∈ Cc (R , C ). Then there exists a unique ζ ∈ Csc (R ) that solves the Dirac equation with initial conditions ζ(0, ~x) = ϑ(~x). It satisfies suppζ ⊂ J(suppϑ) and is given by Z ζ(t, ~x) = S(t, ~x − ~y)βϑ(~y)d~y. R3 Conserved current ∞ 1,3 4 Let YD be the space of space-compact solutions of the Dirac equation, that is ζ ∈ Csc (R , C ) µ ∞ 1,3 4 satisfying (−iγ ∂µ + m)ζ = 0. Let ζ1, ζ2 ∈ C (R , C ). Set

µ µ j (ζ1, ζ2, x) := ζ1(x)βγ ζ2(x).

We easily check that

µ ∂µj (x) = (−iγ∂ + m)ζ1(x)βζ2(x) − ζ1(x)β(−iγ∂ + m)ζ2(x),

Therefore, if ζ1, ζ2 ∈ YD, then µ ∂µj (x) = 0, Scalar product µ For ζ1, ζ2 ∈ YD, the flux of j across a space-like hypersurface does not depend on its choice.

It defines a scalar product on YD Z µ ζ1 · ζ2 = j (ζ1, ζ2, x)dsµ(x). S In terms of the Cauchy data we have Z ζ1 · ζ2 = ζ1(t, ~x)ζ2(t, ~x)d~x. Solutions parametrized by test functions ∞ 1,3 4 For any f ∈ Cc (R , C ), we will write Z Sf(x) := S(x − y)f(y)dx.

∞ 1,3 4 Theorem 8.3 1. For any f ∈ Cc (R , C ), Sf ∈ YD.

2. Every element of YD is of this form. RR 3. Sf1 · Sf2 = f1(x)βS(x, y)f2(y)dxdy.

4. If suppf2 × suppf2, then

Sf1 · Sf2 = 0. Classical Dirac field # 1,3 We will also consider the space dual to YD, denoted YD. In particular, for x ∈ R , 4 ψ(x), ψ(x) will denote the functionals on YD with values in C given by

hψ(x)|ζi := ζ(x), hψ(x)|ζi := ζ(x).

Clearly, Z ψ(t, ~x) = S(t, ~x − ~y)βψ(0, ~y)d~y.

It is convenient to introduce the symbol

ψ˜(x) := βψ(x).

(In a large part of the physics literature, ψ˜ is denoted ψ.) Spatially smeared fields

We can use the scalar product to pair distributions and solutions. For ζ ∈ YD we introduce also the functional on YD given by

hψ((ζ))|ρi := ζ · ρ,

hψ((ζ))|ρi := ζ · ρ, ρ ∈ YD.

Clearly, Z ψ((ζ)) = ψ(t, ~x)ζ(t, ~x)d~x, Z ψ((ζ)) = ψ(t, ~x)ζ(t, ~x)d~x. Space-time smeared fields We can also introduce Z ψ[f] := f(x)ψ(x)dx = ψ((Sf)), Z ψ[f] := f(x)ψ(x)dx = ψ((Sf)). Current and charge We can introduce the current

J µ(ζ, x) := jµ(ζ, ζ, x) = ζ(x)βγµζ(x).

We can write it as

J µ(x) = ψ(x)βγµψ(x).

It satisfies µ ∂µJ (x) = 0. The total charge is Z Q = J 0(0, ~x)d~x. Plane waves We would like to diagonalize simultaneously the following commuting operators on the space ~ YD: Dirac Hamiltonian D, the momentum −i∇ and the polarization −iijkσij∇k. To this end we introduce for p = (E(~p), ~p) with E = E(~p) = p~p2 + m2, " # " # 1 0 χ+ := , χ− := , 0 1

√ " # χs 0 u(p, s) = E + m ~σ~p , p = E > 0; E+mχs " # √ −~σ~p χ u(p, s) = E + m E+m s , p0 = −E < 0. χs We also introduce the corresponding plane waves u(p, s) Φ(p, s) = eipx. (2π)3/2p2E(~p) We have

0 0 0 Φ(p) · Φ(p ) = δ(~p − ~p ), sgnp0p0 > 0, 0 0 iΦ(p) · Φ(p ) = 0, sgnp0p0 < 0. For negative frequencies, that is p0 = −E(~p), it is customary to have an alternative notation " # √ ~σ~p χ v(p, s) = u(−p, s) = E + m E+m s , χs v(p, s) Ξ(p, s) = Φ(−p, s) = e−ipx. (2π)3/2p2E(~p) We introduce

a(p, s) := ψ((Φ(p, s))) Z − 3 i~p~x = (2π) 2 d~pu(p, s)e ψ(0, ~x)

We have Z X X − 3 ipx ψ(x) = (2π) 2 d~pu(p, s)e a(p, s)

p0=±E(~p) s Z X − 3  ipx −ipx  = (2π) 2 d~p u(p, s)e a(p, s) + v(p, s)e a(−p, s) s We have

ha(p, s)|Dζi = p0ha(p, s)|ζi, ha(p, s)|(−i∇~ )ζi = ~pha(p, s)|ζi,

ha(p, s)|(−iijkσij∇k)ζi = |~p|sha(p, s)|ζi.

X X Z ζ1 · ζ2 = ha(p, s)|ζ1iha(p, s)|ζ2id~p.

p0=±E(~p) s

Clearly, every ζ ∈ YD can be written as Z X X − 3 ipx ζ(x) = (2π) 2 d~pu(p, s)e ha(p, s)|ζi.

p0=±E(~p) s (±) YD will denote the subspace of YD consisting of positive, resp. negative frequency solutions. (+) (−) (±) (±) Every ζ ∈ YD can be uniquely decomposed as ζ = ζ + ζ with ζ ∈ YD . (−) We change the complex structure on YD . For ζ1, ζ2 ∈ YD we define the scalar product

(+) (+) (−) (−) (ζ1|ζ2) := ζ1 · ζ2 + ζ1 · ζ2 Z Z = ha(p)|ζ1iha(p)|ζ2id~p + ha(−p)|ζ1iha(−p)|ζ2id~p,

(+) (−) where p = (E(~p), ~p). We set ZD = ZD ⊕ ZD to be the completion of YD in this scalar (+) (−) product. The ortochronous Poincar´egroup leaves ZD and ZD invariant. Quantization of free fields We would like to describe the quantization of the Dirac equation. As usual, we will use the “hat” to denote the quantized objects. We need to use the formalism of quantization of charged fermionic systems. We want to construct a Hilbert space H, a positive self-adjoint operator Hˆ called the Hamiltonian, a normalized vector Ω, which is a ground state of Hˆ and a distribution

1,3 R 3 x 7→ ψˆ(x), (8.2)

∞ 1,3 4 which smeared with Cc (R , C) functions has values in bounded operators tensored with C satisfying

ˆ˜ ˆ∗ 1. Setting ψ(x) := γ0ψ (x), we have

(−iγ∂ + m)ψˆ(x) = 0,

ˆ ˆ˜ ˆ ˆ [ψ(x), ψ(y)]+ = S(x − y), [ψ(x), ψ(y)]+ = 0

2. eitHˆ ψˆ(x0, ~x)e−itHˆ = ψˆ(x0 + t, ~x).

3. Ω is cyclic for smeared ψˆ(x), ψˆ∗(x). There exists an alternative equivalent formulation of the quantization program, which uses the smeared fields instead of point fields. Instead of (3.11) we look for an antilinear function

ˆ CYD 3 ζ 7→ ψ((ζ)) with values in bounded operators such that

1.’

ˆ ˆ∗ [ψ((ζ1)), ψ ((ζ2))]+ = ζ1 · ζ2. (8.3) ˆ ˆ [ψ((ζ1)), ψ((ζ2))]+ = 0.

2.’ ˆ itHˆ ˆ −itHˆ ψ((r(t,~0)ζ)) = e ψ((ζ))e .

3.’ Ω is cyclic for ψˆ((ζ)), ψˆ∗((ζ)). One can pass between these two versions of the quantization problem by Z ψˆ((ζ)) = ζ(t, ~x)ψˆ(t, ~x)d~x. (8.4) (+) The above problem has a solution unique up to a unitary equivalence. We set H = Γs(ZD ⊕ (−) (+) ∗ (−) ZD ). For ZD the standard creation/annihilation operators are denoted aˆ , aˆ. For ZD the standard creation/annihilation operators are denoted ˆb∗, ˆb.

∗ ∗ aˆ (p) =a ˆ (Φ(p)), p0 = E(~p); ˆ∗ ˆ∗ ˆ∗ b (p) = b (Φ(−p)) = b (Ξ(p)), p0 = −E(~p).

Ω will be the Fock vacuum. Z − 3 X ipx ψˆ(x) = (2π) 2 d~pu(p, s)e aˆ(p, s) s p0=E(~p) Z − 3 X ipx ∗ +(2π) 2 v(p, −s)e ˆb (p, s). s p0=−E(~p) The Hamiltonian and the momentum are Z X   Hˆ = aˆ∗(p, s)ˆa(p, s) + ˆb∗(p, s)ˆb(p, s) E(~p)d~p, s ~ Z X   Pˆ = aˆ∗(p, s)ˆa(p, s) + ˆb∗(p, s)ˆb(p, s) ~pd~p, s Z X   Sˆ = |~p|s aˆ∗(p, s)ˆa(p, s) + ˆb∗(p, s)ˆb(p, s) ~pd~p. s The whole orthochronous Poincar´egroup acts unitarily on H. We have the quantized current density

Jˆµ(x) = :ψˆ∗(x)βγµψˆ(x):.

It satisfies µ µ ∂µJˆ (x) = 0, (Ω|Jˆ (x)Ω) = 0. Besides, we obtain the charge operator Z X Z   Qˆ = Jˆ0(0, ~x)d~x = aˆ∗(~p,s)ˆa(~p,s) − ˆb∗(~p,s)ˆb(~p,s) d~p. s We have ˜ (Ω|ψˆ(x)ψˆ(y)Ω) = S(+)(x − y), ˜ (Ω|T(ψˆ(x)ψˆ(y))Ω) = Sc(x − y). Axioms ∞ 4 For f ∈ Cc (O, C ) we set Z ψˆ[f] := ψˆ(x)f(x)dx, Z ψˆ∗[f] := ψˆ∗(x)f(x)dx.

fin (+) We obtain an operator valued distribution satisfying the Wightman axioms with D := Γa (ZD ⊕ (−) ZD ). For an open set O ⊂ R1,3 we set

ˆ∗ ˆ ∞ 4 00 F(O) := {ψ [f], ψ[f]: f ∈ Cc (O, C )} .

We define A(O) to be the subalgebra of F(O) fixed by the automorphism eiθQˆ · e−iθQˆ. The algebras A(O) satisfy the Haag-Kastler axioms. External potential Let R1,3 3 x 7→ A(x) ∈ R1,3 be a given function. The Dirac equation in an external potential A is µ (−iγ (∂µ − ieAµ(x)) + m) ζ(x) = 0 (8.5)

If ζ satisfies (8.5) and R1,3 3 x 7→ χ(x) ∈ R is an arbitrary function, then eiχζ satisfies (7.8) with A replaced with A + ∇χ. If we introduce the fields R1,3 3 x 7→ ψ(x) ∈ C4, then we can rewrite (8.5) as µ (−iγ (∂µ − ieAµ(x)) + m) ψ(x) = 0 (8.6) (7.9) can be obtained as the Euler-Lagrange of a variational problem. The Lagrangian density can be taken as i   L(x) = ψ˜(x)γµ(∂ − ieA (x))ψ(x) − (∂ + ieA (x))ψ˜(x)γµψ(x) 2 µ µ µ µ −mψ˜(x)γµψ(x).

The Euler-Lagrange equations ∂L ∂ψL − ∂µ = 0 (8.7) ∂(∂µψ) yield (8.6). Quantization in the presence of external potentials We are looking for quantum fields satisfying

µ ˆ (−iγ (∂µ − ieAµ(x)) + m) ψ(x) = 0. (8.8) coinciding with the free fields for t = 0. Such fields are given by  Z 0   Z t  ψˆ(t, ~x) := Texp −i Hˆ (s)ds ψˆ(0, ~x)Texp −i Hˆ (s)ds , t 0 where the Hamiltonian Hˆ (t), and the corresponding Hamiltonian in the interaction picture equal Z ˆ∗ ˆ
Hˆ (t) = d~x: ψ (~x)((αi(i∂i + eAi(t, ~x)) + mβ + A0(t, ~x))ψ(~x) :, Z ˆ ˆµ HInt(t) = d~xeAµ(t, ~x)JInt(t, ~x). 2N-point Green’s functions.

For yN , . . . y1, xN , . . . , x1, the 2N point Green’s function are defined as follows:

 ˆ∗ ˆ∗ ˆ ˆ  G ψ , y1, ··· ψ , yN , ψ, xN , ··· ψ, x1  +  ˆ∗ ˆ∗ ˆ ˆ  − := Ω |T ψ (y1) ··· ψ (yN )ψ(xN ) ··· ψ(x1) Ω . Feynman rules for Green’s functions.

1. In the nth order we draw all possible topologically distinct Feynman diagrams with n vertices and external lines. All the charged lines have a natural arrow.

−ipγ+m 2. To each line we associate a propagator S(p) = p2+m2−i0 For internal lines we integrate d4p over the variables with the measure (2π)4 .

µ + − 3. To each vertex we associate a factor −ieγµA (p − p ).

4. If two graphs differ only by an exchange of two fermionic lines, there is an additional factor (−1) for one of them. This implies, in particular, that loops have an additional factor (−1). Scattering operator

 Z ∞  S = Texp −i HˆInt(t)dt . −∞ Scattering amplitudes: matrix elements of the scattering operator between plane waves

 + + +0 +0 −0 −0 − −  Φ(p1 , s1 ) ··· Ξ(p1 , s1 ) · · · | S ··· Ξ(q1− , s1 ) ··· Φ(p1 , s1 ) .

For simplicity, we assume that the momenta are distinct. Feynman rules for scattering amplitudes. To compute unrenormalized scattering amplitudes with N − incoming and N + outgoing particles we draw the same diagrams as for N − + N +-point Green’s functions, where we adopt the convention that the incoming lines are drawn on the right and outgoing lines on the left. The arrow on photon lines can be now chosen in the natural way. The rules are changed only concerning the external lines. 1. With each incoming external line we associate

q m (i) fermion: E(~p)V u(p, s). q m (ii) anti-fermion: E(~p)V v˜(p, s). 2. With each outgoing external line we associate

q m (i) fermion: E(~p)V u˜(p, s). q m (ii) anti-fermion: E(~p)V v(p, s). Each incoming and outgoing antifermion has an additional factor (−1). (This follows from the rule (4) above).