Chapter 3 Quantum Field Theory

Relativistic Quantum Field in Theoretical Physics by

fEJSA HU, ETTI !NSTITUTE Trung Van Phan OF TECHNOLOLGY Submitted to the Department of Physics AUG 102015 in partial fulfillment of the requirements for the degree of LIBRAR IES Bachelor of Science in Physics

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

June 2015

@ Trung Van Phan, MMXV. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created.

Signature redacted A uthor ...... Department of Physics May 8, 2015

C ertified by ...... Signature redacted Jeise D. Thaler Assistant Professor Thesis Supervisor Signature redacted Accepted by ...... Nergis Mavalvala Associate Department Head for Education 2 Relativistic Quantum Field in Theoretical Physics

Trung Van Phan

Submitted to the Department of Physics on May 8, 2015, in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics

Abstract

Quantum field theory is the most well-developed tool in theoretical physics to study about the dynamics at microscopic scales, with particles and quantum behaviors. In this thesis I'll review about the construction of quantum field theory from the S-matrix point of view (building up particles and formulating interactions), then poke at interesting topics that are usually briefly mentioned or even ignored in standard quantum theory of field textbooks. In more details, chapter 1 will be about how a quanta is defined from the analysis of group theory, chapter 2 will be focus on how the language of field to describe the quantum behaviors of quanta is desirable for interactions and arised quite naturally (it should be noted that, string theory can be viewed as a totally different theory in quantum interactions, with conformal symmetric and topological natures, but at low energy scale it can always be reduced back to quantum field theories), and chapter 3 will be about several angles of quantum field theory.

Thesis Supervisor: Jesse D. Thaler Title: Assistant Professor

3 Acknowledgments

I'd like to express my sincere gratitude to Massachusetts Institute of Technology (MIT) for letting me pursuing my dream about studying theoretical physics here. I would also like to thank Prof.

Jesse Thaler and the Physics Department for guiding me with this thesis from start to finish. I'm indebted to Prof. Jesse Thaler, my thesis supervisor, for his understanding, patience, enthusiasm and encouragement during all these years in MIT. To Prof. Leonid Levitov, Prof. Edmund

Bertschinger, Prof. lain Stewart, Prof. Hong Liu and Prof. Washington Taylor, I'm extremely grateful for the past discussions and perspectives from you in different topics in theoretical physics, which some of them are mentioned in this thesis. To all my friends for supporting me during hardship, especially Thai Pham, Duy Ha, Dan Doan, Nhat Cao, Tru Dang and Truong Cai. To my family for always having faith and not giving up on me.

4 Contents

1 Quanta 7 1.1 Spacetime Symmetry ...... 8

1.1.1 Under Spacetime Transformation ...... 8

1.1.2 Poincare Group and Poincare Algebra ...... 10

1.2 Fixed Characteristics ...... 11

1.2.1 Casimir operators ...... 11

1.2.2 Quanta definition ...... 12

1.3 Labeling a Quanta ...... 14

1.3.1 4-momentum Label ...... 14

1.3.2 Details Dynamical Information of Quanta ...... 16

1.3.3 More on Spin and Helicity ...... 18

1.3.4 Quanta in Flat Spacetime and Curved Spacetime ...... 21

2 Field 22

2.1 Construction of Multi-particles Dynamics ...... 22

2.1.1 Scattering Setting ...... 22

2.1.2 Creation and Annihilation Operator ...... 24

2.1.3 Causality Condition ...... 26

2.1.4 The General Form of S-Matrix ...... 28

2.1.5 Internal Symmetry and Conserved Charge ...... 28

2.1.6 Cluster Decomposition ...... 30

2.2 Macroscopic Description ...... 31

2.2.1 Creation and Annihilation Fields ...... 31

2.2.2 Finite-Dimensional Representations of the Lorentz Group. 32

5 2.2.3 Physical Requirements for the Theory ...... 36 2.2.4 Spacetime Rotation and Particle Self-Rotation ...... 41

3 Quantum Field Theory 44 3.1 The Free Theory Lagrangian ...... 44 3.1.1 Massive Quantum Field ...... 45 3.1.2 Massless Quantum Field ...... 47 3.2 Collective Behaviors of Quantum ...... 49 3.2.1 Gauge Freedom ...... 49 3.2.2 Renormalization ...... 51 3.2.3 Long Distance Physics ...... 57 3.2.4 Maxwell's Electromagnetism and Einstein's General Relativity ...... 62 3.2.5 Classical Field Configuration ...... 64 3.3 Functional Integration ...... 6 5 3.3.1 Connection to Canonical Quantization ...... 66 3.3.2 Is Functional Integration always Superior? ...... 70 3.3.3 Mathematical Rigors ...... 74 3.3.4 Effective Field Theory ...... 75 3.3.5 Quantum Anomaly ...... 9 3 3.3.6 W avefunctional ...... 100

6 Chapter 1

Quanta

Unchanged properties under the change of observers and reference frames are of great interests in understanding nature, since they are very useful to formulate an universal theory and also easy to keep track of, experimentally. Since it is expected that the law of physics is the same in every inertial frame, which is a postulation in special relativity (with a depth philosophical reason), hence one can partially define the building blocks of the universe to be objects associated with the set of most the fundamental and fixed characteristics under the transformations of spacetime that relate different inertial frames together (to get the full definition, one has to take into account the interactions). The transformations are described by Lorentz group, which is the result from the other postulation of special relativity (unsurprisingly, based on a constant - the speed of light), and the spacetime translation group, which is seen in flat spacetime. Combining these two gives the Poincare group. From the experiments, it is known that nature exhibits quantum

(discretization) behavior, therefore the fundamental building blocks should be one-particle states and multi-particles states. A quanta is the physical realization of an one-particle state. From the quanta point of view, the physical transformation that unchanges the presenting physics is a classical interpretation for symmetry.

7 1.1 Spacetime Symmetry

1.1.1 Under Spacetime Transformation

Transformation of Spacetime Coordinates

The spacetime symmetry is a global symmetry described by the Poincare transformation, with Lorentz rotation Al and spacetime shift a":

T(A, a):x:-+" a' = AI",xv + a , T(A', a')T(A, a) = T(A'A, A'a + a') (1.1)

Since Lorentz transformation preserves the 4-vector length, the Lorentz parameter satisfies:

-1 0 0 0 0 +1 0 0) (1.2) 0 0 +1 0 0 0 0 +1

To,,AvA , = , --+ A = uir1 ", = AAnan , (Ado)t , = A" = lsoP te hAPt: (1.3)

To have the connection with unitary, the determinant det A should be 1. Also note that:

n/A "oAvo= qoo -+ -(A o)2 + (A'o) = -I, (A00) 2 > (1.4)

0 There are disjoint regions of the real axis for possible A 0 , and for linking requirement with identity

0 transformation, one needs to have A 0 > 1. From the Lorentz proper orthochronous subgroup, 0 with det A = 1 and A 0 > 1, to go to the other part of the Lorentz group, spatial inversion P and time reversal T are employed:

3 po 1 , Pp= 'P2P> _ 2 7 1 (1.5)

From now on, the proper orthochronous Lorentz group is refered as simply Lorentz group, unless mention explicitly. Also, Poincare group is with the the proper orthochronous Lorentz group, not

8 the full original Lorentz group. The infinitesimal transformation of the Lorentz group:

At', -+ P' + w", , wA = p, w", = r""w ,p, ; ap -+ 16 (1.6)

The Lorentz transformation of the metric indicates that w.,, is anti-symmetric:

0 77t = 77PAP AU = TipcG(P + w"i)(5" + w ,) (1.7)

= + wI, + WVA + ... -- w,, (1.8)

Transformation in the Hilbert space

In quantum terminology, any collection of properties can be represented by a ray of normalized vectors in the physical Hilbert space. The transformation in spacetime coordinates is realized by a unitary linear transformation, with similar composition rule [36]:

1IT) -+ U(A, a) IT) , U(A', a')U(A, a) = U(A'A,A'a+ a') (1.9)

Note that, there's a caveat here. The associatity of the symmetry group representation on the physical states is not always exactly equal, because the transformation T (a general symmetry transformation, not just restrict to the spacetime Poincare transformation) take a ray to another ray but don't really put a restriction at the possible phase arise:

U(T')U(T) = eOT1,T) U(T'T) (1.10)

Because of linearity (also, anti-linearity), the phase should be independence of the state which the transformation acts on. The name for a representation with a general # is the projective representation [36]. The symmetry group cannot tell by itself whether the physical states furnish an ordinary or a projective representation, but its topology can tell us whether the group has any intrinsically projective representations (supporting non-zero phase # in what way). But since any symmetry group with projective representations can always be enlarged in a way without changing its physical impications (for example, go from single-covered to double-covered or even multi- covered) so that its representations can all be defined as non-projective, hence set 0 = 0. From now on, unless specify, the choice of U is always that of the representations after the enlargement

9 was done: U(T')U(T) = U(T'T) (1.11)

The spatial inversion and time reversal are the extensions of Lorentz rotation, described by:

= U(P, 0) , 7 = U(T, 0) (1.12)

If a state is the eigenstate of the Poincare transformation, then the same still holds after spatial inversion and time reversal (in more details, there's only a phase diffent between these).

1.1.2 Poincare Group and Poincare Algebra

If there is no transformation, it is expected that U(1, 0) is the same as the identity. Under an infinitesimal Poincare transformation, the corresponded transformation of quantum state can be written with independence operators J" and P", which should be Hermittian so U is unitary:

U(1 + W, = 1+I) -iJ" - iP + ... , J"= -J" (1.13) 2

The J" for Lorentz rotation and P" for translation shift are called the generators of the Poincare group. One can associated physical meaning to these operators. From quantum mechanics, since the energy operator, also known as the Halmintonian H, is related to the translation generator in the time direction, therefore H = P'. Similar arguments can be applied to the 3-momentum

P = {P1 , P2 , P3 } and the angular momentum f= {J23 , 3 1 , 712}. The remaining opertors don't have a classical analogy since they come from special relativity, thus we will borrow the name from there, the boost 3-vector K {J0 1 , J 2 , J0 3 } [36]. The algebra of these generators can be read-off by considering a following transformation:

U(A, a)U(1 + w, e)U'(A, a) U(A, a)(1 + -iwitV + AP'J)U (1.14)

U (A(1+ w)A- 1 , Ac - AwA-1a) = 1+ 2i(AwA-1),,Jv - (Ae - AwA-'a),P1' (1.15)

Hence, the operators JI" and PP are transformed as:

U(A,a)JIlvU-(A,a)= A"A,"(JP' - alPv + avPl) , U(A, a)PPU-l(A, a) A,"PV (1.16)

10 -=H, - = -1, f-I = f0, - 1 = -k (1.17)

H,-1 = -H , P - p TJ_ 1 =- ,9K,-7 =K (1.18)

Take A -+ 1 + w' and a -+ E' infinitesimally, then one arrives at:

2 1WJft"J""+EiPo'i2W,J,"EP",PP]w"PP(1.19)[pJ'",wwLJP'-EUPP,=

The commutation algebra of the generators of the Poincare group is found to be:

i[JW", JPa] - qVPjA01 - 7J - 7 /J" + 7O" , (1.20)

i[P", J"'] = r 1VPP - nr P" , [Pt, P"] = 0 (1.21)

Lorentz generators J", in general, obey the 0(1, 3) algebra, but the restriction to the proper

orthochronous subgroup make it SO+(1, 3) [17, 361. The spacetime translation algebra, from the

generators PY, is trivially T4 . Combining these, one arrives at the Poincare algebra SO(1, 3)+ 0T4 .

Operators that constructed from the Lie algebra generators which commutes with all of these

are called Casimir operators, and the eigenvalues associated with Casimir operators are fixed

characteristic of the transformation with generators in that algebra. In other words, the eigenvalues

of the Poincare algebra's Casimir operators are what define the fundamental building blocks of

physics, and the argument should applies to a quanta.

1.2 Fixed Characteristics

1.2.1 Casimir operators

The Casimir operators C1, 02 of SO(1, 3)+ 0 T4 (which commute with all generators of the sym-

metry group) are the length-squared of the 4-momentum P-0 and the Pauli - Lubanski 4-vector WA [17, 26, 32, 36]:

C1 = -P 2 = (P) 2 __ P 2 , C2 = W2 , W = A"U"PJPU , WVV P, = 0 (1.22) 2

11 2 It's straight-forward to see that C1 is nothing but the mass-squared M in special relativity. The mass-squared, however, can be zero, positive or negative. The negativity of the mass-squared seems to violated the causality of special relativity (as from the energy-momentum dispersion relation, a quanta with negative mass - a tachyon - should move faster than light), however, when constructing a field theory out of the individual particles and interactions (perturbative description), one realizes that a tachyon is not only non-localized when moving faster than light [7] but also indicating an unstability of the configuration where the full (nonperturbative) theory is formulated around. Therefore, we will only consider the cases where the mass-squared is positive and zero. Also note that, the mass is always chosen to be non-negative, so that the positive energy branch is associated with particle creation, and the negative branch is for particle annihilation.

1.2.2 Quanta definition

Massive Quanta and Spin

Consider a massive quanta C1 = M 2 > 0 and go to its rest frame, then the meaning of C2 in quantum mechanics can be seen:

PP = (m, 0) + W = Mj , C2 =1W2 = -M2P =M 2 s(s + 1) (1.23)

Hence, C2 encodes the information of the spin s, a well-known concept in quantum mechanics describing the spatial rotational nature of the quanta with s E 1N [32].

Massless Quanta and Helicity

For massless quanta, both C1 and C2 vanishes (not quite, it's actually that non-zero C2 is never seen in nature, so for physical purposes, let's just consider C2 = 0 only), and from the vanishing of W"P,, since the only null 4-vectors whose products with a given non-zero null vector vanish should be those propotional to that vector, therefore:

Wh = hP , P" = (PO, 0 ,,0 PO) -> h = (1.24) |PI

12 The ratio h is the helicity, which can be viewed as the angular momentum in the direction of motion. To see the possible value for the helicity, note that in the chosen frame above:

0 Wo J=P3 , W 1 =P ( +K 2 ) , W2 = P"(J2 - K 1 ) (1.25)

2 The algebra of Wo, W1 , W2 is closed and same as E2 , isomorphic to SO(2) 0 T [32]:

1 [W1, W 1= 0 , [W , J3] = iW1 , [W1, J ]=-iW -+ h - (1.26) 2 2 3 2 2

Although the helicity comes in pairs (with opposite sign ), it changes sign under spatial inversion, and since that transformation is not connected to unitary, therefore the value of h is a fixed characteristic. The helicity for massless quanta plays the similar role as the spin for massive quanta, therefore, Jhi can also be called spin. The different is that each helicity h is one-dimensional while each spin s is (2s + 1)-dimensional, and we will see this later.

Particle Type

In summary, there are always two fixed characteristics of the Hilbert physical states, which can be used in the particle definition (different characteristics for different species): the mass-squared

C1 = M 2 and the spin s c 1N (for massive quanta) or the helicity E Z (for massless quanta).

For interacting theories, there should be more unchanged characteristics (doesn't have anything to do with spacetime symmetry), such as the charges, which will be discussed later. After take in to account of all those characteristics, one has the particle's definition IF (for example, an electron, a proton, or a neutron). The subset of the complete characteristics can be used to define families

(such as baryons, mesons, leptons). It is not enough, however, to study the dynamics by just knowing the particles. It is important to know different dynamical states of the very same quanta.

Therefore, one needs labels to further specify more information of the fundamental degrees of freedom (the particles) in the theory.

13 1.3 Labeling a Quanta

1.3.1 4-momentum Label

A quanta is partially defined by the fixed characteristics under Poincare transformation (the collection of these are colled IF; we will add more to it later from the information of the interactions), and can be labeled by the eigenvalues of a complete set of mutually commute operators. The natural choice for the label is the eigenvalue of the 4-momentum p", since the spacetime translation group T4 is trivial, which means this choice of label is good. However, the label p (which will be shown later) is not a complete description for the dynamical states of a quanta, so let's denote others be a general label a:

P"I;p,u -) =p";p,) , U , U)(,a)4, Im UTE; p, -) = e Ip) (1.27)

Since all information about Pl is known, let's consider a Lorentz transformation, with:

P4U(A) I'T; p, U)U(A) (U-1(A)PPU(A)) IT; p, a) = U(A) (A %P") [IT; p, a) (1.28)

= A'p"U(A)I ; p,a) - U(A); p, a) = C,(A, p)IT; Ap, a') , U(A, 0) = U(A) (1.29)

Since the combination is linear, in principle the a label can be choose in such a way that the matrix C,(A, p) is block-diagonal, as the state 14T; p, a) with a within any block by itself furnish a representation of the Poincare group [36]. It is then natural to identify the states with the components of a irreducible representation of the Poincare group.

Wigner's Little Group Representation

For each value of -P 2 (a specific value of C1, the mass-squared M 2 ), choose a standard 4- momentum kP and express all other 4-momentum of the same length by a Lorentz transformation, then the states can be related by a normalization facter N(p):

p J=LP,,(p)k' , I4;p,a) = N(p)U(L(p))14;k,a) (1.30)

14 Applied an arbitrary Lorentz transformation on the above state:

U(A) 'IF; p, o-) = N(p)U(AL(p)) W; k, o) = N(p)U (AL(Ap)) U (L-1(Ap)AL(p))II; k, o) (1.31)

1 (L-1(Ap)AL(p))k = L '(Ap)Ap = k -+ W(A,p) = (L-1(Ap)AL(p)) , W,kv = kL (1.32)

The Lorentz subgroup that leaves the 4-momentum invariant is called the little group, and one can see already see the relation between this and the Pauli - Lubanski vector, in the sense that

they both respect the very same E2 algebra, which we will discuss in a moment. The little group transformation can be written as:

U(W)I; k, a) = Dr,(W)I|; k, o-) (1.33)

It can be seen that the coefficients D(W) furnish a representation of the little group:

S Data(W'W)I T; k, a') = U(W'W)|IT; k, o-) = U(W')U(W)|IJT; k, o-) (1.34) a'

= >11Da(W)Daa,,,(W') IIF; k, a') -+ Data(W'W) =3 Dai"(W')Da/a(W) (1.35)

Use this in equation (1.30), one arrives at:

U(A)jT;p, o-) = N(p) D,',(W(Ap))U(L(Ap)) IT; k, a') (1.36)

ND l,(W (A, p)) I ; Ap, a') (1.37)

This is known as the Wigner's little group representation [24, 36].

Normalization

To determine the normalization, choose the usual orthonormalization in quantum mechanics (the

6-function arises and the completeness relation follows because the states are eigenstates of a complete set of mutually commute Hermittian operators) [361:

('; k', 'I T; k, o-) = - )6a'a6i-' -('-+ Dt(W) =D(W) , (1.38)

15 Consider another inner product, with p = L(p)k and p' = L(p)k' (same Lorentz transformation):

(T; p', o-'j T; p, a-) = N(p)N(p)t (I; p', o'lU- (L(p)) U (L(p)) IiT; k, o-) (1.39)

= N(p)2D (W (L-1(p), p) )6(3)( k) (1.40)

Since at p' -+ p, the transformation W(L-lp,p') -÷ 1, so:

2 3 (';p',o'l I; p, u-) = IN(p) ( (' - O) , (3) p (1.41)

The trick is, under Lorentz transformation, the 4-vector length stay as a scalar, thus [22, 36]:

0 3 '- p k 6( )(k' - L) -+ N(p) (1.42)

-+ U(A)I'; p, o-) = (Ap)0 D,, (W(A, p))II; Ap,o') (1.43)

The only remaining problem now is finding the representations of the little group. The exact form of D(W) should depend on the definition of the quanta, therefore, in general:

Do, (W(A, p)) = D(, (W(A, p)) (1.44)

1.3.2 Details Dynamical Information of Quanta

Massive Quanta Labeling

2 For mass positive-definite -p = M2 > 0, the little group is SO(3), isomorphic to SU(2)/Z 2 so the representation theory of SU(2) naturally including that of SO(3), with unitary representations can be broken into the direct sum of irreducible representations spin-s D() of (2s + 1)-dimensional

SU(2), corresponds to (2s + 1) different spin states, with s E -N. The specific value of the a can

be associated with the eigenvalue of one of the little group's generators. Under an infinitesimal

rotation Rij = 6 ij + E9i [36]:

Do(1 9)- 6 + ( ) , = -S5, , a= -s,-s+1,...,s- 1, s (1.45)

16 (J() iJs), = (JlS) Jls)) = ',c, 1/(s T o)(j i a + 1) , (j (s) = (JlS)), = OUalc (1.46)

The rotational generators in the spin-s representation is given in details, above. To calculate the element of the little group - Wigner's rotation, one choose a standard transformation L(p) to carries the 4-momentum kA = (11, 0, 0, 0) to p", and this can be found straight-forwardly:

L =(p) L7 t o or , Lo(p) = a b read-off: (1.47)

The Lorentz transformation rule for a massless particle of spin-s can be read-off:

U(A)IT; p, a) = Ap 0 -91) (W(A, p) T;A, / (1.48) 0' o

Massless Quanta Labeling

For massless quanta -p 2 = 0, the little group is ISO(2), also known as the Euclidean group E2, which is composed of rotation and translation. It's a little bit different from SO(2), where one has only rotation. To see how the translation can arise, choose the standard 4-momentum k (k0, 0, 0, k0), and note that the little group transformation satisfies [21, 361:

WL k" = k= -+ W = S(a, 3)R(0) (1.49)

1+ (a2+#2) a 13 -(a2+32) 1 0 0 0 a 1 0 -a 0 cos0 sinO 0 S(a,/3) =, R(O) (1.50) 13 0 1 -3 0 -sin0 cos0 0

2 2 (a2 + 2) a / 1-0 (a + 0 )/ 0 0 0 1

From equation (1.12), the corresponding transformation on the physical Hilbert space:

U(S(a, O)R(O)) = I+iaNiiN 2 +i0J 3 , Ni = + k, N2 = -JI +K$2 (1.51)

The product of any group element can be work out, and one can already see that it's ISO(2) instead of SO(2) [32, 36]. It can be seen directly from the algebra:

[J3, N1] = iN 2 [13 ,N 2 ] = -iN1 , [N 1 ,N 2 ] = 0 (1.52)

17 These results are very similar with what was found by studying the Pauli - Lubanski vector, given in equation (1.24) and (1.25). The unitary irreducible representations of the group E2 are labeled

2 by the eigenvalue n' = (ni, n2 ) of generator N = (N1 , N2 ) and h of generator J3 . For n > 0 one gets an infinite dimensional irreducible representations, which can be interpreted as particles of infinity spin, but since there is no such particles ever observed in nature, these representations are of no physical interests [261. For n2 = 0, the group E2 effectively acts as SO(2) rotation, and its representations are one-dimensional, labeled by real, integer of half-integer eigenvalue h E !Z, which is called helicity. The transformation of massless particle is now finished:

Dal, (W) = e 60, U(A),h;p) = (Ap) 0 e h; Ap) (1.53)

The transformation is independence of a and 13, and the freedom of these parameters is related to the problem of gauge ambiguity, which will be discussed later in details.

1.3.3 More on Spin and Helicity

Multi-valued Problem of Representations

There's a "sneaky" feature of the continuous group that we ignored when nailing down the possible values for helicity [36]. Indeed, every continuous group has the possibility of having multi-valued representations. But it turns out that the existence of multi-valued representations is not without restriction - it is intimately tied to connectedness, the global topological property of the group parameter [32j. For example, in the case of SO(2), the group parameter space (which is the unit circle) is multiply-connected, which implies the allowance of multi-valued representations. Thus, it is possible to determine the existence and the nature of multi-valued from an intrinsic property of the group. The occurrence of double-valued representations can be traced to the connectedness of the group manifolds of symmetry associated with the physical four-dimensional spaces, and that's the topological reason why the helicity can only be an integer or a half-integer, instead of any possible real number. For the group SO(3) and SU(2), there's a clear algebraic reason restricting the spin to be of jN [32, 36].

18 Topology of Spacetime Symmetry

Let's settle the point with helicity of massless particles, one and for all. Any real 4-vector VP can

be used to constructed a Hermittian 2 x 2 matrix:

V+V3V 1 -iV 2 V3V2=-det v (1.54) V = VO- = , V1i + iV v 0oy

In this language, a Lorentz transformation A can be realized as a Hermittian 2 x 2 matrix A [361:

(A,4(A)V)a-, = AvAt , A(A'A) = A(A')A(A) , det A = 1 (1.55)

The required properties of A means it should be a special linear complex matrix SL(2, C). It is

important to note that although the algebra of A is the same as A, the group SL(2, C) is not

the same as SO(1, 3)+, and this can be seen from the fact that A represent the same Lorentz

transformation. Mod that out (identifying +A together), one has SL(2, C)/Z2 , which is now the same as the Lorentz group. Now that the setting is done, we can study the topology of the Lorentz

group (the Poincare group topology is just a trivial extension of this, since the T4 group is nothing

more that adding R' fiber bundle). By polar decomposition theorem, any complex non-singular

matrix can be written as:

A = AeB A tA=1 , Bt=B ; dctA=1 -+ detA=1 , TrB=O (1.56)

In the components form, one can see the topology of A is S3 and B is R3 :

S d + ie f + ig d2 + e2 + f 2 +g 2 =1 ; h= C a - ib (1.57) -+ig d-ic a+ib -c

The topology of the group SL(2, C), therefore, is S3 0 R3, which is simply-connected. For

SL(2, C)/Z2 , identifying A with -A is the same as merging the unitary factors A and -A (since eB is always positive), then the topology of the Lorentz group SO(1, 3)+ can be understood as

3 3 3 S /Z 2 D R = RP(3) D R , and the geometry is not simply-connected anymore. The manifold

is doubly-connected since the 1st homotopy group r 1 (RP(3)) = Z2 , so a double loop can be

19 continuously deformed to a point. Consider a transformation trajectory 1 -+ A' -+ A'A - 1 twice:

(U(A')U(A)U-1(A'A)) = 1 -+ U(A')U(A) = +U(A'A) (1.58)

Here, we use the caveat mentioned earlier, as the representation is before the enlargement is done and the phase is set to 0. The sign can be understood from the phase of the Lorentz transformation when acts on a physical state, integer-spin always see +, while half-integer can see

- in some cases. It is best to see this from equation (1.52), since after applying a full 2wr rotation, a phase appears:

2 A 2 ,p = p , 0(A 2 ,,, p) = 27r -+ U(A 2 ,)IT, h; p) = ei 7rhO(A,p) IT, h; p) (1.59)

If h is an integer, then the state transform back to itself, bit if h is an half-integer, then the

4 state gets an opposite sign. Because of the doubly-connected requirement, then ei 7h should be 1, therefore the helicity cannot be more fractional than half. Even for higher spacetime dimension, the arguments hold for both massive spin (s E IN) and massless helicity (h E 1Z). and to see this, note that SO(1, D)+ ~ SO(D) 0 HD. The hyperbolic D-space HD has a trivial topology of RD, therefore the 1st homotopy can be calculated entirely from SO(D). But it is known that

7r (SO(D)) = Z2 for D > 3 [11, thus the Lorentz group at higher dimension theory is also doubly- connected, which only supports spin and helicity no more fractional than half. About the spin states and the helicity states of a quanta, these can be understood from the projective representations of the special orthogonal groups. For SO(N), for N odd, it has one spinor representation, categorized by N-i2 half-integers pi -2'> 1, and for N even, there are mirror conjugated spinor representations, pi > - [141. categorized by z2 half-integers -_

Still,did we left out something?

The answer is yes, actually we did. To see this, let's take a look at the massless particle again, and to be precise, its helicity h = -. This operator is not boost-invariant, therefore Lorentz-invarance of helicity is not generic [371. The realization leads to the discovery o a new Lorentz-covariant massless particle type known as the continuous-spin particle (which is labeled by a spin-scale p with units of mass in 1 + 3 dimensional spacetime). This type of particle is not (yet) found in nature, but still of physical interests [11, 23].

20 1.3.4 Quanta in Flat Spacetime and Curved Spacetime

Let's conclude this section by summarize the important points, and a remark. From spacetime symmetry, it's possible to define a quanta (can be a fundamental particle or a composite particle) to a certain degree, with fixed characteristics and labels for the particle's dynamical information.

To represent massive quanta, the following physical state can be used:

I : M > 0, s ; p ) -÷ I;p, -) (1.60)

With massless quanta, the physical state is of the form:

IX : M = 0, h ; p) --+ I, h; p) (1.61)

Note that, all arguments in this section are done in flat spacetime. In a general curved spacetime, quanta, or particle, is usually a concept of a locally flat region, which is meaningful for the observer only if its characteristic wavelength (an estimation for the size of the quanta in quantum mechanics, which is the Compton wavelength for massive particle and the normal wavelength for massless paritcle) is less than the curvature scale at the position of the observer. It should be noted that in some special cases, such as in AdS space, particles can be defined similarly by finding the irreducible representations of the AdS algebra, but that's different from flat space algebra. Indeed, although the local Lorentz symmetry is fine as the spin is still good for labeling, the 4-momentum

and also the mass are not well-defined and position-dependence since the spacetime translation is not a symmetry of the theory of general relativity anymore. This is one of the reasons why quantum field theory on a curved spacetime, even fixed and non-dynamical, is extremely hard (but very thought-provoking).

21 Chapter 2

Field

A single quanta is just a perturbative description of the universe. To go beyond that picture, one should try to study the collective behaviors of many quantum with interactions (multi-particles dynamics), and realize that the language of field comes out naturally. It should be noted that, field formulation of the quantum world is so desirable, not only because it's naturally arise, but also it's a lot easier to dealt with, compare with the language of creation and annihilation operators. It should be mentioned that, while most of the time theoretical particle studies tend to use the former language, the later one is still very popular in condense matter physics of interacting system, when the interactions are not quite complicated.

2.1 Construction of Multi-particles Dynamics

2.1.1 Scattering Setting

To simplify the notation, for a single massless quanta state:

II: MAp= 0, h; p) -+ F: AMi= 0, s =Ihl;p, -= h) (2.1)

If the particles in the theory are not interacted, then one can always build a multi-particles state as a direct product of single quanta states. To short-hand the exhausting labels, let's just use H to keep track of the multi-particles state (the sum integral only include configurations that do not

22 simply differ by just permutation of particles):

(J;d= Jd3f;id39P 2 ... (2.2)

The Poincare transformation, the normalization and the completeness relation naturally follow. The (perm.) takes into account the possibility that a permutation of one of these multi-particles states is the same as the other, and the sign is totally depend on the quantum statistical nature of the particles [36]:

U(A, a) (;p, -).= (A)..ea(+ D1 (W (A, p,).. (IF;p, o1.. p1.. (2.3)

K'( p', 1'I)... (1;pio)...) = (3)( (perm.) (2.4)

( ) ='6= , 1 = f d='')(E'I -+ = fd-'I') (E'I E) (2.5)

To make a connection between theoretical predictions and experimental data, it is important to have the same setting. Interactions between particles are usually understood through scattering experiments, which can be described by S-matrix from the theory side. The preparation always requires a definite particle contents, which means an effectively non-interacting multi-particle state, at a far past and let the particles interact inside a scattering region of the size much much smaller than the distance between the particles initially, then at a far future, when what come out are away from others, the measurement takes place. The S-matrix encodes the information of the very same process, telling the probability amplitude for the transition between the initial and final states, which is nothing but multi-particles states that are approximately the direct products of the effectively non-interacting degrees of freedom (can be fundamental particles and composite particles) in the theory at long distance scale. It is important to realize that some interactions cannot be seen directly, at least at low energy, for example, the communication between single quarks, because at the usual energy of experiments, these are confined and cannot exist alone. The interactions have to be observed indirectly from the scattering of composite particles with quarks inside, such as baryons and mesons, which can appear as a single quanta. In general, S-matrix cannot be used to describe short-distance quantum process (might be useful for weakly interacting theories, but usually fails for strongly interacting ones).

23 2.1.2 Creation and Annihilation Operator

Creation Operator

Let's define creation operators, which arises in the context of non-interacting multi-particles very naturally. The creation operator, for a given quanta of specific labels, is define to be an operator that simply add a new particle to a state:

(2.6) aT;p,, (1; pi, G1)(4 2 ;P2 , 72 )... = (; p )('F1; pi, U)(' 2; p 2 , 0 2 )...

With this, under the Poincare transformation of the non-interacting theory:

U0 (A, a)aF.,,U6-1(A, a) = ei(AP)a (A0 D(, W (A, a Q))pfAp, (2.7)

Define a vacuum state IQ), which has no particle and is usually assumed to be invariant under

Poincare transformation and also the spatial inversion and the time reversal. Any multi-particles states can be built by acting many creation operators on IQ), and due to the quantum statistical nature of the particles, the adding order might be important [361:

XFP,0 '2a2*..) = (T 1;pi,91 )(T2;p2,9 2)... (2.8)

There's only two possible quantum statistic for particles of the same definition (identical particles), because if one switch them twice, the state should be back to the original, hence the effect of each switching can only gives a factor of +1 or -1. Boson statistic is +, and fermion statistic is -, and the sign Sperm.(n) for the permutation of n identical particles, label ordering 1, 2, ... , n, can always be kept in track by finding the integer r < n which label the particle switch place with the first one, and the sign Sperm.(n-1) for the permutation of the rest:

Sperm.(n) ( )r* Sperm.(n-1) (2.9)

24 Annihilation Operator

The annihilation operators can be defined as the Hermittian adjoint of the creation operators, and from the normalization of physical Hilbert states:

( i ; P', c)...( ; p',, u,4) ap ;p,, ('P ; Pi, o-l)...(i ; Pr, -n) (2.10)

= (e+;p, u))(9 '1' , o)...(";P , -,m) ('P ; P1, a1)-...( ; Pn, -n)) (2.11)

ni m

6 E per.(n-1)r _ - Pr) H5iperm ( - Pperm.(i))m+1,n (2.12) - r=1 perm.(n-1) j=1

Jm+1,n Y r-1perm.(n-1) r 6 - Pr) (2.13) r=1 perm.(n-1)

x (+;p1 I..(+ ( ;;,o1..W;p_, r1(iP+,J+)-( Pn, an) (2.14) .',.('T

Therefore, one arrives at the realization of annihilation operators in the Hilbert space that it takes particles out of a state, like the description in its own name:

awp;p,,j9) = 0 , ap,;p,, ('i;P1,u)..( ;Pauo1 ) (2.15)

Z()' 1 0ua,( 4 )(p _ Pr) (P';Pi,.).--('+;Pr-,ao-)(J ;Pr+1,) r+)--('F ; Pn, on) (2.16) T-1

From this, under the Poincare transformation of the non-interacting theory:

Uo(A, a)aq;p,U -1(A, a) = eiAP) A 0 D(', (W1 (A,(p))2a;Ap, (.17)

Interaction from Creation and Annihilation Operator

It's trivial to see that creation and annihilation operators of different quanta must be commute.

To see the commutation relation between these of the same quanta, apply an annihilation operator

on equation (2.6) and an creation operator on equation (2.16):

an'' p ~- p) + P1, 9~1)... ( ; pn, n) (2.18) ;'' ;p, ( 1,11) .. ; Pn, n

25 N 2 +Z()r (( -Pr) ('I ;p , c)('I Pr-,u-1)( ;Pr+, r+)...(I ;Pn, an)) r= 1 (2.19) a ap,;ap ;y,-, (T; pi, UI)...(J'; pn, Un) (2.20)

4 SZ( )r+1jo, 3( )(P/ _ P+r) I('F ;1Pui)... (' ;Pr-i,0r-1)(F ;Pr+l, r+) ... ; Pn,on) (2.21) r=1

4 -+;PO T- aq,;P,,,aQ;,,aI [alp ;p,,-', aXP;Pu] , j3 )p- (2.22)

In addition to the above relation, one can also find [361:

{A, B} = AB 4 BA , {at 7 ,,, at } = {aW ;p,&, af ;,,}= 0 (2.23)

Since any state in the physical Hilbert space can be reached by adding in or snatching out particles and the connection (the matrix elements) between any two states can always be assigned to any value, therefore, with creation and annihilation operators, any operator on the physical Hilbert space can be construct, so the interactions in the theory can be described. With the algebra found in equation (2.22) and (2.23), in any given operators one can have all creation operators are moved to the left and all annihilation operators to the right. This arrangement is called normal ordering.

2.1.3 Causality Condition

The evolution of the quatum world depends on the Halmintonian H = HO + V, with HO is the free theory Halmintonian that including all the particles (can be both fundamental and composite particles) and V describe the interactions, all of these can always be written in terms of creation and annihilation operators of normal ordering:

H, HO, V d (J d p 1 a ;P)(, j api pjo) (2.24) nmn *1, 42,. ' '..o .....= j=1 k=1 1=1

x hmrn, homw, Vmn ((T'1; P'" Ul)...( qf' P; 1'm), (y1; P1, a,)---(.. P77,7 ) , (2.25) nm ?n=0n=O The requirement of having real energy means the Hamiltonian should be Hermittian, both the free theory and the interactions. For S-matrix calculation, one can always take that V is effectively turn-off everywhere except inside the scattering region, or having the interactions die out slowly

26 when approach the past and future infinity, where the initial in-coming and final out-going state

are set [22, 36]. Use the later approach, S-matrix can be expressed as:

S = (E ;-) =(ESi) ,E;~) = |Eit =-oC) I ) = IE7f,t = -+oo) (2.26)

The S-matrix operator is nothing but the renormalized evolution operator:

S = E(+oo, -oo) = eiHotf e-iH(tf -ti)e -iHoti (2.27) tf=+oo,ti=-oO

Since we want to build a relativistic theory, the Poincare invariance should be preserved, hence S-

matrix should be unchanged under Poincare transformation of the in-coming and out-going states.

In other words, S-matrix commutes with all generators of the Poincare group in a free theory:

1 0 Uo(A, a)SU- (A, a) = S , [P = Ho, S] = [Po, S] = [Jo, S] = [KO, S] = 0 (2.28)

In the interacting picture, S-matrix can be expressed using the Dynson series [22, 36]:

V(t) = eiHotVe-iHot , S -Tefj dtV(t) (2.29)

,= ( jdn dt1...dt(EIT (V(t1)...V(t) I7i) (2.30) n=O To make S-matrix Poincare invariance, the interaction density should be a Poincare scalar:

3 Uo(A, a) V(x)U- (A, a) = V(Ax + a) , V(t) = f d 5V(5,t) - S = Te-ifdaxV(x) (2.31)

dxl...dex(BfIT(V(xi)...V(xn))IEi) (2.32) = 0= 4 n=O However, even with this, the S-matrix still doesn't fit the bill yet, because of the time-ordering

operator is not Lorentz invariant, in the sense that it loses its meaning for space-like separation.

Therefore, a condition for interaction density should be impose, for the sake of consistency, and it

is called the causality condition:

2 (X' - X) > 0 -+ [V(x'), V(x)j = 0 (2.33)

27 2.1.4 The General Form of S-Matrix

The Poincare invariant feature of S-matrix theory indicates that it should vanish unless the total

4-momentum of all particles disappears (in-coming momentum and out-going momentum are of opposite sign in the summation), since under an arbitrary Poincare transformation U(A, a) on both the initial and final states, there's always a single phase with a-dependence pops out and it should disappear so that S-matrix isn't depend on the choice of a. This is nothing but the momentum conservation law, which is also realized in classical physics:

eiapPA P = 0 , P,= p, (2.34)

Therefore, a general S-matrix can be written as:

-n,=os - i2w )(ps, - pE1)Ma , (2.35)

If the theory has no interaction V = 0, then S-matrix is simply , so the information of interactions are all encoded inside Ma+sf. In a general interacting theory, S-matrix should be unitary since it connects two complete sets of orthogonal states, therefore:

4 JdE'S~~S&+~ -~ i(~ ~ - Af f4)- 2-71 J4 dE5 (P=, _ P=f '~,MI.

(2.36) By taking Ej = Ef = 7, one arrives at the optical theorem:

Im MIa -IrfdE'5(Ps, - p6) M .1+s|2 (2.37)

2.1.5 Internal Symmetry and Conserved Charge

Because the particle zoo in nature is very crowded, there are too many possibilities for the dynamics can be encoded inside the S-matrix, then to put more constraints on the possible interactions

(and make the theory easier to keep track of), one can employs internal symmetry, the kind of symmetry that has nothing to do with the Poincare group. Unlike Poincare symmetry which acts on the spacetime and is realized by each quanta individually, the internal symmetry mixes different particles together and leaves the S-matrix unchanged. It is also desirable that the internal

28 symmetry can still be seen when the interactions are smoothly goes to vanish at far distance past and future, for the initial and final states, as the symmetry transformation in free theory works:

Uo(I)SU-1 (I) = S (2.38)

Let's focus on the symmetry that can be realized as linear transformation between particles (nternal symmetry realized in nature is mostly of this kind, for example, in nuclear physics under the interchange of neutrons and protons):

Uo(I')Uo(I) = Uo(I'I) -+ D(I')D(I) = D(I'I) , Dt (T)D(T) = 1 (2.39)

UO(I) (F1; p1, Oi)(4 2 ;P2 , 0 2 )...) = Dq, q(I)Dpj.. (I';piFi)(P';p2 ,cr2 )... (2.40)

1' 2'**

Consider the cases when the symmetry group is a Lie group, which can be described by a continuous

0' parameter, then it can be written in an exponential form, with the symmetry generators,

Hermittian Q'. This generator should commute with all Poincare generators of the free theory, and, through the commutation realtion, put the restriction on the interactions:

UO (I(oa)) = eiQaoa , [P = HO, Q] = [PoQ] = [,Q"] = [KO, Q"] = 0 , [V, Qa] =0 (2.41)

Hence, Qa should be the conserved charge in the theory, and any process should make the charge unchanged. To see this, consider the symmetry with one-dimensional parameter 0 that gives different particle species independence charges, then the S-matrix invariant under that symmetry indicates charge conservation as the total charge of the in-coming state is the same as the out- going state, follows in the same way as momentum conservation arises from translation invariant in equation (2.34):

DTx10)= 6tp q -> q- = q~f (2.42)

And indeed, charge conservation can be seen in nature and in many different physical processes, such as the conservation of electrical charges in electromagnetism interactions.

29 2.1.6 Cluster Decomposition

Another requirement for the interactions is, quite a philosophical one, if the events are sufficiently

far apart, then these should be uncorrelated. This is a fundamental principle of science indeed, because it's allow us to make predictions and formulate theories without the need of knowing

everything in the universe [361. It's called cluster decomposition (or, sometimes, locality), and it

allows S-matrix to be factorizable into different scattering processes that happened at scattering regions far away from others:

S i =EE, S-+,;S= s..Sa , =(7 1)(B 2)...(En) , E. (E')(E')--(') (2.43)

This is indeed a very strict condition on the possible Halmintonian of the interacting theory, hence also for the Halmintonian of the free theory and the interactions. For realization of cluster

decomposition, the coefficients for the direct product of creation and annihilation operators of these three in equation (2.25) should be proportional to a single 6-function of 3-momentum 1361:

hmn, homn, Vmn ((O; P' ,) N o ), (P1 P1, o1)...(I'n; Pn, 9-n)) (2.44)

-- u $ ), (1; P1, o1)...' nPn, 0'n) 6(3) __ m-Pi)rn, 1Omn, "Drnn (e"; , ( i=1 i=1 (2.45)

In other words, there's no 3-function in h, ho and f. The need for a 6-function in cluster decom-

position can be easily proved by using equation (2.30) then going from the momentum space to

the position space, and the restriction for that 6-function of the 4-momentum to be the only one

can be seen from the topological nature of the perturbative Feynman diagram (which replaces the

algebraic language of creation and annihilation operators by a geometric language of lines and

vertices) at all order of expansion [361.

30 2.2 Macroscopic Description

2.2.1 Creation and Annihilation Fields

Since T already specifies the quanta definition, so the on-shell condition is known, therefore, one

can always drop p0 out of the notation. Define the creation

'11-(x) =f d p a aui(xI|';i, -), Ir(x) =f dpa ,.vi(x IT; , o) (2.46) 01 a

Basically, we group the creation and annihilation operators in spacetime symmetry irreducible

representations, which are specified by the label 1. The function tensor coefficients ut and v, can

always be chosen so that the fields have nice properties [36]:

Uo(A, a)'Uo-1 (A, a) = ED(A-1)Tj(Ax+a) , D(A')D(A) = D(A'A) (2.47) 1'

From equation (2.7) and (2.17), one arrives at:

Ulf (Ax + a I T; Ap, o-')D ( (W(A, p)) = P Di(A)ei(AP)aui(xI1p; , c-) , (2.48) a1

Iv/ (Ax + a IT; Ap, -')Ds (W(A,p)) = D (A)e-(AP)avi(x14; p5 -) (2.49)

Under transformation alone, the two function tensor coefficients change to tensor coefficient:

ui(01';j ,ar) = u(AI; ,-) -* ui(x1'; a,) = (27r)-2eiPu(T; , -) , (2.50)

=-f+(x) (27r)- ZZld3Pata e 7t, (IF; ,a-) , (2.51)

-+ ui,(I; Ap, aD')D, (W(A, p) EDri(A)u(P;i7,-)DPO ; (2.52)

vi(0IW; , o-) = vz(T; , o-) - v(x|T; , o-) = (27r) -eAiPxv(IF; ', a) , (2.53)

qf-(x) = (27)-2 1 d1pa e- Pxvl(P;7,U) , (2.54) '4' a 1 f

31 viF(P; Aip, -')D/, (W (A, p)) = P Dii1 (A) vi(I; p5 o-) (2.55)

2.2.2 Finite-Dimensional Representations of the Lorentz Group

While D(s)(W) represents only the little group, D(A) represents the full Lorentz group with generators J and K. The non-compact Lie algebra SO(1, 3) of the Lorentz group can be reduced to the direct product of two non-mixing compact subalgebras SU(2) 0 SU(2) by separating in the following way [32, 36]:

A = -(J+iK) , [A A] = iEjklAl ; = -(J-iK) , ,5 k] = iEkBk , [Ai, B] = 0 (2.56) 2 3 2

Thus, any finite dimensional irreducible representation of the non-compact Lorentz group can be written as a pair of integers or half-integers (A, B) with A and B specifying the spins representing the operator A and B. The simplest non-trivial irreducible representations of the group are the two- dimensional (1, 0) left-handed and (0, 1) right-handed, and these are fundamental representations so that any finite dimensional representation of the group can be obtained by the decomposition of repeated direct products of these [32], and the results in components [36]:

A1+A 2 Bi+B 2 (A 1 , B 1)®(A2, B 2 ) 0 0 (A, B) ; AaIbI,ab= Jaab'b , Ba'b',ab= a'aJ, ; (2.57) A=IA1-A 21 B=IBI-B21

(J)))a/a = a'a , (J1) 2 A), = 6a,a i /(A T a)(A a + 1) (2.58)

The (A, B) representation is irreducible with respect to the Lorentz group as a whole, but reducible under the rotation subgroup [24, 32]. To see this, note that J = A+ B, and the tensor product of spin A and spin B, in general, can be reduced to objects of rotation spin s [361:

IA - BI < s < A + B (2.59)

Although irreducible representations of SO(1, 3) are also irreducible representations of SU(2) 0

SU(2), some important properties of the presentation, such as unitarity, are not presered, since the generators A, B and K cannot simultaneously be Hermittian [32]. In this particular representation, the generator K is anti-Hermittian, therefore the finite dimensional representations of the Lorentz

32 group are non-unitary, which is expected from the fact that non-compact groups do not have nontrivial finite dimensional unitary representation, therefore cannot be directly realized as physical states since all symmetry operations should be realized as unitary operators on the physical Hilbert space[32. This is not a lethal problem, however, since here we are dealing with fields instead of physical states. Also note that, for any spin s, there's infinite many way to choose (A, B) to gets to the final results, so one has the freedom to choose the simplest one, for example, to have D(A) of the spacetime transformation form AAA... as much as possible. For integer spin s then a good choice is (1, 1), for half-integer spin s it's can be A + , - left-handed or - , + right-handed or + , - - , + Dirac.

Massive Field

For the field of massive quantum, choose the standard 4-vector k" has vanishing spatial components k = 0, then:

up (T; , a) = D t (L(q))u(;Oo) , v),(t;, ,I-) =IF;D i(L(q))v(J;, c-) (2.60)

Now, let's the Lorentz transformation to be a rotation W(A, p) R, so that the standard 4-vector is left unchanged, then with j(s and j to be the angular-momentum matrices in the representations D(W)(R) and D(R):

(Jls) = (-)7'-+ 1j(S),,, ; (2.61)

u11(4; 0, a-')D, (R) Dpi(R)u(I;( , c-) -+ up (4; 0, -') J = Su (T; 6, a-) 0~ 1 to, 1 (2.62)

V3 (T ; G, D-')D t(R) = Dz'(R)v,('I; , o-) -+ vv(T; d, o-')(J )f = - 3 J 1 (T; U, a-) (2.63)

The eigenstates of J are called the polarization tensor, and from the analysis of the little group we know that these correspond to 2s + 1 degrees of freedom. In (A, B) representation, one arrives at the famous equations for Clebsh - Gordan coefficients [36]:

4 > a3'b(W; 0, ) > a tab'(X; J, c-) + Ua'b( ; G, O-) , (2.64) ' ab

33 2 ( V ' a'a) Vab',(P,,u - .. d Va'b(T&; (2.65) alb 0 a 1 ba~; G, cr)

1 -+ Uab('; G, o-) =CAB(so-; ab) Vab(T; ', 0) "nab(T; , -o-) (2.66)

For a general finite momentum, the two tensor coefficients (with our choice of convention) are:

U) = - Uab(T; , ;T ,-)a - Vabe' U0,,) / (-;0 -) Uab >1(e+'jjA), (C jB)O)bCAB(su; a'b') a'b' (2.67)

Massless Field

For massless field, equation (2.52) and (2.55) can be simplified with (1.52):

0 ul (T; Ap, a-)e''O - (Ap) >Dii(A)ui(';, ) (2.68)

vi(P; 4, -e--ioo(,A) , o-) (2.69) p 0 Dij(A)vi(T;

Choose the standard 4-momentum k = (k0 , 0, 0, k0 ), so:

kO Dii k D U1 (V; , or) PO ( L( p) iiN; , 0-) vi/(TI; , C-) 0 > D1,1(L(p))vi(r'; k-, 0-) (2.70)

The polarization ternsors are assigned as the eigenstates of D(R), which should be two in total, as

seen from the SO(2) part of little group. The full little group transformation is E2 , with rotations and translations in a two-dimensional spatial plane, which is described in equation (1.48):

u1(4'; Ak, o)e" >3 E Jq ( R())ui(P; yj, c-) V(X;AkO-)e-i = ,j(R(O) vj((; ,2-)

(2.71) 'ul(T; Ak, o-) = 3([(S~c, 3))ui(P; Y, a) vI;Ak,-) = > i'i(S(a,3))vI(IF; -,ao) (2.72)

From the polarization tensors, one can get the tensor coefficients, with a prefactor 1 for nor- m2 malization.

34 Restriction on Massless Field Construction

However, it's not possible to find the tensor coefficients satisfy these equations for any general representation of the homogeneous Lorentz group. To see this, consider (A, B) representation with an infinitesimal transformation on Uab [361:

- ii (Jl, ab+ aa(I))V (2.73) Jia'b',ab = k k + oa'a ka'aubb

-+ (J1A) - jA)aa ua'b(C; k, o) = (J'B) Z B))a b(I k, a) 0 (2.74)

Same goes for v, these requires the two tensor coefficients to disappear unless a = -A and b = B. Thus, a massless field of type (A, B) can only be used to described particles of helicity a- (A - B). For later interests, let's consider massless fields of helicity 1 and t2. Instead of using a vector (1, -), one needs an anti-symmetric tensor (1, 0) D (0, 1) to get massless a = 1, and instead of using a symmetric tensor (1, 1), one needs an 4-rank tensor (2, 0) D (0, 2) to get massless a t2 [361. However, if the objects isn't exactly a Lorentz 4-vector and a Lorentz symmetric tensor, then these are still possible to be massless, and we will see that later.

The Building Blocks of Polarization Tensors

Since there's only two polarization tensors for each massless quanta, corresponding to the highest and lowest possible value for the angular momentum along the propagating direction, therefore the polarization tensors of a field with many spacetime indices can always be built trivially by a sum of direct products of the polarization tensors from fields with fewer spacetime indices. The most fundamental polarization spinors and vectors are:

= (1, 0) , _ =(0,1) ; e t (kt) (0,1,ti,0) (2.75)

For example, if we want to have the polarization tensors for a symmetric tensor of spin 2, then:

0 0 0 0 0 1 i 0 E (k, ) = d'(k, )eL(f, t) = (2.76) 0 i -1 0 S 0 0 0

35 However, like mentioned before, one should note that in general they didn't have the right transformation properties to be massless, under a little group transformation in equation (1.48) [211:

e TiYE(, ) = (A-)E"(Ak, +) + (a i )k ; (2.77)

e*2i6 PE(k) (A-)",(A)",eP(Ak, ) + kl(A v )v + kv(A-1v ) , (2.78) 1 1 Va(a = i) c4((,k(a i3) 2(Ak)" (2.79) ,,F2k 0 4(kO)2

2.2.3 Physical Requirements for the Theory

The four Requirements

In order to put quantum mechanics and special relativity together, with some philosophical ques- tions in consideration, the dynamics of the theory, which is encoded in the interacting Halmintonian (the free theory Halmintonian and the interactions) should satisfies four main requirements 124, 361:

1. Hermicity

2. Causality 3. Conserved charge 4. Cluster decomposition

Quantum Field Theory pops out

All requirements can be satisfied with the language of field, and what is known as quantum field theory does pop out from here. Note that, Hermicity is the easiest requirement since it can can always be employed when writing down the field dynamics, so we won't discuss any about it.

Checking Cluster Decomposition

Construct the Lorentz scalar interaction density out of the creation and annihilation fields:

VWx mZ S.l S--- .. (X ) ,j (2.80) '' rn 1 n

36 (JJD1 T,,(A-1) D j.(A~1))g., .. = g ., 1 T (2.81) j=1 1 I...I 11...In i=1 The interactions in the full Halmintonian, from equation (2.31), (2.51) and (2.54):

3 !m - - 3 ) va (W';'1 01)--WM; Pm, Omr), OF1; P1, O~1)---.(Fn; Pn,7 ~n) = (21r)- -P-i_ ) (2.82)

..4,1.. (V'; -5, a'.) vl, (XF ; pA, 9-k)) (2.83) X H 3=1 k=1 . I,...,l' 1 i,..,1m This expression automatically satisfies cluster decomposition. The non-interacting free theory Halmintonian gets this requirements trivially.

Quantum Field

In general, the creation and annihilation fields are not commute, even separated space-like. There-

fore, instead of just always treat them separately, one can try to combine them into a new field, called quantum field, and make the dynamics depend on it and its Hermittian adjoint. For example:

'I'(x) = T- + AI'- --> 'Jab(X) = KI'+ AT- (2.84)

The causality requirement in equation (2.33) can be satisfied, if one can show that it's possible to have the commutation relation, with considerations of the quantum statistics:

(x - X') > 0 -* [P1(x), JTI/,(x')] = 0 (2.85)

At first glance it might seems impossible for such a simple combination can solve the problem with causality, but by some extensions with a new type of quanta known as anti-particle, then it's actually the case. We will discuss more about this later. Also note that, causality can be seen from the free theory Halmintonian easily, because of the on-shell condition for each quanta.

37 Checking Charge Conjugation

The conservation charge Q, which is Hermittian, should be commute with the whole Hamintonian, from equation (2.41). If the quanta carries a non-zero charge qT:

[Q, A ] = qa , [Q, a;g,] = -qap;,-+ [Q, TI (x)] = -Fq,'F'(x) (2.86)

For the whole Halmintonian to be written in i, it's necessary for the field to have a simple commutation relation with the conserve charge:

[Q, TI(x)] = q,'I(x) , [Q, It(x)] = qqI'(x) (2.87)

And one can keep track of the free theory and the interactions with operators of no total charge:

2.-+ q + q 2 +.- - q.. -qp - ... = 0 (2.88)

Equation (2.82) can be taken on by introducing the concept of anti-particle, and like mentioned before, it is crucial for causality to be realized in the theory. The idea is to have a doubling of particle species (which means same mass and spin) carries the charges with opposite signs:

[Q, at )] -qafcg , [Q, ac;g,,] = qapc;, -+ [Q, 'I+] = q'Ic+ , [Q, IV-] - - (2.89)

Generalize by redefining the quantum field in equation (2.79), and this is the form we will use from now on:

Ii(x) = Ki+ + A'- -+ 'Fab(x) -- ab + AT'ab- (2.90)

If the particle is its own anti-particle ap = apc, then it should carry no charge and we get back to equation (2.79). The reason why we want the anti-particle to have the same mass and spin with the normal particle is, since we want to put the particle annihilation (creation) field and anti-particle creation (annihilation) field in the same quantum field, the on-shell condition should be the same, therefore the mass and spin should be in agreement.

38 Checking Causality

We will only consider massive particles since the extension to massless particles is straight-forward

but usually needs lengthy case-by-case analysis. Let's look at the quantum field in more details, with creation and annihilation opertors

Tab(x) = (27r)- J d3a ;veab( ; , ) + AjcU(e-ab (P;p~, a)) (2.91)

The commutation relation can be written in terms of the spin sum E(4I; p):

3 [ x , i(X -2i- J 1 (P 2eip(x-x') IA12ei(x-x') , (2.92)

Eab , ; i) 2p 3 Uab(;,, o- (; j, cr) 2p 3Vab(; P, ab,(W;P, ) (2.93)

2 E >3CAB(SY; cd)CAB(sA JA '' ; (2.94) 01 cd c'd'

0 Eab,a'(P;P) = Eaab'(; P0 , A -E1 (aba'b'(;Af + 2P E2a,alb, , (2.95)

E-a~''1 -P - (- 2(A+I) E1ab,a'b1(q;P1 , 22a,a',61 -A -- (-)2(A+B)+ E 2 ab, a'b,(O; ) (2.96)

For a space-like separation, choose a Lorentz frame so that x0 x'0 , then the commutation relation

becomes much simpler [361:

3 1 [W(, 1 /- 2 (_)2(A+B) 12 f d jT [qPab(X), q'" -ilk-A \EhI) I ab a'b'('(\tiDV)20 (l_3 i (2.97)l i :\b'(X)J

_ (2.98) +(112 F ()2(A+B) 2) 2 ab,ab _j(3) -

This should be vanished when F 4 , therefore:

2 2 IK1 = -(-)2(A+B)A 1 2 T _)2(A+B) -1 12 _A (2.99)

If A + B is an integer then the sign should be +, and if A + B is an half-integer then the sign

should be -. This is nothing but the relation between the quantum statistical of the field with

boson (+) full integer and fermion (-) half-integer nature. With this, the causality requirement

is solved, if the full interacting Halmintonian are made of T, and Pl (cannot be only Ti or Tt,

39 because of Hermicity). Note that, the relative phase arise from the relation Ir,12 - A 2 can always be eliminated by the redefinition of the creation and annihilation operators, and also the overall

scale of the field, hence, in the end [361:

'j(x) = '' +(-) 21c- , 'ai(X) =T+ +() 2Tca- ; (2.100)

I'(x) = (27r)- Z d3(a pePxui(T; , or) + (-)2aac;g6e-xvi(4;5,cx)) , (2.101) Ti(x) = (27r- d a qiug,; a) + ', )2 creg- ~Vs(; 0, )) (2.102) TI'ab(X) =(27r)- ziYf dPfA Ilr;l&Pxab(aIVz7, )a+) (2.102)

Other interesting properties of the field that should be mentioned:

{'i(x), I''(x')}+= 0 , {J/(x), J),(x')}= 0 , (2.103)

{'ab(X), Ja'b'(X')} = 0 , {4Ia(X), ',,(')}I =0 (2.104)

Anti-Particle and CPT theorem

Consider the massive case only, for the same reasons as the previous subsubsection. We start by

constructing a quanta out of spacetime symmetry, then arguing the dynamics of multi-particles

and arriving at the collective behaviors encoded in quantum field with the four physical require-

ments. The concept of anti-particle arise natrurally when considering about charge conservation, which is theoretically used to restrict the possible interactions and experimentally observed in

nature. There's a relation between the normal quanta and the anti-quanta known as the charge

conjugation, or charge inversion, Co which reverses all charges of the the internal symmetry. Thus, the charge conjugation should be anti-commute with the charge operator, but commute with the

Halmintonian, the momentum, the angular momentum and the boost 3-vector in the free theory.

Along the discrete symmetry of spacetime given in equation (1.16) and (1.17), spatial inversion P0 and time reversal 7To, since these are all commute with the Halmintonian, so there're only phases

different in the quanta state between before and after these unitary change of basis transformations:

C0p;P1 or) =,q,, y f; , or) , PoKT; k, 0) =ntl4Fc; -, or) , TOX; , a) =It C; --, -o) (2.105)

apc;1.a, Poa P-- = ta. ,Ta 1= Coal(-)-aq.ta 3_,.XCo7 (2.106) = n

40 Brute-force calculations from equation (2.102), the combined transformation COPoT is an anti- unitary transformation (To is anti-unitary, which is known from quantum mechanics, while PO and

CO should be unitary) change the field in the representation (A, B):

CO 'I(,B) (X)C ,~c ()-2A-a-b-spf(B ,A) t() (2.107)

Po )'x) 61 - , ),(A+B-s 4 B,A)(_, ) , (2.108)

(y 0 ) T4 '(AB)O-1 = ,(-)A+B+a+b-2sq(A,B)

t (CoPo))_1=)(x),(C1Po_)) 7(-)2B,,(A,B) CP =c (2.110)

Set qf't = 1 for any particle [36] (for a charged field, the charge operator can be employed to construct an unitrary transformation which modifies the phase of the field and also the discrete symmetry operator, hence jct). Since any Poincare scalar written in terms of the quantum field

4 1 A,,B', F, ,,'B1) (might be with derivatives) must has EZ Ai and >js Bi are integers, therefore the S-matrix, which is related to the Hermittian scalar interaction density in equation (2.31), is invariant under CPT transformation:

(CoPoTo)V(x)(CoPoTo)- 1 = V(-x) -+ (CoPo7T)S(CoPTo)- 1 = S (2.111)

It should be noted that, instead of setting q't =1 for any particle, one can keep it and show that for every scattering process with the same incoming and ongoing particles, the S-matrix is only shifted by a constant phase, and it's not affected the physics of theory, therefore can be ignored effectively. The invariant of S-matrix under CPT transformation is known as the CPT theorem, and the theorem holds for any order of applying the discrete symmetry transformations, since only an overall phase is produced as the order is changing (T is anti-Hermittian, so any on the right of it should be complex conjugated). A more rigorous mathematical proof in the context of axiomatic quantum field theory yields the very same result [15].

2.2.4 Spacetime Rotation and Particle Self-Rotation

There's an interesting relation between the physics at macroscopic scale and microscopic scale that cannot be seen from classical physics, know as the relation between spacetime rotation and

41 particle rotation. The idea is that if we interchange the position of the two identical fermion particles, the new configuration will be different from the original, and it's actually the same as rotating one of the fermion around only a single time. To see that, note that one can always characterize the transformation into two different type, the active transformation (which acts on the field, generating the self-rotation) and the passive transformation (which acts on the spacetime, generating the change of particle's positions). We will show this interesting realization, at least schematically. The active transformation can be expressed as:

IF (x) -+ Uw(A)'P(x)Ujj(A) (2.112)

Also, write down the passive transformation:

qJ(x) -+ T(A)T(A-lx) (2.113)

Equate the passive and active transformations, one arrives at the relation between the two:

TT(A)U,(A)T(x)Ugjj(A) = T(Ax) (2.114)

Consider a half-integer spin fermionic particle, then desire relation to be shown is the equality of switching place the two particles (which correspond to the passive spacetime rotation 0 = -r) and rotating one of these around once (the active particle rotation 0 = 7r). Note that, for a fermion:

1 1 T 1(0) = A-'(0) , A- (27r) = -1 , A- (47r) = 1 (2.115)

Illustratively, two particles can be represented by field insertions at two different position, say, x. For a clearer description, use the test functions f+localizes at xo and f_ localizes at -xO so that f+(-x) = f- (x), and the spatial rotation simply exchange these 2 functions. Then, let's passively rotate them an angle 7r:

I(Xo)XF(-xo) Jdxjif+(x1)P(x) Jdx 2 f-(x 2 )(x 2 ) (2.116)

-dx'f'(x1)V(x1) fdx'2f'(X2)qj'(X2) (2.117)

42 = dxifi(x)A-'(7r)1(-x1) fdX 2 f+(x 2 )A-'(7r)P(-X 2 ) (2.118)

~ Aq 1 (27r)P(xo)'(-xo) = -I (xo)4(-xo) (2.119)

The result is exactly equal, if we actively exchange these two by a rotation of angle 27r:

41(X)X2(-X) - UT,(27r)AP(x)U--1(27r)w(-x) = A-1(27r)qI(x)T(-x) = -I(x)T(-x) (2.120)

Indeed, there's a very nice way to keep track of the spacetime rotation and the particle self-rotation of fermions, by the topology of a rubber band [101.

43 Chapter 3

Quantum Field Theory

Quantum field theory, undoubtedly, is the best tool for physicists nowadays to understanding the nature at microscopic scales. Start from scratches with a philosophical question of what it means by a particle, and then let the particles interact with others through some guiding principles with great depth of how nature works at quantum level, a field description with quantum nature arise.

In this chapter, we will learn how quantum field theory is formulated by a field Halmintonian and

Lagrangian, and consider some exciting ideas about the theory and interesting applications. We will not go into many details, only briefly discuss the core ideas and intriguing examples. This part can be enjoyable if the readers have some certain understandings about quantum field theory priory.

3.1 The Free Theory Lagrangian

The quantum field is constructed by summing over all particles of the same specific definition with different labels. Each quanta is a physical Hilbert state of the free theory on its own, an on-shell state. From the mass and spin given in the quanta definition, also restriction from the possible polarization states, one can derive a suitable equation of motion with constraints for the field that gives rise to all possible labels of the particle. Then, the field Halmintonian and Lagrangian can be arrived at that describe the found equation of motion with expected constraints. These, in the free theory, should be quadratic in field (due to propagation and the addictive nature of the

Halmintonian when adding more particles in the theory), and should has no more or less than second derivative for bosonic field and first derivative for fermionic field (because of the position of

44 the propagation pole and its residue). The specific form of the Halmintonian and the Langrangian is fixed after taking the four requirements into account along with some physical considerations.

We will not give the details treatment since it can be done easily case-by-case. Rather, we will only give the results from some general cases that are of many interests in theoretical particle physics.

Also, it should also be noted that, in principle, by using the optical theorem given in equation

(2.37) (in details, for a process with a single particle of interests propagates in the middle of it) and the polarization tensors of the particle, then propagation properties of the field can be read-off, and one can get out the free theory Halmintonian and Lagrangian. It's not always straight-forward to use, however.

3.1.1 Massive Quantum Field

Massive Bosonic Field

Consider the massive bosonic field T, ... . of mass m and spin s E N of the Lorentz representation ( E), then the field must be a totally symmetric traceless rank-s tensor:

rfAVA3 ..., = 0 , (3.1)

. T(s)(2r)-3d35(aw;(';j,ab;e-Pxvt...s(P;5,co))(3.2)ei*x'u...,-)+-+i

The value of the Casimir operator C1 is encoded in the equation of motion, and the Casimir C2 requires that all the lower spin values be eliminated, which impossing the transversality condition [9, 29]:

I =-8N ,(L 2- ,l...11(x) 0, OPT2...4L(x) =0 (3.3)

The number of degrees of freedom can be counted to be 2s + 1, which is expected. The free theory

Lagrangian for quantum field T(s) can be written down, with some symmetric traceless tensor auxiliary fields 4 [27]:

LO = d ()( - M) + 2(a2 (s))2 (3.4)

2 s(s - 1)2 (s-2)I (S2)(L - a2 1)1(s- ) + b2 (4 ( 2 2s-1 2 2(35)

45 E c (. 11 ]O-q) - aq2j)s-q ( -q))2 -- M2,(s-q) (s---1)) , (3.6) q=3 k=2 1

2 aq q(2s-q+1)(s-q+2) b (s-q+2)2 q(s-q+2) (s-q+4)(2s-q+4) 2(2s - 2q+ 3)(s - q+ 1) 2s-2q+7 q 2(s - q+ 3)(2s - 2q+ 5) (3.7) This is known as the Singh - Hagen Lagrangian for bosonic fields. It can be checked that the Lagrangian satisfy equation (3.3).

Massive Fermionic Field

For the massive fermionic field T,1.../, of mass m and spin s E N + 1 of the Lorentz representation

+ , - then the field must be a totally symmetric rank-(s - Dirac tensor-spinor: 1 7AI-2-... n =0 , n=s- (3.8) 2'

,41..., (27)- d3P a ~ U,1... P(;C, uec-) + a i Pxvpi...n(I; yo)) -+ )

(3.9) The equation of motion and the transversality condition can be read-off [9, 291:

0 = 948, , A(i1- -0)111... /(n) =0 , 0" TP'-2... P.(x) =0 (3.10)

The number of degrees of freedom can be counted to be 2n + 2 = 2s + 1, which is expected. The free theory Lagrangian for quantum field T(n) can be written down, with some symmetric traceless tensor-spinor auxiliary fields <) and <' [281:

LO- A LL)dx'n' -+ 2c49- 1 (&((+)) (3.11)

+ 2n2 fll)(if + a1Mk)

b2+ 24(-n-) - b2 ( + a2 AMq)<(- (3.13) n q-1

+(-)q ( Ckbk) ( 2 "-q) (0(n-+1)) _ g (n-q) + aq_ TMIM -) (3.14) q=3 k=2

46 -bq 1(,$'(n-q),(n-q) + ("~q4)/(n-q)) - bqI(n-q) (i + aq A1,)

2 n + 1 bq- (q - 1)(2n - q + 3) _ 2(n - q + 4) (3.16) S n- q+1 2n - 2q+3 ' 2n - 2q + 7 This is known as the Singh-Hagen Lagrangian for fermionic fields. It can be checked that the Lagrangian satisfy equation (3.10).

3.1.2 Massless Quantum Field

Massless Bosonic Field

Similar to the massive bosonic field, but the free theory Langriangian is significantly simplify, since all of the auxiliary fields decouple except for the one with rank-(s - 2). Combine the symmetric traceless rank-s and rank-(s - 2) together into a single symmetric double-traceless (not single- traceless) tensor A ... A [12, 291:

rfr[IV 77..OT=0 (3.17)

1 Lo f dX( T1.'.-T...s +)(N1t2...11)2 + 2 , TPVlJ . PO'W 'P... Ps (3.18)

s(s - 1) V32prh,3...2 + s(s - 1)(s - 2) (aPqW . )2 (3.19)

This is known as the Fronsdal - Fang Lagrangian for massless bosonic fields. Counting the number of the degrees of freedom, one gets 0 for s = 0 and 2 for all s > 0. There's an ambiguity in the theory, which can already be seen at quanta level as the freedom in the transformation S(a, 3) given in equation (1.49) and demonstrated for massless spin-1 and spin-2 quanta in equation (2.77) and (2.78) as "strange" pieces proportional to the propagation direction (we will discuss about it more later). That's the gauge freedom that leaves the Lagrangian unchanged:

TA1-... A(X) -+ TP,...,(+D..( ,

The gauge parameter A is a symmetric single-traceless tensor of rank-(s - 1). For later interests, let's consider the cases of massless spin-1 and spin-2 field, which is later known as photon electromagnetism field and graviton gravitational field, for later interests. After some integration by

47 part, the massless spin-1 and spin-2 field free theory Halmintonian and Lagrangian can found:

4 -+ A , - 4L1 FtvF Fp, = OpAv - vA, , SAY = o,A(x) ; (3.21)

Tpy, - hjv , L 1=)O"h"POhvp + &"hPV0phPv + 21hOlh + hO'Ovhj, , (3.22)

0 2 2~V&h~rhg h = Vhv, 1j # 0 , 6h,,(x) = 0,,A,(x) + &A,(x) (3.23)

Massless Fermionic Field

By taking the massless limit of the fermionic Singh - Hagen Langrangian, all auxiliary fields decouple from the theory except for the ones of rank-(n - 1) and (n - 2). Combine the symmetric traceless rank-n, (n - 1) and (n - 2) tensor-spinor field together into a single symmetric triple- traceless (not single- or double-traceless) tensor-spinor 'I' ... , then one arrives at the Fronsdal - Fang Lagrangian for massless fermionic fields [8, 291:

'-I"'I/A P 4... A - 0 (3.24)

L 3=d(P-1 ...+ i 12../n . (3.25)

4 -

3 ~.-An) (3.27) 2 ( VJ i3/PP3 . An3_.(9PW

The number of degrees of freedom is always 4, for all spin. The ambiguity gauge freedom has a symmetric single-traceless tensor-spinor gauge parameter:

An (x) -F1...-- qP, 1 .... ,(x) + 6'fP 1 ... y (x) , (3.28)

J A1 ... a(x) = 0(/1AP2 ... in)c(x) , 7AAI/ 2 ... pn-I = 0 (3.29)

48 3.2 Collective Behaviors of Quantum

3.2.1 Gauge Freedom

The degrees of freedom in each quanta can be understood through the polarization tensors, which is encoded in the tensor coefficients of the field, a collection of all possible physical particle modes. As it is mentioned before, one cannot build a massless particle out of a general representation of the Lorentz group, and at quanta level, it can be understood that the polarization tensors are not transformed under the little group as it should be as Lorentz tensors in some cases. Nevertherless, the particle, although not really a Lorentz tensor are indeed realized in nature. In order to make the theory with these massless particles consistent, one has to associated the strange "pieces" that comes out after little group transformation of the polarization tensors to be unphysical, hence must be ruled out of the theory. Thus, the ambiguity in defining that massless particles becomes the ambiguity of the whole theory.

Gauge symmetry

Consider the cases of massless spin-1 and spin-2, the transformation in the polarization tensors given in equation (2.77) and (2.78) can be seen from the field, photon A,, (A = Ac) and graviton hl, (h = hc), as:

U(A)A" (x)U-1(A) = AAv(Alx) + O"A(x) , (3.30)

U(A)h"(x)U- 1 (A) = A,"A,"hPv(A-'x) + Ak(x) + "IA"(x) (3.31)

Note that, from the quanta point of view, the unphysical degree of freedom is just the longitudinal part induced to the polarization tensors. However, from the field point of view, the strange "pieces" are 0,IA(x) for the photon field and O(PM<(x), and in general, the specific forms of these are horrendous. For example, the shift in the photon field is both Lorentz transformation dependence and field dependence:

/A(X) = -i(2r)-2 E ~ /d 75 1((a - (ae - ioui3aW> ei); (3.32) f V~Ovcr 0()a+A; 4 .aeipx 'AAXI

Therefore, instead of having the unphysical degrees of freedom to be a specific tensor function A and A', make them a generic tensor function, and then agree with the results found in equation

49 (3.21) and (3.23), which come from the massless limit of massive particles free theory. Thus, these equations are used, legitimately, to describe the free theory of massless spin-1 and spin-2 particles. The ambiguity of the theory, which is the unphysical degrees of freedom, is known as the gauge freedom. The freedom is local, since it can be different at distinct positions. By couple the field with this gauge freedom (the gauge field) to other field in the theory, a symmetry associated with that freedom can be realized, hence the theory has gauge local symmetry. This is indeed desirable, since it can be viewed as the internal symmetry of theory that helps to put a constraint on possible interactions. Note that, gauge symmetry is the symmetry realize at the level of field, as the collective behaviors of quantum.

Gauge fixing

Gauge fixing arises naturally as the technique to get rid of the gauge freedom and take out only the physical part, so that meaningful dynamics can be read-off from the theory. It is found that the Hilbert space of a gauge theory is defined by BRST symmetry (a symmetry that replacing the gauge symmetry by adding ghosts in the theory and let these absorb all the unphysical information, such as the longitudinal polarization tensor of the gauge field) or to be precise, BRST cohomology

[2]. The method of using BRST cohomology to do gauge fixing is known as BRST quantization, and it has interesting topological origins [381. Note that, this complication comes from the fact that we want to use the language of field to described the theory. In operators and states, one can always specify the physical Hilbert space from the very beginning, and build out consistent interacting theory with creation and annihilation operators. However, the good thing with field formulation is that one can imposed internal symmetry easily, and that symmetry is not there for the only purpose to cancel the unphysical degrees of freedom of the gauge field. Another way of thinking, the internal symmetry is already in the theory in the first place, and massless gauge fields, with all unphysical polarization tensors ruled out, is what come along as the indication of the gauge symmetry.

Is Gauge Symmetry Physical?

The gauge freedom is associated with unphysical degrees of freedom. One can always artificially add a gauge symmetry in the theory, since it's just a redundancy of the description for the dynamics.

Although one can thought of gauge symmetry as a way to constraint the possible interactions,

50 which has a very physical implication, but it should be noted that the very reason why we need restriction in the first place is that the mathematical origin of the formulation has more than what the physical theory needs. So, the final verdict is, gauge symmetry is not physical.

3.2.2 Renormalization

Renormalization associated physical meanings to mathematical concepts can be read-off from a theory. There are many uses of renormalization in theoretical physics, such as in statistical field theory [16], but here we're only interested in the context of quantum field theory.

Process of Renormalizing

In the language of perturbative quantum field theory, which encodes the dynamics of particles and fields in a diagrammatic form known as the set of Feynman diagrams, divergence usually arise because of loops since the integration over the momentum can go up to gigantic scale with no limitation. In order to make the calculation well-define, one usually let the divergence at loop levels to be cancel by the physical quantities at tree level (in technical details, bare terms and counter terms), and that process of cancellation is known as regularization. In other words, regularization is how the theory can be made finite, and all the calculation for the quantum contributions at any tree or loop orders can be given with finiteness.

The validity of the perturbative description of the calculation with quantities after regularization is that as one goes higher in loop order, the contribution decreases, and these all converge to a specific result. But usally that's not the case, since perturbative series in quantum field theory are usually not convergent but rather asymptotic, hence after a certain loop order, the contributions grow larger and larger. Most of the time, the super-complicated and time-consuming loop calculations don't allow us to reach up to that point (indeed that's the case with quantum electrodynamics), but in a strong interacting theory (such as quantum chromodynamics), some calculation goes on that side track at 3-loop order [30]. If the calculation is known at all orders, one can use analytic continuation, such as Borel transformation, to make the final result finite, since indeed the divergent is usually an artifact of the perturbative description. Usually, quantum field theory isn't that nice, and we have to make predictions base on the calculation that regulated all higher loop orders after where the divergent starts (it's not

51 mathematical rigorous at all, but still pretty much the best we can do, and as expected, the predictions can be very wrong, such as the quark-quark potential in quantum chromodynamics predicted from perturbative side of the theory).

It's important to understand that regularization removes the physics at UV region, not IR region. There are two different type of divergences, one at IR region and the other at UV region, and both of them can be physical. While UV divergence can sometime be unreal due to the lack of knowledge in the degrees of freedom in the theory at high energy scale, IR divergence is always physically meaningful, and it's indeed play a very important role in effective field theories since the IR behaviors should agrees with that of the original full theory. Thus, infinity isn't always bad, it's just that when one needs to make a connection to observables, the results should better be finite.

There are cases when the physical quantities actually goes off to infinity or becomes badly defined, but that's usually when the nature of that quantities cannot be understood normally any more as the physical systems has a change in phase, such as the quark mass, which is can be read-off physically as the pole of the quark field propagator, lost its meaning as the quark cannot exist as a single quanta normally under the confinement energy scale.

After regularization, the calculation is finite, but there's still one more step to come out of the hard-core maths to the reign of the physics world. Renormalization scheme is the way we choose to define physical meanings to specific calculated object, and with the physical quantities are renormalized, we conclude the process of renormalization. It should be noted the observables are not scheme-dependence, at all, and even if the choice of renormalization is bad with huge divergences appear, usually these are all cancelled out the others hence the observables are finite.

Indeed the divergences are artifacts from splitting up the physics at different energy scales.

If the number of physical quantities need to absorb the divergent is finite, then in principle, it can make good predictions as we climb up in the energy scale, at least as long as we reach where one of the quantities blown up, such as in quantum electrodynamics for the gauge couplings at the Landau pole energy scale. If the number of physical quantities is infinite, it's lost the predictive power, but still work in a certain range, and indeed, this is the key principles of the effective field theories. The theory in the later case is said to be non-renormalizable.

52 Keeping track of Regularization

For an integration Jy of a graph 'y with external momentum p and internal momentum k which are integrated over, define the Taylor expansion operator t of order N in the external momentum:

J .1(pi, (2,7r)4...I ( ...;O,k i,...) (3.33)

lN n! 'n DPnI n=O The counter term of a subgroup can be described by the operator tY, then regulated integration is defined up to a polynomial in the external momentum of degree D-, which is the freedom of regularization choice [5]:

D~y i-t = (1 - tY) J_ , J-Y -+ j-Y + !fd..,p .. (335 n=O

In dimensional regularization, the divergent can be followed with the dimensional variation E -* 0:

00 00 J S a.," , j(7) = (1 - ty)Jy =, j(y) an j(y) +d (3.36) n=-D-f n=O

Now consider an arbitrary Feynman diagram G, then a subgraph -Y C G (can be the full G or the empty 0) is a renormalization part if the degrees of divergent is non-negative D ;> 0, which means superficially divergent. Two subgraphs are disjoint if:

71 n72 = 0 (3.37)

If one is contain totally inside another, then these subgraphs are nested:

71 C 2 72 C 71 (3.38)

The subgraphs are overlapping if these share lines and vertices:

!( 1 n 2 = 0) ( !(_Y1 'C 2) ( !(2 c -Y1) (3.39)

53 A set of non-full renormalization parts y # G is a forest, a set with all non-full renormalization parts is a full forest F and sets with only disjoint and nested non-full renormalization parts is non- overlapping forests Ta. Weinberg's power counting theorem states that by eliminating potentially divergent contributions from a full graph, the leftover should be absolutely convergent. Therefore, after regularization, one arrives with a finite answer with the Bogoliubov's R-operator [6]:

RGIG fl(I - t,) IG , RGIG = (1 - tG)RGIG (3.40) 7EF

The order of carrying out the regularization for subgraphs should be inside out, but it's not appli- cable to overlapping subgraphs. However, one can get rid of that problem, since any divergent of the Feynman diagram cannot be non-local, and one arrives at the Zimmermann's non-overlapping forests formula [39]:

RGIG = (1 - t G) ( ( ))IG (3.41) a tE.Fc, With this formula, one can keep track of the regularization, for any Feynman diagram. This is a very crucial part of the standard Bogoliubov - Parasiuk - Hepp - Zimmermann renormalization method.

An example: Quark Mass Ambiguity

As mentioned before, renormalization scheme is the way we interpret the mathematical results physically. A bad choice of scheme may lead to a bizarre answer, and we will demonstrate it through the axample with quark mass, by considering a set of diagrams that's can be calculated at all loop orders, and analytical continuation can be imposed to get a non-perturbative result.

Quarks, at quantum chromodynamics scale, are at confinement phase, which is a collective behavior that's not allow a single quark to be alone in the theory, therefore, it can only be found in bound states, such as baryons and mesons. Indeed, if one try to define a quark as a particle in a conventional way, and read-off the pole mass from its relation with the MS mass (which comes out from just regularization of the Langrangian in the MS scheme), then an ambiguity of the size of the quantum chromodynamics scale arises if the selected energy modes for quantum contributions is not treated with cares [30].

Consider a particle subset of Feynman diagram in quantum chromodynamics, the bubble chain

54 (a gauge vector field chain with fermionic bubbles) correction to the propagation of quark. Indeed, the bubbles sum diagram is unique in any order of perturbation theory that gives gauge invariant contributions to the flavor or color structure of the theory, and it has the most power of

nf (the number of active fermions running in each bubble). The fundamental ingredient after

regularization, the bubble (in Landau gauge) is [30]:

Dab(p,a.) = Z - PPV) 6 ab ( OOCS)l1- e (3.42)

2 g 4167r 30 a, 5 2 11 4- )00 =-n+-CA (3.43) 92 = 47a, = -

Taking a Borel transformation of the sum of all bubble chains (ignore the 6 ab):

0X) n+ 00 n 1 f(a) = fna" = due-uF(u) , F(u) = f_ 16(u) + fnu ; (3.44) n=-1 n=O

GtwPas l4v) E (/0o nnn( _ 00 4 (45 GpV (p, as) (=2 (p) - P2 nn" 41r P = 47 (3.45)

167r 2 (9 2(G ( p, U) = 2 ("2 PPv) ( ec ) (3.46)

The mass pole nmz can be read-off, after sticking the bubble chain sum to the quark propagator

with the MS mass fi [301:

2 4 + i4V'u(p) 2G ab (3.47) 11 (p ffi, U) -iCF k i(p)'t(fi + 2 J (27r)4 (p + k) 2 + (3.47)

2 - CF ( ecu6(1 - u)F(u)r(1 - 2u)+ 67ro 2 F(3 - u) + ... (3.48)

-u) (u)F(1 -2u)( CF A2 c U6(1 -+ M,('u) = (n6(u)- /o(j ) 7( )+ ... ) (3.49) 67r,30 AM2 17(3 - u)

The 6(u) part comes from transforming the factor 1, which corresponds to mr = f- at lowest

order, to Borel space. The omitted terms contain both the terms where there is a pole structure

of the form 1 rendering it regular at u = 0 and terms that regular in u, which are not needed for

the analysis of interests. Prom the F-function structure, the strongest pole, which is closest to 0, is at u = ., corresponds to a renormalon with 2"n! growth. The mass pole can be further

55 simplified to express the effect of that pole in more details, and inverse Borel transform it:

CF pei 2 0\ 42_ MA(u) = 11 6 7 3 (M ) + MP(OS) due -as MA (u) (3.50)

Because of the u = j renormalon, there's an ambiguity which is realized by analytical continuation above or below the pole, hence half of the residue around u =1:

AmP(a) = Res (MP(u = ) = 1(2i17u)i ()= CF = C C

(3.51) Since the pole mass has this ambiguity, one should avoid this choice of renormalization scheme for mass. Note that, the mass pole uncertainty doesn't depend of the use of in and it's independent of the regulated energy scale p, which is the evidence for the fact that it comes from the bad choice of separating physics at different energy scales, and to be precise, IR region [30]. To cure the ambiguity, one has to introduce a new energy scale R, and in general a scheme change gives:

mp = m(R) + Am , Am = R alnk ( ) (74 (3.52) n=1 k

The mass m(R) can be chosen to be free of renormalons, if Am properly substracts the pole mass renormalons. The physical picture of the new renormalization scheme is that the energy scale R can be considered a floating cut-off which sets the scale for absorbing the IR fluctuations

(that causes the instability by dressing up the pole mass) together with the pole mass to yield a well-defined mass m(R).

It's not trivial why the origin of renormalons is in the IR region. To understand this, one has to note that the MS scheme separates short and long distance physics for logs correctly, but for powers it relies on setting the scale of integration to zero, which is forced from the very definition of dimensional regularization and the scheme itself. This treatment leaves residual sensitivity to power divergences from including the wrong regions of momentum space in the quantum field theory integrals, which results in renormalons [30]. The problem wasn't in Wilsonian picture of renormalization, however, since Wilsonian has a hard cut-off so in general the calculations are very difficult (not to mention that the symmetries will not be preserved order-by-order). One can always start with the MS-scheme and perturbatively go toward the Wilsonian picture in a

56 Lorentz and gauge symmetries preserving way, and indeed, it is the case with the so-called

MSR-scheme [30].

3.2.3 Long Distance Physics

The long distance physics in quantum field theory is dominated by the exchange of massless particles. For massive particles of mass m, the estimated interaction range from the uncertainty principle in quantum mechanics is nothing but ~ -, hence dies off at long distance. It can also be seen from the tree-level Born approximated potential, which the potential is decayed exponentially

~ mr (along with a negative power of r) with the separation r. This should be the case, since the physics of quantum field theory always decouples the massive degrees of freedom (dynamically inactive) as one goes to an arbitrary small energy scale, passing the mass thresholds. Here, for integer spin field, we will discussed about the decouple of spin-> 3 as a possible soft-exchange from the theory at IR region. A generalization to massless half-integer spin field can be done, also yields the decouple of spin-> 2' at long distance scale [3]. The implication is that particles with spin more than 2 cannot have couplings that survive at low energy limit.

S-matrix with Massless External Quanta

Consider a scattering process in which a massless -y particle is emitted with momentum q and integer helicity states h, then under a Lorentz transformation, the S-matrix should be unchanged

(in general, the S-matrix for emission and absorption of several massless particles can be treated similarly). The S-matrix is simply refer as Sh(q, p), with p stand for all the momentum and spin states of other particles, well, schematically [35]. From equation (1.53):

Sh(,p) = ite*O(AS(Aq, Ap) (3.53)

Use the standard 4-momentum trick with qO and po, so that a Lorentz transformation AO can carry these to the momentum q and p of interests. Since in a general physical process, the only Lorentz transformation that leaves both q and p invariant is the identity, so the carrying transformation

Ao is uniquely determined by q and p [35]. Let's try to write down the S-matrix in the following form, with the polarization tensor of any massless quanta with integer spin can be built purely

57 from the known polarization vector given in equation (2.75):

1hI qa(, =t 1) t ( Aa..,,(q, p) ,E'(ql) = Rl,(q)cv (3.54) j=1

Some useful properties of the polarization vector can be lists [35]:

CP1t(q) = CA ,) (0qheI(q) = 1, Ef(gjIE,1 (q) = 0 , (3.55)

EZ(qE'I(q)h h = 'rf" + qIqV I q12+ q'q" , qI = (Iq1, -q') (3.56)

Also, equation (2.77) can be rewritten in a more usable form as:

A iAv (q) = ih(A q-) (3.57)

With these, one can construct the standard "M-object" from the standard S-matrix:

E(q o))Sh(_, PO) 1h (q , po) = 2Ig- ( " (3.58) j=1

Define the general "M-object" by Lorentz transformation, hence it's a Lorentz tensor:

hi Mlh (, P) = (Ao)"j, ( q,T P)) A"hI' ( qO, , PO) (3.59) =1

jh Ihi IhI (Ao)PkVk Ap)) Mh7.~P~h (q, Ap) (Ao)lj V (q' P)) HQ (q, A,PP) MhI (IhI A36) -(Fl k=1 j=1 (3.60)

This expression for the M-tensor satisfies equation (3.54) for all possible q and p. To see that, rewrite equation (3.57), then rewrite equation (3.59):

-ihO AO q, qI(Ao OV~*P C'(q) = e (A1(, p) - E (q) (3.61)

Ihi

-- ( h 1 ...ijhl(qj p) (3.62) j=1

58 ihe A-(-) -(, J| qhq ( )(A )O t e (q0) -- 0hh(o, po) (3.63) M k=1 Using equation (3.55) and equation (3.54):

A 0 (3.64)

-+ (jj !() M ... = e(A ) (qic t(ql.))M. (AI, po) (3.65)

j=1 k=1

1t J(q) q, ) = A SI(qIo po) (3.66) -- + E

_2-I (JJhqJj1 ~ MhB. A~hI qP)=

Setting A = A- 1 , then one arrives back at equation (3.54). Now, after knowing that the "M-object" is just a Lorentz tensor (even symmetric), so the S-matrix Lorentz transformation infinitesimally

can be read-off: A ,1 + W", ,(3.67)

Sq, )= ihO(Aq) IT' - (Aq) PiAvOEh (Aq) t q,

= 'viI eiO(ASh ( q, Ap) - WE ( IE )A'i... ,,, (-, p) (3.69) -I/21 j=2

To get the right transformation for S-matrix, then [35]:

ihi qh )(AJhMl)... Afhp (Aqp) =0 (3.70) j=2

This imples the on-shell gauge invariance, since from the equation above, S-matrix should be

invariant under a regauging of the polarization vector with an arbitrary parameter:

EI(q --+ El(q-) + Ah(q)qP (3.71)

For a massless vector field with spin-1, then one arrives at the Ward - Takahashi identity:

q'Ai,1 = 0 (3.72)

59 For a massless tensor field with spin-2, then:

qlEI iA2,v = 0 -+ q"A'i 2 11 q" (3.73)

But because of equation (3.55), a shift in M,, proportional to q,, will not change the S-matrix, so one can always get the choice of M-tensor so that:

q"Ma22MV= 0 (3.74)

Soft-Exchange of Massless Quanta

To describe the long distance physics, one needs to go to the IR region of the quantum field theory.

For a high-energy scattering process Ej -+ E, the emission of a soft-massless quanta of integer

helicity h and momentum q- -+ 0 from a particle of mass mj and spin sj (use rg to specify incoming

+1 and outgoing -1) in the process, as the internal line is off-shell (very close to on-shell) with

mementum pj - q and spin state 0' and the external line is truely on-shell with momentum p and

state o-, contributes to the S-matrix:

(pj+q)2 +m 2 2pjq pjq (3.75) Note that, there's also possible contrubition to the scattering S-matrix from emission of photon

from the internal lines, but these aren't singular at the soft-limit, therefore we neglect [35]. Indeed,

2 2 external particles are on-shell, so when q --+ 0, the propagator diverges as (pj + q) -+ n , while

for virtual particles, since these aren't on-shell in the first place, so in general divergent of that

form doesn't arise [21]. From equation (3.70):

, ..)j8 e ( =0 (3.76) > > 3( ... q,) 1 j=1 k=2

Thus, the only possible form for the M-tensor so that the above can be true for any input:

q"'M,1...Lh(sj;pj, o-j, o') Vq -- -) ( k7j, ) (3.77) k=1

60 The constraint now becomes:

|h| Z)f ,h"((s;pjo-, 0 (3.78) j , k=2

As the soft-momentum goes to 0, one expects a smooth change so that the scattering process should know nothing about the interaction between the high-energy particles and the massless soft-particle, hence there must be a factorization, which can only arise if [351:

|hj ff hI)(s.;P. p, a',,) = f (h|)(S'; P_) ) f( (s; pj) =0 (3.79) j k=2

The value of f(IhI)(sj; pj) can be defined to be the renormalized charge under the coupling with the massless particles. With M-tensor is now known, the soft-interaction vertex can be read-off [35]:

_2i (2r)%,,CT (pjEc~t)jhi V(h) - ) h __f -( (p)=1p fy (sj; pj) (3.80) (27r) /'2Ij 2p 0 Let's look at the interaction ,with the exchange of spin-i and spin-2 particles, of two particles with mass ma and mb, momentum Pa and PA, spin-1 coupling ea and eb, spin-2 coupling ,/87Gg. Let the momentum transfer t = -(Pa - Pb)2 -+ 0, then the S-matrix is dominated by a single spin-1 and spin-2 exchange, and can be calculated at tree-level (ignore 6C,,) [351:

___12 2 mamb) S(Pa, Pb) = 41 t eaCb(PaPb) + 87rGg ((PaPb)2 _ 2 2 (3.81)

Choose the frame where particle b is at rest, then:

S(Pa,Pb)= eaeb + 87rGg(2po - a gmb (3.82) '7rt 4ra 0)

As the massless spin-1 particle is photon, and the massless spin-2 particle is graviton, hence ea is the electrical charge and the effective gravitational mass f is:

fila = g(2p - , p (ma, i) -+ ma = gm7a , mb = gmb (3.83)

61 With the physical meaning of the charges is related to classical physics, one can take a look at equation (3.79) and interpret the found constraint. For spin-1, it's the conservation of electrical charges (the whole arguments can be generalize to non-Abelian gauge group):

1 1 7 jf ( )(s;p) = 0 -+ Zf )(sj;p,) = f )(s; p) (3.84) in out

For spin-2, the only possible constraint for ff2 ) is that it should be universal, so momentum conservation leads to satisfaction trivially. This is the interactions point of view of the equivalent principle in general relativity:

2 2 njtpff (sj; p) = 0 -* f} (s2;)p.) = f ( ) (3.85)

For spin-> 2, in general there's no possible f('), or in other words f(>2 ) = 0, therefore these massless particles should be decoupled from the theory at long distance scale, low energy limit.

3.2.4 Maxwell's Electromagnetism and Einstein's General Relativity

Maxwell's Electromagnetism and Einstein's General Relativity can be seen from the part of the

Lagrangian contributed by the dynamics of the massless spin-1 photon field and the massless spin-2 graviton field, given in equation (3.21) and equation (3.22). Also, long distance physics, which closely relates to the reign of classical physics, does give-off the nature of electromagnetics (charge conservation) and gravity (equivalent principle), which is described in equation (3.84) and (3.85).

The Maxwell Lagrangian is exactly given, while the Einstein-Hilbert Lagrangian can only be seen perturbatively. However, the important underlying nature of these two classical theory can be understood through the symmetry (in which the classical field equations are read-off from), and indeed, the gauge symmetry of the electromagnetism vector potential is the same as the gauge freedom of the photon field, while the diffeomorphism of spacetime, which can be encoded in the metric, agrees with the freedom in defining the graviton field. Therefore, quantum field theory gives rise to electromagnetism and gravity. It should be noted that the quantum field theory for perturbative gravity is non-renormalizable. It's still work well as an effective field theory of low energy scale, but to look more into the UV region, one needs a new method. Superstring theory is the way to put quantum field theory and gravity consistently at perturbative level.

62 An Example: Geodesics from Field Theory in Curved Spacetime

The particle-wave duality in quantum mechanics comes from the fact that a single particle behaviors are described from a wave equation. In classical field theory, a particle can be described by a localized field configuration, and the classical field can be realized from the quantum field theory point of view, even with the discreteness of quantization. In this example, we will use this realization of quanta in a classical field language to verify that quantum field theory in a curved spacetime background indeed does have the features of general relativity, and in details, we will verify that a free particle should move a long the geodesics. We will consider massive particle only, and skip the generalization to massless one.

In general relativity, the reasons why a particle is moving along the geodesics can be understood from the Lagrangian point of view. Since the action is proportional to spacetime interval, by extremize it one gets the geodesics:

S[s] -m ds = -m JdrL = -m f dT V/gpTOX9XV (3.86)

d 2 Xp dX" dXO 1 dL dXA S+ rp - = 0 (3.87) dr 2 aO dr dT L dr dT '

d 2Xp dX"dXO dr ~ ds - 2 + F'3 = 0 (3.88) dr2 " di dT

Now, let's look at the problem of classical field theory in a curved spacetime. Consider a real scalar field action, the equation of motion can be read-off:

SA[ - _'fd4(g," O4"( + m2 2) (3.89)

+ (09 - M 2)4) 0 , E]4 = 1 &( 9g"J4) (3.90)

Schematically, a localized wave-packet can be used to describe a particle. With A is the size of the and r is the curvature length scale of the spacetime background, then a good description for a particle from the classical field point of view is:

so A A (x)4 ei =1 > - ~-+ 0 , kA= -01,0 , Vk, = Vk, (3.91) E

63 The equation of motion is now change to:

1(,A -&O#+... =0 A " - m2 + + (3.92)

At the leading order -, one arrives at the on-shell condition of the particle:

(3.93)

At the next leading order , the size of the wave packet A(x) has the evolution:

1 k ,, ln A - ( k) (3.94) Vc,--

From the metricity condition, one arrives at the equation of geodesics flow:

V,,8 = 0 V, (k2 = -m 2 ) = 0 = 2k"V,k = 2kV ak = 2VkkA (3.95)

The position of the wave packet can be seen from the flow with respect to the proper time T, then once again, the geodesics equation arises:

3 dX" d2X 1 dXa dXl (3.96) dT dT 2 aO dT dT

3.2.5 Classical Field Configuration

Since the physical Hilbert space is completed, therefore any physical field configuration can be

made, in general, by superposition of multi-particle states. To make a connection the classical

world of physics, let's construct a classical field configuration, and for simplicity, consider a free theory of massive scalar field:

H = d X_((O)2 + (G)2+ m242) 2] ~' (3.97)

We want to build a classical field state:

(1)1) = ID( 0)1 )) (3.98)

64 One can always try to construct the Halmintonian and Lagrangian from the creation and annihilation operators, but since it's a lot simpler to deal with functions, usually one choose to start with

the Halmintonian and Langrangian first, find the degrees of freedom and the associated canonical

momentum, and then use canonical momentum to quantize the theory, thus get the operators and

states language back again. The field and the canonical momentum, represented as creation and

annihilation operators:

rl(y) = -i da 23x2(2 (a# - at g)eigx (3.100)

In a functional form, the canonical momentum can be written as the derivatives with respect the the field, and vice versa:

Z') -+ _(x)#) =- ) , b(X)1) = [<(5),-i,(,)i|) f(i')](3.101) = ((l -

Hence, it's straight-forward to build up a classical field state from the vacuum IQ):

I|) = eif d3 (x)r(x)I) = exp d(Id ) d 2(2w) (ap.- aLe) (3.102)

To see that, let's act 4i(Y)-operator on it:

3 3 = JDr ) (7r -I_z,(: f d ( )7(i) IQ) = JD17) (wj.(x)eifd3 Y )(Y) 7(Y) (3.104)

= f Dw Kr(i()ef "(i)|I) = 5(Y)eid3 I(')H()IQ) (3.105)

3.3 Functional Integration

As mentioned before, the S-matrix language from particle construction of quantum field theory

isn't good for short distance study, but with field formalism one can always theorize the theory

65 first (usually with the gauge symmetry, since it puts the constraint on possible fields and interactions one can write down) and read-off particles out from it then try to match with observables in experiments, instead of start with the particles and then guess the underlying degrees of freedom. Canonical quantization is the method to get the quanta out of the theory, and generally very useful for understanding physical interpretation at microscopic scale. The calculations with quantum fields are fairly simple for weak interacting theory, such as quantum electrodynamics, and the predictions come out are generally good, but the same cannot be applied for interacting theory, such as quantum chromodynamics, since the perturbative description is ill-defined. Hence the analytical results, even just approximated ones, are hard to see, and it's desirable to have a numerical method for quantum field theory.

Functional integration is a different way to define a quantum theory, where instead of quantum fields as operators and physical Hilbert states one has to deal with the quantum fields as functions, which is more familiar and easier to keep track of. It's not only good for analytical purposes, but also can be formulated - to be precise, reformulated - numerically in a very well-defined way, with some guidances from constructive quantum field theory as the theory is

Euclidean. Short distance physics can now be seen with correlation functions in functional integration, where insertion of fields can be anywhere, as the need of the non-interacting picture for initial and final states in S-matrix is not needed. In other words, functional integration is usually a more convenient choice of describing a quantum field theory than the canonical method.

It should be noted that, in most cases of physical interests, functional integration quantization and canonical quantization are equivalent. The quantized nature of functional integration is encoded in the the quantum field itself, and the counting of quanta in non-interacting theory can be track down by the number of quantum field insertions inside the integration of functional. For interacting theory, one has to connect it back to canonical quantization to see the quanta more clearly.

3.3.1 Connection to Canonical Quantization

Let's derive the functional integration from canonical quantization, and then see the most important aspect of canonical method, the commutation relation, again from functional integration point of view. In a general physical system, start with the the choice for degrees of freedom, the action

66 S in a spacetime volume V"3 can be written down since the behavior of the system, by definition, extremize the action. Hence, the Lagrangian density L can be read-off, and the Halmintonian density W at a time t in a space-like hypersurface slice Vi comes right after by introducing an unit timelike vector N,, = (1, 0), then thus canonical quantization.

The Importance of Space-like Hypersurface

Indeed, only a space-like hypersurface can provide initial conditions for the time evolution. This statement is true in a general curved spacetime. If the initial hypersurface is not space-like, then some events on the surface will be in causal contact with others, implying that field values cannot be chosen independently. This violates the requirements of phase space that the variables might be chosen freely on an initial hypersurface. It's not only a bad idea to do quantization on a non-space-like surface, but also impossible to formulate a well-posed initial value problem, which is true in both classical and quantum physics.

Another way to put this, is that cannonical quantization can only make sense on a space-like hypersurface. Typically, a slice of constant time is already a space-like hypersurface, but sometimes it's not, such as the choice of time in Schwarzchild metric spacetime. Strictly speaking, there's no need to use a slice of constant time, and just a space-like hypersurface is enough (of course, it can always be defined as a space of constant time in appropriate coordinates). Quantizing on a time-like hypersurface means trying to quantize on a surface of constant position in space there, which has to be bad. There's also a technical issue with non-space-like initial hypersurface, that the Laplace operator (for the fields equation of motion in usual cases of physical interests) is no longer elliptic, meaning that it gives a spectrum with singularities, thus cannot be inverted.

Derivation of Functional Integration from Canonical Quantization

For simplicity, let's just consider a theory in flat Minkowski spacetime with the degrees of freedom are N scalar fields

N Svl,3 = d12 ,4 I =oaE =N, (PI -+W = ill-tI,. -r (3.106) nr=1

67 Canonical quantization means imposing the commutation relation operator-wise:

[

fln(x)J#) = -- C00.(E)I#n) , Dn(X)brKn) = -~ ns|T) (3.108)

From the Halmintonian density, which is the operator form for the evolution of the quantum theory with respect to time, one gets the change with time in the definition of the fields and its canonical momentum, and also the eigenstates:

H(t) = f d3 N(t, X) (3.109) Jv?

Sei f dTH(T) -i fot drH(r) -+ drH(r) (3.110)

fln.(t, 9) = eifi -r)Hnef> H(r ) -+ idrH; t) -i fo d-H(rTiw) (3.111)

Equation (3.108) can be used to proof that:

(#1, ... N; tj 71, ... 7n t, -i, 7I1, 7-- ) = 7( ei27)f d-2n(f ) )n1( (3.112)

The inequal time inner product can be read-off, with the time ordering is implemented inside the function integration measure Do and D7r for each history of configuration [181:

(3.113)

M-1 N M-1 t' - t tj+1 - tj = Jt = (3.114) im f D~) ( fj (#+; tj+1 I ) , Al j=1 72=1 j=0

M-1 N M-1 N M-1 = im j1 D D-. +1; t+1 7rj ; tj ) (7rj; tj| # j; tj ) ) (3.115) j=1 71=1 j=1 n=1 j=0

M-1 N M-1 N fv3 dajrN jxr(#i -#iV)-H(tj)jt - (27r)-N lim D7r ) e j=1 n=1 j=1 ' 1 (3.116) #';t' N - N -f 4 d (ZEQ ir Ot-7- = (27r)-NJ (m n)(1[ e'7 fV1,3 2 (3.117) n=1 n=1

68 The most popular form of the Halmintonian density is quadratic in the canonical momentum:

1 W = 17rnCnrnm [01.,qN17m +O0(71 170 ) (3.118) 2

Hence, the Gaussian functional integration can be evaluated, with stationary constraint [181:

O/ i fy1,3 d4X(E N 17rato._7j)., (#'1, ... , #'n; t'| 1 ,1 ...,10,; t) = f N 10n) 4,0 13dX[ IC[1,.,N](3.119) 9 n=1 (2 7r)N v'e(4)(O) f,3 dOTr nC[i,. NI(

But at stationary solution of the canonical momentum 7r in terms of the fields #, one get back the Lagrangian, which is written with the fields only:

at#O = O, , -= -1ana -+ 7rt#n - K)S = L (3.120) n=1

If the operator C is independent of the field, then one arrives at the building block of functional integration, which states that the oscillatory weight of a history of field configuration should be the action: N 4' ;t' (0' ... n; t'i #01,., o; t) ~ D ) es' (3.121) n=1 4;t

There will be a disagreement between the canonical method and the conventional functional integration if the operator C is field-dependent, as the weight associated with each history should be history-dependent. However, this is just a problem of definition (can always modify the original formulation of functional integration so that it has history-dependent), thus for any canonical method there will be a corresponded functional integration.

Commutation Relation in Functional Integration

The order of operator in the equal-time commutation relation of a single field type of interests

(inequities only yield 0) of canonical method corresponds to the order in time of the insertions in the functional integration:

[

69 The correlation value of the insertions can be calculated:

m D o5Di(#f; t + t|Ib(t + St, )HI(t, ) - f(t, F'>1(t - ot,)I#; t - 6t) st-+0lim f17(3.123) f Df Doi (of; t + &| #1; t - 6t)

f Df D#4D7re vr ( )f (Y) - #i(Z)) f DfDjDwe (of W)(3.124)

Using the method of discretization with P x P x P grids of size 13, with label = (ji, J2, j3) runs from 0 to P =(P, P, P), on V 3 , then:

f ] d dd7rleNr f 7' -+ -(3.125) f Hl; 6 d} d#7 dirieT-

_fi d dfdwid# d# dirte'94eT ( le ( - r'(# - (3.2) f o d ( dddd 7r/dddd( Wd 7(i 'f i eN ' -e - (3 .12 ) (3.127) f#d/4de de5 / f fen9(4- ) de+ (##))

f d o*, d#/5,' (dora &-+ 4) (i i'el -

f d#d#d' (os(# - )36( -

f di d.. d.5-S( - -. =- f o -+ i6N()(- f) (3.129) f ' - - -)( 7')

This is precisely in agreement with the canonical method.

3.3.2 Is Functional Integration always Superior?

It should be noted that, although functional integration is easy to keep track of, sometime it's not as convenient as the canonical method, such as in understanding the polology nature of the S-matrix, since the completeness of the physical Hilbert space gives on-shell particles interpretation natural, while the correspondence in the functional integration is not that straight-forward to see. One should be flexible with the two methods to read-off interesting physics from quantum field

70 theory.

An example: Polology

The study of pololology in S-matrix (or, correlation function of insertions with non-vanishing factorization) gives nonperturbative physics of the local theory, scanning the mass spectrum as the energy-momentum scale goes up. Let's start by looking at a n-point insertions and factorize it (by on-shell unitarity complete insertion) [36]:

G(xi, X2, ...,X) = (Q|T (01(Xl)02(X2)...O(X.)) IQ) , IF =g,2 )(IF I (3.130) f2 VpT+ mq,

J 1 e1ir(QIT 01(x)...O,(x,)) I')(PI (o,+( + 1)...O(X.)) + ... (3.131) (27r) 3 2p,+M2 I)

The assumption is that the theory has spacetime translation symmetry, so that the 4-momentum

P is a good quantum labeling. Note that, choosing x' = j - x 1 for j < r and '= xj - x,+1 for j > r:

eir =0(min{ti, ... , tr}) - max{t,+1, .--, tn}) (3.132)

=(ti - tr+1 + min{0, t', --, t'} - max{0, t+ 2, ... , t}) (3.133)

For now, let's just assume that the ground state is Lorentz has no 4-momentum, as PQ) = 0 [36], then:

(Q T (01 (Xi) ... O,(X,)) I) = (QIT (ePx1-i 01(x1)e-iPX ... e Or (x,) e-iPXl e-iPX)) (3.134)

- e-iPqx (Q IT (ePaO1(x 1 )e-iPx .. .eiPX O,(xr)eiP1) IXF) (3.135)

= e-iPqx1 (QT (01 (0)o2(X'2)...Or(X')) )IT (3.136)

Using the 0-function integral representation:

f()' dw i _-iWX 0(x) =1 -e d (3.137) 071- 2iw 71 Define At = min{0, t' 1,} - max{0, t' 2 , ... , LJ, then work in the momentum space:

df 1- 0n(PliP2, --- , P.) = Ryg d4Xieipii... dd4X' (3.138) S(27r)3 2 V/p + m4 f f f 27r w

x e-iP'(x1 xr+1)e i(p2+-+r)xl ei(Pr+2+-+Pn)xr+1 e-i(t-tr+i)e-WALt (3.139)

x (QIT (01(0)02 (X'2)...Or(X')) IT)(4 IT(Or+1(0)r+2 ('+ 2 )...On(X'))IQ) + ... (3.140)

3 2 1 f d f e d i (3.141) 2/pE + m2 1 (2-r3) Jf 27r w

x(27r)() Pj _=P (3 - j=e i j= Ej - Vp + n4 - w) (3.142)

r n1 x dPi+ Mi + f E Ex( T (01(0)02(2)-...E0,(',))IT) (3.143) j=r+) j=11 j+1r+=

x (T IT (Or+1(0)0r+2(r+2)..0n(X'n))IQ) +( . ; pj , q= P (3.144) j=1 j=r+

1 E =r~ E - E ,)A t d 14 n d r'e-ET (3.145) /= i ( 27r)3 j 1rif2 p ,+ m2, j I Ej -E

(2ffr)j(3 (A) j (Ej - Ej (3.146) j=1 j=r+1 j=1 i=r+1

x (QIT (01(0)O 2 (X'2)...Or (4') T)(1 IT (Or+1(0)Or+2((Xr+2 )...On(X'n))IQ) + ... (3.147)

4 _i(2wr) 3(4)(p + q) I = p 2 - Mr,2 (QIT 0 1(pI)...Or(pr)JI)(T'IT (Or+(pr+1)...On(Pn))I) + ... (3.148)

Unitarity of the theory means the completeness of (on-shell states) Hilbert space, and the used

notation in the calculations above is short-handed, and the better one should be:

( 3 )(W - (2 3 2E1q |T;p)( I; p (3.149)

The poles of the n-points correlation in momentum space are at the mass of a particles in the

theory. This is a nonperturbative physics, since the mass is not only for fundamental particles

(appears in the Lagrangian) but for all the possible ones (such as bound states and

multiple-particles, even unstable particles). For a single fundamental particle insertion

72 factorization, the pole arise when the particle goes onshell, and this is nothing but the Lehmann

- Symanzik - Zimmermann reduction, with a nonperturbative derivation. In that description, S-matrix can be read-off as the residual (up to a prefactor cj = (QJDj(O)Jp)) near the pole of onshell insertion. For details:

2 n-1 n -

Gn(P1, P2,..., P = -Z pj) = J7G2 (p,-pj)Sn(pI, p 2,..., pn) , G 2 (pj, -pj) = 2 z + j=1 j=1 m3 (3.150) The momentum of the theory can be constructed from the Lagrangian as the conserved charge of spacetime translation. For a free scalar field 4D theory, it's simply:

-fd3H()V() , (3.151)

H = Jd35Q 2( ) + (V()) + m2 X2)) , GX) = & () (3.152)

Note that, the assumption that the vacuum state is annihilated by the momentum operator P, in general, isn't always true. Essentially, in quantum field theories, the kinematic structure is fixed by the canonical commutation relations between fields and their canonical conjugate momentum.

In non-relativistic quantum mechanics there is a finite number of degrees of freedom, and so the

Stone von-Neumann theorem tells you there is a unique Hilbert space and choice of operators on the Hilbert space (let's just take it as the mathematical fact). The theorem, however, does not hold for infinite number of degrees of freedom, and actually there are infinitely many inequivalent choices for the Hilbert space for quantum fields [34]. In flat spacetime, the requirement that there exists a state such that PIG) = 0 picks out a unique Hilbert space, and one can check it, in principle, with lattice quantum field theory (to avoid horrendous analytical calculations):

(QIPIQ) = JD2N eisP (3.153)

In curved spacetime (which in general have no killing vectors), it is unclear how to single out a unique Hilbert space, and this is a big difficulty [341.

It's important to note that, already in a free scalar quantum field theory, the ground state has a bad divergence for P'JQ) (PIQ) = 0, well-define). However, it is possible to redefine the

73 Halmintonian and Lagrangian in the way that these contribution goes away, and the theory realize Lorentz symmetry as expected from observations and experiments. Supersymmetry actually kills-off this ambiguity, as there's no freedom to shift the Halmintonian around. One can think of it as a trick to some how regulate unwanted physics of the theory, and then if supersymmetry cannot be seen, give it a breaking mechanism so that the effect of superpartners become unobservable.

3.3.3 Mathematical Rigors

From the hopes of having a more rigorous framework of formulating quantum field theory, axiomatic quantum field theory and constructive quantum field theory were born. As at the moment, we cannot say that the two theories are very rigor and successful, but still, it's important to note that, even quantum field theory is usually "sloppy" with maths, it still give us more valuable understandings of the nature.

Axiomatic Quantum Field Theory

Since until now there are several times we use the axioms in axiomatic quantum field theory, it is necessary to mention the full list of Wightman axioms for real scalar field theory (generalization to more complicated theory can be done similarly), to do the theory more justices [311:

1. There is a separable Hilbert space V, the states of the theory are described by unit rays.

2. There is a unitary, positive-energy representation U(A, a) of the Poincare group on /.

3. There exists a Poincare-invariant vacuum states IQ). 4. The quantum field 4(x) is an operator-valued distribution, for x E Rl 3 .

5. The states are of the form F(x1)...(x2)IQ) E 71.

6. The field '1(x) transforms covariantly under U(A, a).

7. If two fields are space-like separated, then the fields either commute or anticommute.

8. The space of invariant vectors IQ) is one-dimensional (uniqueness of the vacuum).

Constructive Quantum Field Theory

In general, the Lorentzian signature is problematic, so it's tempting to change the Lorentzian time to the Euclidean time by an imaginary factor, and the spacetime change from R' 3 to a more familiar R 4 . In constructive quantum field theory (there are many other interesting things in this

74 topics, but not of our interests), from the point of view of the functional integration, the Euclidean field theory is nothing but the analytical continuation of the field theory in Minkowski spacetime

[19]. However, there are mismatches between these, for example, there's no Hilbert space (in the conventional sense) in the theory with Euclidean signature, and instead of the S-matrix in

Lorentzian quantum field theory, one has correlation functions in Euclidean quantum field theory.

Osterwalder-Schrader quantization gives a set of axioms (we will not mention them in details here) for constructive quantum field theory, so that for an Euclidean theory there's a corresponding well-defined theory in Minkowski spacetime. Unfortunately, although rigorous constructions for two and three-dimensional 04 -theory, Yukawa theory and Gross-Neveu theory were found, there are still many open problems in four-dimensional theory and beyond. Nevertheless, indeed, this trick is used, even overused, in theoretical studies, such as in lattice quantum field theory for well-defined behavior of the far away contributions (oscillatory becomes decay) and string theory to define a Riemann surface (for perturbative calculations of stringy worldsheets). Even without mathematical rigors, the found answers are very interesting, and some even fit with reality, for example, the spectrum of baryons and mesons found with lattice quantum chromodynamics. The method of Wick's rotation to go from Lorentzian time to Euclidean time will be used in later discussions, so be prepared for the "sloppiness".

3.3.4 Effective Field Theory

Functional integration provided an intuitive way to think about the process of taking out the degrees of freedom in the quantum field theory, by explicitly doing the integration. One can integrated out a field totally from a theory, or get rid of the contributions from quantum fluctuation at the energy scale outside of the interval of interests.

An example: Effective Theory of Quantum Gravity

Let's consider an effective theory that can be obtained by simply integrate out the field, from quantum gravity coupled with a real scalar field. For simplicity, deal with it in a general

Euclidean curved spacetime (Wick's rotation the Lorentzian time t -+ -it) [201:

S[1, gPV f d2lvy/-(gjvO lA + V4)) d -t f g-4P

75 V = V(x) , F = -Og + V , 094 = --- ,(Vg""uO4) (3.155)

The function operator F has eigenfunctions

The wishful thinking is that those gives a basics for any arbitrary function 1b (that's not always the case, though; for example, the whole spacetime manifold of the Schwarzchild blackhole):

00 100 S[4,9g""v= 2E c2A, (3.157) n=O n

The (vacuum) functional integral in this basis can be calculated as:

7 /, 00 -1/2 Z =-Db-S f= < e-n"4 = f A) (3.158)

Let's have a remark here. In the Euclidean manifold, it's natural to impose zero boundary conditions on the basis functions 4D,, (the choice of regularization). To calculate anything, such as

S-matrix, put insertions in points in the bulk of the regularized manifold, using this very definition of functional integral. The logic is, if the regularization is of any good, the correlation comes from applying the described rule should be insensitive to the boundary conditions [20].

There are cases where this trick brings new insight to the table, such as quantum anomaly.

The effective action of the metric, which comes from 1-loop physics (integrating out the scalar field degrees of freedom), can be read-off:

Seff[g"v= In Z = 2 A 2ln det F (3.159)

This functional determinant should be regularized so the final answer is the renormalized one

(well-define and get rid of divergence problems). To do that, one srart with an auxiliary Hilbert space,which has a complete basis of generalized vectors jx), normalized as:

(Xj X') = 6(x - X') , Jd2z2v)(XI = 1 (3.160)

76 The task is to determine a Hermittian operator 0 that has precisely the same eigenvalues An as F, that is:

01JFn) = AnIJn) , (lml IXn) = J d2 WX4m(X)1n(X) = jmn (3.161)

Hilbert space, in general, is different than quantum mechanics. The appearance of a Hilber space and of the Dirac notation does not mean that the vector IT) are states of some quantum system.

One uses the Hilbert space formalism because it simplifies shits when calculating the renormalized functional determinants. It is possible but more cumbersome to work directly with the partial differential equations. Using the decomposition of unit operator, one can rewrite the eigenvalue problems in the coordinate basis in the following way, then do the matching [201:

Of d 2wXIIXI) (x'I I) = AnII.) Tn (x1O d2wXI Ix') (x'I X) (3.162)

+ dJ2w (xO|x')4fn(x') = An (x| IF) A 7, , (x) , XI' = /4 , (3.163)

(xIOx')4 = g114(x)(-Eig(x) + V(x)) (g-1/4(x)6(x - x')4) (3.164)

Before defining (rigorously) the regularization for the quest of finding the functional determinant of 0, let's study right about the trick (hand-waving) that helps physicists to find the finite part of the functional determinant using the (-function:

00 (6(s) = Tr(6~) - A- (3.165) n=O

The trick is similar to how the Riemann's (-function is used for getting a finite result (except poles), by analytical continuation (the relation between analytical continuation and renormalization will be mentioned later):

00 ds(6(s) = d e-lf" Ad = - es In A, In An (3.166)

C0 00 => In det O = in fA = ZA -d,(6(s) (3.167) n n 8=0 The function (6(s) is usually regular at s = 0, so the derivative should be finite, which is good.

This equation can be thought as the choice of regularization for the functional determinant of

77 operator 0 (this trick can also be employed for finite dimensional system).

The calculation of (6(s) can be reduced to the problem of solving the partial differential equation for the heat Kernel. Given a Hermittian operator 0 and a complete set of eigencevetor IJT), the heat Kernel operator is defined as:

00 K(7) = e-OT Ee~AIn)(nlZ , k(0) = 1 (3.168) 71

The proper time T (> 0) is auxiliary and will disappear from all physical results. The trace of the hear Kernel gives the (-function of interests:

00 Tr k(T) = E nk(T)IT) = E- , (3.169) n1 n

F(s) J 6T5-1dT = A' eTAr- dr Re(s) > 0) , (3.170)

(6(s 0wA 1 [00 0) 1-A ( - T00 T ( )= F(O)_= J- "-idrT = F(s)] TrTk(T)T-dT (3.171)

From the definition of the heat Kernel, given above, the evolution along the proper time T:

dK(T) = -OK(r) , (xIk(T)IX') = K(x, X'; r) (3.172)

- dTK(x, X'; T) = d2XII(x61 xd")K(x", x; T) (3.173)

At the intitial time-slice:

K(-r 0) = 1 - K(x, x'; 0) = 3(x - ') (3.174)

Since the trace of the operator doesn't depend on the choice of the orthornomal basis, hence:

TrK(T) J d2,xK(x, X; T) (3.175)

The steps to find the functional determinant of 0 with the matrix element t0 x') can be summarize as follow: (i) solve the heat Kernel equation with the initial condition, then determine

K(x, X'; T); (ii) subtitute the expression the the trace of hear Kernel into zetag(s) and calculate

78 the convergences; (iii) analytically continue the obtained function (6(s) to get out in det 0, and the determinant functional is found straight-forwardly after that. Let's try to solve the heat

Kernel equation, in flat Euclidean spacetime, with V(x) = 0:

1 4 (xO = F = -- ")4 = 9/ (-) (7-1/46(x - X")4) = 2(6(x - x")4) (3.176)

The heat Kernel equation is simple enough:

dTK(x, X'; T) = & K(x, x'; T) , K(x, x'; r) = f -eik k(k, X'; T) (3.177)

-+ dk(k, x'; r) = -k 2 K(k, X'; T) , k(k, x'; 0) e-ikx' (3.178)

The solution for the heat Kernel, unsurprisingly, is just a Gaussian:

k(k, X'; T) = e_,k 2 _ikx' - K(x, X'; T) = exp - ) (3.179)

For the general curved spacetime, this is a very hard problem. Therefore, instead of attack it nonperturbatively, consider a perturbative description around a flat spacetime configuration:

171W = 6pL , gpv = 7711 + h, + g'L" = , + hl"" , l, I < 1 (3.180)

The operator can be rewritten [20]:

1 4 (XIOx)4 = -g- 0, (g 1/2gv"(g1/46(X - X')4)) + V(x)6(x - X')4 (3.181)

= EZ (6(x' - x)4) + s[V , V(x)]4 , s[h"", V(x)] = h + F + P (3.182)

The operator h and F are linear, even nonperturbatively order [20]:

hA = hV",0, (6(x - x')4) , P4 = Oh"'&,1 (6(x - x')4) (3.183)

The operator P has a quite horrendous full form [201:

3 P(x) = 4~) 4 - 1 g1gg"0,,,haoO ,h", ,h - 4 h&"ga6Ophc1 (3.184)

79 1 gg 01, haOvhzA - V(x) , P4 = P(x)S(x - x')46 (3.185)

The heat Kernel can be solved order-by-order of perturbation (in .):

K(T) = Ko(T) + Ki(T) +K 2(T) + .. , drK = (E] + s)(3.186)

Well, basically (drop the 6-function, notationally):

dKo(T) = KO , dTKI(T) = Ki(T) + Ko(r) Ko = 1 , K>1 = 0 (3.187)

The solution can be read-off:

Ko(T) = e ki(() =ko(r)O(r) (3.188)

1 -+ 0(7) = dT'k- (r')sko(T') ki(T) = / d'r'Ko(T - T')ko(T) (3.189)

To find the trace of the heat Kernel, one first need to calculate the matrix elements of ko and ki:

(XIKO(T)y) = (xIeEIly) = eTl6(x - y) 2 2 (3.190) (2w)d Wkw _4+&2-)

Hence, one arrive at the boring Gaussian:

4r (3.191) xJO(r)Jy) = - -(4lrT

Calculation for higher orders of K., is very cumbersome, in general. For the next leading order:

k 1 = kit, + k1r + kip (3.192)

Let's just do the last term, since it's easiest (and others are similar):

(xlKiplx) dr'(xIko (T - T') PJo(T') IX) (3.193)

= dTd2w yd 2u(XI o(T - T')Iy)(yIPIZ)(ZIko(T')IX) (3.194)

80 2e ),-7 'Els(r r dT' 1( dr'e+7e xP(x) (3.195) 0 I (4lT)w J (47r(T - r')) (47rT')WP

The diagonal element of k1 p will need more work [20]:

2 2 d wk 11(r- T)k+ik X'+r-r) P(k) d wk eikxPk (XIKlh IY) 4r dT 7P(X>J= I (2i_)w (2(w .) (3.196)

For the other parts of k1 , after tiresome brute-force calculations [201:

(xIKirjx) = -( / dT'e O6( - hl" (3.197)

" ) (xIKlhIl) = - 1 j d'eI x h + T ) 2 o 0 h (3.198)

The trace of the heat Kernel, at linear order:

irk(r) J d2wX(xlko+Ak1x) = 1 (47r-r)w d2 wx 1- 1 ,-rV+ ( (3.199)

From dimensional analysis and diffeomorphism invariant of the action, for a general curved metric: R TrK (3.200) = (47rT)w J d2wX (1 +'r

At the next next leading order (as one can see, controlled by the proper time r) [20]:

(R-v R- if1 (-Tg)V +T2V 1k(T) = 1 d2w XV4I+ 'f 2 (-rEg)R (3.201) (47rT)w I

2 eu(1u)zdu +T Rf3 (-TL g)R+ 2R,,f 4 (-TEg)R" ) , fi(z) = (3.202)

-1+ fi(z)-1 , fi(z) I fi(z) - 1 f (z) f2(/z) = fi(z) h (Z) f3(Z) = f1(z)2+ (3.203) 6 2z z2 3 8z 8

The f-function contains a glimpse of nonperturbative physics, at all orders of r (take T go to 0

81 inside f-function to get the T2 -order result). In conclusion, the trace of the heat Kernel:

TrK( (4 1)=) d2wxv 1 2RV)+T(V2 VR 2+ 0(73) (7T fQ2 6 120 60iv~y~k) f (3.204) (3.205)2 (dra,(x)2wXF1+2_a(X)O(T3)

The function a,(x) and a2(x) are the famous Seeley - deWitt coefficients. Since one will need to integrate T from 0 to oo, so the Seeley - deWitt expansion (the result above) in general cannot be used to calculate the corresponded (-function (at least, normally, but analytical continuation

actually might works) since it's only valid for small T. The behavior of the heat Kernel at small T gives UV region physics, as an effective field theory. The fastest way to see this is to note that r has the dimension of x 2 and thus small T means short distance (or high energy scale), describes things like vacuum polarization [201. On the other hand, large value of T tells about IR region physics, which is related to particles production [201. Note that, for a slightly curved (almost flat) metric, the expansion should be written with the power counting of R and V instead of -:

1 fd2xv 1+ R-V T+ TrK(T)(V1 VR+ IR 2+ RRPv )T2+(R3,V3)d (47FT)'w J\\ 6 /\ 2 6 120 601 (3.206) The (-function of interests can be used to read-off the effective action:

1 8- 1 (s=(s) 1 Tr K(T)dT , Se55[g"] = -- dSP(s) (3.207) s=O

For s = 0, the integral diverges at T -+ 0, hence the definition above is application only for those s for which the integral converges. The value of (p(s = 0) is then obtained by analytical continuation procedure, removes all the divergences in the effective action, without any justification, hence does not reveal the physical meaning of those divergences.

Define the (-function for small s by regularizing the integral through an explicit cut-off instead of perferming the analytical continuation, for the sake of studying the divergence origins. Consider massless field V = 0 (note that, UV physics at an energy scale a lot higher than the mass does

perceive the particles effectively massless) in d = 4 (or w = 2). For small T, it's best to use the

82 Seeley - deWitt expansion, for cut-off Tc:

(S) = ( d4 x/ Ts- 3 dT + RTs-2 dT (3.208)

(4wr) 2 F(S) f f foc

+( R2 + RAvR )] T5- 1 dr + O(Ts)>0). (3.209)

Change the cut-off scale to T' < -rc, and define:

2 A') = f T-d ~ , B(T') f dT-dT ~'-1 , C (T') T dT~ In( (3.210)

The for r' -+ 0, the leading divergences is of the structure:

(Ps) = (41r) (s) d4x /(A(7') + RB(') + 1 R 2 + 1RVR")C(T') (3.211)

For small s, the F-function has the structure:

1 S + O(s2) (3.212) F(s)

The contribution of the part of UV physics (between r, and Tc) to the Euclidean effective action, the regularized effective action as the cut-off scale approaches 0:

Seff [g' ] ~ d4X (a-2 + bRr'-1 + c(1 R2 + I RWRt" ) lfl(Tc') (3.213)

The backreaction of the quantum field on the gravitational background causes a modification in the Einstein's field equation. The total action for the gravitational background is the sum of the free gravitational action, Einstein-Hilbert SEH [9v], and the effective contribution Seff [gill] as quantum corrections. The classical action of general relativity SEH [gv] contains the cosmological term A and the spacetime curvature R, and the renormalization procedure for those is implemented as follows: assume that the free gravitational action (without the backreaction from

83 quantum fields) has terms quadratic in curvature [20]:

SEH,bare[1v] dd4xV R-+2Ab ay2 R2 + RR") (3.214)

With the bare constants Ab, Gb, ab, and these values are not observable since the quatum fields always present and cannot be decoupled from the theory of gravity. The modified action for gravity is the sum of the free action and the effective quantum contributions:

S[g[v"] = SEH,bare[9gW + Seff g"I f d4X l r b + (3.215)

+(-I + B (Tc))R+ (CT) ab) (~R2 +I ,Rv (3.216) 167rGb 1927r2 R+ 327r2 120 R2+60RyR"(.2)

The renormalization procedure assume that the bare constants are chosen so that they cancel the divergences in the effective action:

A _ Ab A(r') 1 1 B(') O C(T') 2 2 8G 8TG 32W 2 ' 167G 167rGb 1927r , 327r a (3.217)

Set the cut-off to T' = 0, then the renormalized constants are interpreted as the observable values.

Now, let's go down to the IR region, and since it's not a good idea to use Seeley - deWitt

expansion anymore, one should try that with R and V as plate-holder of order expansion (which

is only good at slightly curved spacetime). For simplicity, still choose V = 0 as the matter field is massless, and then the (-function:

jIis J d2wx17/j dTTs1 I(1 + R+ 2Rf 3(-Tg)R + T2R wf4 (-QTEg)R" (3.218) For d 2 (w = 1), there's a simple relation for the metric:

Ry= IgZVR Cd(s) I- ]))R2xx/ 0 dTr'R(Rf3(--Eg)R+ f4((-r4 (3.219)

84 The corresponded effective action (to do the integration, change the variable from T to -Trg):

2 2 Sef[g"] = -- J d x jR j dr (Rf, (- T,)R + 2f4 (-rig)) 967fd x/RGi--R (3.220)

With the Green's function of the Laplacian Elg, then one arrives at the gravity Polyakov action:

1 2 Seff[g"] = 96 dxd2 y Vg(x)g(y)R(x)R(y)GE(XY) (3.221)

For 4D result (d = 4, w = 2), the answer is:

Sejj[gI"] ~ [d'x1 -R n )R+...R,,InR" (3.222)

The mass scale A2 is only introduced for a dimensional reasons. The logarithm of the Laplacian operator:

In ( ) j d(r(2 ) (3.223) \ p2 (A 2 +M2 -9 +M2)

Note that a change in the mass scale p ' would add a term ~ R In b to the action, thus changing the constant in front of the R2 term. This constant hence is scale dependent and runable. This is the manifestation of the renormalization group of the theory (which can be explained in more details from statistical field theory [161). The coupling value at a given energy

(normalization point) must be determined experimentally and then the dependence of this constant from the energy is obtained by solving the renormalization group equation.

Top-Down and Bottom-Up

As we move toward more and more general theories, things become harder to compute, for example, hydrogen energy levels with quantum field theory rather than nonrelativistic quantum mechanics and elliptic orbits of planets with general relativity rather than Newtonian gravity.

Generalization is very interesting, but the same can also be said about the other direction, toward finding the simplest framework that captures the essential physics in a manner that can be corrected to arbitrary precision, such as an expansion in v/c < 1 in a non-relativistic quantum field theory. This is the guiding principal of the effective field theory. For the construction of any effective field theory, one should take into consideration of [30]:

85 1. Fields -+ Relevant degrees of freedom. 2. Symmetries -+ Interactions and possibility of broken symmetry. 3. Power counting -+ Expansion parameters, leading order description.

The power counting in an effective field theory is just as important as gauge symmetry.

Effective field theory is based on the realization, with both physical and philosophical reasons, that the description of physics at some energy scale m should not be dependence on the the dynamics of much higher energy scales A >> m. By integrate out all the degrees of freedom but the relevent ones in the physical situations of interests, the calculations can be simplified. However, this real insensitiveness of nature to high energy physics indicates that to probe short distance physics at low energies, usually the only way is to increase the precision.

There are two main ways to build up effective field theories, the top-down method and the bottom-up method.

In top-down, the high energy theory is understood, and the questis to look for a simpler effective theory at low energies. Usually, we integrate out (the phrase corresponds to explicitly doing the integration of the high energy field modes in the path integral formulation) and remove the heavier particles, and this procedure yields new operators and new low energy couplings. More specifically, the full Lagrangian is expanded as a sum of terms of decreasing relevance

LhighZ~ . The Lagrangians Chigh and 4,, will agree in the IR. region, but will differ in the reign of UV. Some examples of top-down effective field theories are [30]:

1. Integrate out heavy particles (top quark, W, Z, and Higgs bosons) from the Standard Model. 2. Heavy quark effective theory for charm and bottom quarks at energies below their masses. 3. Non-relativistic quantum chromodynamics or quantum electrodynamics for bound states of two heavy particles. 4. Soft-collinear effective theory (I and II) for quantum chromodynamics processes with ener- getic hadrons or jets.

There are some usual aspects of top-down effective field theories one should aware of. For effective theories that built out of quantum chromodynamics, a separation of scales is strictly needed to distinguish physics that are perturbative in the coupling c, (p) (evaluated at the scale

86 p = Q) from effects that are non-perturbative in the coupling (evaluated at a scale close to AQCD < Q) [30]. Also note that, the summation E, L() is a power expansion of the counting parameter, but there are also logarithms which appear with the arguments that are of the ratio between mass scales or the power counting parameters. In a perturbative effective field theory with a coupling like a,, the renormalization of L(") allows us to sum the large logs a, ln (MI) ~ 1 low M when m 2 < m1 [30]. Indeed, any logarithms that appear in qyantum field theory should be related to renormalization in some effective field theories [301. About the precision, the desired precision will be tracked by where to stop the expansion (how far we go with the sum on n).

In bottom-up, the underlying theory is unknown, and the construction of the effective field theories is mainly without reference to any other theory. Even if the underlying theory is known, construction of the effective field theory from the bottom-up is still desirable whenever the matching is difficult, such as in cases where the matching would have to be nonperturbative in a coupling and hence is not feasible analytically. One can construct K1 L(") by writing down the most general set of possible interactions that consistent with all symmetries, using fields for the relevant degrees of freedom. Couplings are unknown but can be fit to experimental or numerical data, such as output from lattice quantum chromodynamics. Some examples of bottom-up effective field theories are [30]:

1. Chiral perturbation theory for low energy pion and kaon interactions. 2. The Standard Model of particles and nuclear physics (the argument for this is that Majorana neutrino masses can come from a dimension-5 operator). 3. Einstein gravity made quantum with graviton loops.

In bottom-up effective field theories, usually the E, expansion is in powers, but there are also

logs. Renormalization of L(" allows us to sum large logs ln (m) (M2 < i) together [30]. Even when mi and mi2 are not masses particles, the same is still true, as that logs in quantum field theory are summed up with some effective field theories [30]. About the precision of this effective description, the desired precision tells us how high do we go in the sum over n before stopping.

General Representation Independence Theorem

The change in the degrees of freedom in an effective field theory is very important, and the general representation independence theorem tells us how to keep track of the modification, so

87 that the rewritten effective field theory is the same as the original one, at least on-shell.

Make the degrees of freedom satisfies 0 = XF(X) with F(O) = 1, so that one can Taylor expand the field around # = 0 with the leading term being q = x, hence the one-particle representation of quantized # and quantized x are the same. The statement of reparamentrization independence invariant is that calculations of observables with L(O) with quantized field # should be no different to the theory L'(x) = L(XF(X)) with quantized field x. To see how that works, consider a scalar field theory, with q < 1 keeps track of the order [301:

1 12 4 6 3 L = 2 - m 2 2 - _ - + (q2) 2 2 Aq 1gg r0 2q Lk (3.224)

The last term can be dropped by making a field redefinition # -+ q + Wg2q3, or using the equation of motion to replace Ei0:

]Oq- m 20 - 4A#3 + (77) = 0 (3.225)

The new Lagrangian in both case of substitution are the same:

1 1 6 L' = 01'00,0 - -m02 - - 77'g,0 + 0(9772) (3.226) 2 2

Explicit computations of the 4-point and 6-point tree-level and loop-level Feynman diagrams up to the order 0(;) from L(O) with quantized # or L'(x) with quantized k arrive at the same results.

The general representation independence theory states that, the field redefinition that preserves symmetries and have the same one-particle states with the original theory allows classical equations of motion to be used for simplification of a local effective field theory Lagrangian without changing the predictions to the observables. A schematic illustration for this theorem in a field theory with complex scalar # can be shown as follows. From the effective Lagrangian

LEFT = E 7 7",,(n) (the expansion parameter 7 < 1), let's try removing a general first order term 1T[O]D24 from C') that preserves symmetries of the theory, with T[4'] is generally a local function of various fields '7. In other words, removing linear terms D24 in the effective field theory.

88 The Green's function with sources J can be obtained by functional derivatives of the partition function with respect to sources:

0 Z[J] f J Do exp i d (L( ) + ()- 2TD)+ 2 7 TD2q+ Jk +0(92))) (3.227) i k

It's straight-forward to see that removing the term IjT[V)]D2 0 is relevant to redefining the field

#* = #*- rT[/] in the path integral:

Z[J(= i fD0expd - YT - ) (3.228)

+n(L) - ITD21) + 1 qTD21+ Jy4 - JoqT + 0(772) (3.229)

One can see that there are three changes: the Lagrangian, the Jacobian and the source term JP.

Without changing the S-matrix, we can remove the change in Jacobian and the source and only left with the change of variable in the Lagrangian L (this is precisely the statement of the generalize representation independence theorem). The piece 5L needs #* + 77T to transform like 0* so that the symmetries of the theory is preserved:

L I(D#)*(Dq) - m2* + (...) (3.230) 22

1 12 1 2(D,#')*(D,#') - M2#'*#' - T(D 2 #' - m 2 0') + (...)' (3.231) 2 2 2

2 2 The -! 7 7 TD #' term from L(0) after redefining the field cancels j71TD #', which is good. Since the effective field theory Lagrangian at all orders q contains all terms allowed by symmetries, all operators in (...)' are already present in (...) as the field redefinition also respects the symmetries.

Thus couplings are simply redefined, and this poses no problem, since the values of couplings of an effective field theory aren't fixed. In other words, we still have the same effective field theory.

The redefinition of # differs from the original one at the lowest order, so lowest order corrections of L(0) (which are also symmetry-preserving) can be all absorbed into L() couplings. L() corrections go to higher orders in the Lagrangian, and terms linear in D 2 #0can all be taken out from L(). Using the same idea, it's possible to cancel D 2 0 to an arbitrary power out of LM by

89 just replacing it using the equation of motion (because of the kinetic term, D2 4 should always be there in the theory), which is also relevant to redefining the fields.

About the change in the Jacobian,let's recall that the Fadeev - Popov method:

det (0'D,) = DcDcexp (if ddX6i-a1D,)c) , -+ 6 1 - q')c (3.232)

Since the effective field theory is valid for the energy scale A < - the ghosts will have mass - Ae, and hence decouple, just like other particles at this mass scale that were left out. Note that dropping ghosts can change the couplings.

To see this, consider a field redefinition:

cq* T=0'* - A'*('*) , -+ (1 - 77E + 2TIAO'*O)c (3.233)

Rescale c -+ crf-2 to have the correctly normalized kinetic term. It then becomes:

c(1-1 - l + 2A#'*#')c , -= Anew (3.234)

The mass term of the ghost showing that it has the expected mass Anew, which is huge as q -+ 0.

Since ghosts always appear in loops, they can be removed like heavy particles and only contribute some corrections to the couplings.

For a Green's function of n-points scalar fields, after the redefinition:

G(n) = T QIT(#(X)...5(Xn)...)IQ) = (QT((O'(xi) +qT(xl))...(#(Xn) + qT(Xn))...)IQ) (3.235)

Here the ... can include insertions of other fields in the theory, and for now let's use real q for notational simplicity. The change of sources can be shown to drop out of the S-matrix from

Lehmann - Symanzik - Zimmermann reduction (field rescaling and field renormalization cancellation, no pole, no contribution to the scattering) [301:

JJddiePi-(OT((x1)...#(xr)...o) 0)(7JY~r ) (P1P2...ISIPjP+1...) (3.236) + ,.2 I_

90 Renormalization in Effective Field Theory

Let's take a look at renormalization in more technical details.

Regularization is the technique to cut-off UV divergences in order to pull out an finite results.

Different regularization methods introduce different cut-off parameters (such as hard cut-off A'Uy) dimensional regularization d --+ d - 2E and lattice spacing). In principle, any regulator can be acceptable, but for the matter of convenient, it's better choose the regulator to preserve symmetries (for example, gauge invariance, Lorentz symmetry, chiral symmetry) and also preserve power counting by not yielding a mixing of terms of different orders in the expansion, the calculations become easier [301. It is because in general, operators mix with other operators of the same dimension and same quantum numbers (with a matrix of counter-terms) due to quantum effects. Good choice of regularizations are usually with mass-independent regulators (strictly speaking, a new mass scale may still appear but in a way that doesn't directly change the power counting factor, and it's mass-independent in the sense that it doesn't see the thresholds of particles' masses in the theory)[301. Still, it should be note that even if the regulator doesn't have these desired properties (for example, supersymmetry is broken by dimensional regularization), one can still use counter-terms to restore symmetries and power counting, therefore making calculations more simple [30j.

Renormalization, with a specific scheme, gives definite physical meaning to each coefficient and operator of the quantum field theory, and usually introduce some renormalization parameters along with it (such as ft in MIS, p2 =- for off-shell subtraction scheme, A for Wilsonian). The relation between the bare value abare, renormalized aren and counter-term 6a coefficients a in different schemes of renormalization (UV cut-off with integrated momenta p, ATV <; IpI and MS dimensional regularization) are related, in the following way schematically:

abare(Auv) = are"(A) + Sa(Auv, A) , abare () aent) + 6a(c, [t)

The definition of renormalizability, in the traditional sense, is that if a theory has divergences from loop integrations which can be absorbed into a finite set of parameters at any order of perturbation, then it's renormalizable. However, in the context of an effective field theory, renormalizability becomes more relax, as a theory can still be good and make nice predictions if

91 it's renormalizable order by order in its expansion parameters [301. This allows for an infinite number of parameters, but only a finite number at any order of the expansion. Also, it's interesting to note that, if a theory is traditionally renormalizable, it does not contain any direct information on the new physics possibly arises at higher energy scale, usually around some mass thresholds [301.

Decoupling Theorem

Consider making an effective field theory by integrating out the massive degrees of freedom, if the remaining low energy effective theory is renormalizable and a physical renormalization scheme is used, such as off-shell momentum subtraction, then all effects due to heavy particles of mass scale M appear as a change in the couplings or are suppressed as y [301. Since the MS scheme (which is very convenient for calculations) is not physical as it is mass-independent, physically doesn't see the mass threshold, one must implement the decoupling argument of the theory by hand, removing particles of mass M for [t < M.

A good example is that, in the MS scheme of quantum chromodynamics [30]:

3 _ g 5 11 4 13(g) = p d g(-p) = 193 bo + O(g ) < 0 , bo = CA - 4nFTF (3.237) dp167 3 3 3

The quantum chromodynamics fine structure constant behaves assymptotically free as the run can be read-off from the lowest order solution:

as =A)- - as(pt) = + as(p) In (3.238) 4x 1 + a, (11)h27 A

Define an intrinsic mass scale: As27 ) (3.239) ACD ba8 (eu

From equation (3.237), this scale can be seen to be independent of the choice for p. The fine structure constant is now can be written in a very nice form, , which specifies the energy scale when quantum chromodynamics becomes non-perturbative (~ 200 MeV):

as(p) = 2R (3.240) bo In (p/Amsfe

92 Note that AgD depends on bo, thus the number of light fermionic flavors nF, on the order of loop expansion for 3(g), and also on the renormalization scheme as one goes beyond 2 loops.

There's a problem that comes from heavy quarks contribution to bo for any A from the point of view of the unphysical MS renormalization scheme and that contradicts the decoupling theorem at low energy scale compare to the masses of these quarks, therefore decoupling should be implemented by hand by together integrating out and changing the fermion number nF which is allowed in the loop effectively as [z passes through a quark mass threshold. Specifically, nF = 6 for p > mt, nF = 5 for mlb < [t < mt, and so on.

The matching condition, of perturbative diagrams and couplings, between effective theories after removing the massive particles can be read-off by studying closely the physics around the transition mass scale m(~ pm = p() = A)), from the fact that the S-matrix elements with light external particles should be in agreement between these two theories. The leading order condition for couplings, which is nothing but the continuous condition of the run at the mass threshold, can be shown to be a,,(Im) =

The general procedure for matching effective field theories L(1) - L(2) ... - L(n) from top-down method for mass thresholds going from higher to lower scale rn > m 2 > ... > Mn can be summed up as follows [301:

1. Match the theory Cl) at the scale mi onto C(') by considering the S-matrix.

2. Compute the /-function and anomalous dimension in theory 2, which does not have the particle 1, to run the couplings down from the evolution equations, then run them.

3. Match the theory C2) at the scale mi onto 0 ) by considering the S-matrix.

4. Compute the /3-function and anomalous dimension in theory 3, which does not have the particle 2, to run the couplings down from the evolution equations, then run them.

5. Follow this procedure for any number of additional steps required.

3.3.5 Quantum Anomaly

Another interesting topic in quantum field theory is quantum anomaly, the different between classical symmetry and quantum symmetry with topological origins, that cannot be seen

93 perturbatively (S-matrix argument) but pops out naturally nonperturbatively (regularization, renormalization and functional integration measure).

Violation of global Noether current's conservation is OK, while for violation of gauge current

(caused by quantum anomaly) it means the theory should be inconsistent. To see this, consider a simple gauge theory and turn on an expectation value for the gauge field A,. This is supposed to be a spontaneous breaking of the gauge symmetry (as the gauge field gets a vacuum expectation value), and the effective action after integrating out the matter fields should be invariant under gauge transformation:

eiseff[A] eis[',D,AsD (3.241)

Change the background by gauge transformation A, -+ A. + iDtE:

isff[A'] _ ,,Sfi[A] f D (,is[,A] _ f fis (s[,A] _ eiS[,A]+fd4xiDeJP"' 4

- D4(f d4xiDe,,J,) eis[D,A] _ Df ( f d4xiEDAJP) eis[,,A] = e d4 x(D,1J) (3.243)

Thus, if (DJ) # 0, then the effective action should be gauge dependence, hence the gauge symmetry is explicitly broken. The broken of gauge symmetry means the longitudinal freedom of the massless vector is not the redundancy of the theory anymore, hence violating unitarity and

Lorentz invariant (which indicates the ambiguity should be there). The breaking of global symmetry, however, doesn't mean any consistency of the theory, therefore should be fine. Indeed, the violation of global symmetry gives observable realizations.

What are the implications of quantum anomaly? First, the symmetry at the level of the action

(classical point of view) is accidental, and in reality (which is quantum), no such symmetry exist.

Second, even if the action doesn't seem to be symmetric, if quantum effects give corrections so that there is a new symmetry arise, one should conclude that the symmetry must be there.

Finally, the role of regularization and renormalization (the origin of quantum anomaly) is undoubtedly critical in the righteousness of quantum field theories, restricts possible models in phenomenology. It is important to remember that, quantum anomaly cannot be seen purely from the "powerful" S-matrix theory arguments (it can still be calculated perturbatively from the Feynman diagrams, though).

94 As a matter of fact, anomaly is not a failure of bad choice for regularization scheme. It's just that there's no regularization so that the symmetry can still be preserved when studying quantum effects, and that's a physical interpretation. Anomaly admits a topological intepretation, given in term of Chern character, by the celebrated Atiyah - Singer idex theorem

[41, and we will show that.

Use Wick's rotation to get to the simple Euclidean signature:

ix = 4 , 4, , (3.244)

iY0 =Y 17,'Y6 Y ,t= =7172737Y 4 , T -T (3.245)

The effective action for the gauge fields can be read-off from the functional determinant:

e-Seff[V] J is[*,A = det(i ) (3.246)

Since the determinant is nothing but the product of eigenvalues (the quantized nature), therefore it's crucial to have a non-ill description of an eigenproblem. Consider an Abelian gauge theory, the Dirac operator in Euclidean space is Hermittian [41:

= Antn(x)Xi(f) , P'y5 0. (x) = -A7nY 5 (x) , {'Y, 75} = 0 (3.247)

One gets the orthogonal complete normalized basis {4'n}. The orthogonality in the non-zero modes V',, _L y5 , is due to Pt = P. For the zero-modes, this is not true, and one can use chirality projection to separate them:

An (750n.1 4) = (754n| 1A'n) = (73tnJ#i n)= -An (-N'V2 0n) ; (3.248)

1 P -=(1 75) , +7V1. (3.249)

Expand the spinor field with respect to the found eigensolutions:

P(x) =7a,(x) = a1 (xl 4') , '(x) = Z n/x) = a (4'| x) (3.250) nl (XIVn :d1Ot() d 0

95 The prefactor denote the (infinite dimensional) transformation matrices between the eigenfunction space and position space, and it's play no roles in the later calculations or the physics of the theory. The functional determinant can be evaluated simply with a Grassmann integral: D'I'D' = d4X+il9Fdn , (3.251)

det(ilg) = f djnjd esEnAn"nan IAn) (3.252) n T 7Fn

A local chiral transformation with an infinitesimal parameter a3T' [4]:

T(x) -+ V'(x) = IF(x)e~1(x)5 , 'I(x) -* T'(x) = T(x)e~f x)5 , (3.253)

4 a'C= Cnnam , ='i= Dnmdn , Cnr = 6nm - if d X'V (X),3(X)Y 5 m(X) , (3.254)

4 Dnm =nm - if d X t(X)/3(X)75 Om1(X) = Cnm (3.255)

Grassmannian measure transforms with the inverse determinant. To see this:

(i = jij)( , ... i Jil5l... Jinin = i ... in dt J , f(+) = ... + - (.i .. (i + ... (3.256) n

1 1 f(x) ... + - .. ( +- ... = ... + -dt J'Ei...in(iI...( + ... (3.257) n, n

But for Grassmannian numbers, derivation means integration, hence:

d(j ... d(i = dot Jdxj1 .. .dXi -+ (dot J)-dC< ...d(, = dxjI ...dxi (3.258)

Therefore, the change in the measure pull out a contribution to the action:

da',db' dct C dot D dandei (3.259)

71.U

(SlnmJn2id4XX)()O()Y, 5 'prn(X)) 1 i = dada (3.260) 71

e~if~x~'i/1fr5(k Jda,,dbn = J(O3) JJdand&1 , J (0) - e-2i fdxIJ) (3.261) n

96 The Jacobian density can be written in a more compact form [25]:

3 =-Tr = 6( 4)(0) (3.262) n

Keep the transform parameter to be a constant, for global transformation, then one get the index [4]:

(740'1 ()")= 0 , D = /P+ , 4 (3.263)

d4X (X) - 5 (X) = d4X Y (X) Y5 On',t(X) 75V7 (X) (3.264) I n n

4 (3.265) f d x~tn+(X)On+(x) -- Zf d4x?4_ (x)4,0 _ (x) n n1

= dim ker D+ - dim ker D_ = index D+ (3.266)

The definition of the index of an operator:

index 0 = dimker(O) - dim ker(Ot) index0 = f d4x lndex(O) (3.267)

Regularization, with regulator function f(s) can be used the pull out the anomaly [25]:

f(0) = 1 , f(oo)= 0 , sf'(s) = 0 (3.268) s=O COo

Consider global transformation, and anomaly can now be seen from the Jacobian density:

Tr (7 5[v4, i"][-Y, fo]) = -16ie""Po , (3.269)

2 d4kf"(k2 ) = igr2 f dssf"(s) = -igr2 dsf'(s) = ir (3.270) Jo00

2 = Jim k Tr )e)ik(x) 3(x) M+0I(2jr)J 4 (Y5f (- (3.271) M2) )X--+y

4 lrmf d k4 (Y5f ( - A/I~]2) , k" --+ MkV (3.272) M-+00 j(27r) T ) - 2 - _21kADP lim m4 d 4 Tr - i+ ] ~ k2 (3.273) M-+00 (2r (y5f

97 if d (k 2 1

2 i(-)4 f"(k)Tr (4 91 + ,]F,)) 327r2 4vpFvFpc (3.274)

This non-zero change (known as the Chern-Pontryagin density x - . [251) corresponds to of the non-conservation of the gauge current. Note that, from Atiyah - Singer index theorem, one also agrees [4]:

index D+ 1 2 dxel'" Tr(F,,Fp,) - Tr(F2) = Ch(F) (3.275) 3i 2 ki 87r2

This give rise to the chiral anomaly (also known as singlet anomaly) in this simple Abelian gauge theory, with the index density related to the topology of the gauge group (the Chern number):

1 index D+ - - 12 Ch(F) , d * j5 = 2i Index D+ (3.276)

Hence, the anomalous axial current, also can be generalize straight-forwardly to non-Abelian gauge theory [4]:

(DPJ ) -272 F jttvP;, Tr (Ta{Tb, Tc}) Al = (A/) (3.277)

The prefactor i disappear when one goes back to Lorentzian signature. In general, anomaly can be read-off from the Atiyah - Singer index theorem. If the Dirac operator containing the

Yang-Mills gauge potential A = APT &P, then the characteristic class is given by the Chern character [4]:

Ch(F) = Tr eF dimF + Tr F + ( )Tr(F 2 )+... , F=dA+A 2 (3.278) 27r 2! (27r

The Atiyah - Singer index theorem states that [4]:

index D+ = Ch(F) = Tr(Fri) , OM = 0 (3.279)

It's important to remind that all calculations of quantum anomaly can be seen, from heat Kernel regularization or perturbatively from a Feynman diagram, perturbatively, in Pauli-Villar regularization scheme (might be a good story for another time). Anomalies can be understood as both UV physics (UV region regularization) and IR physics (zero-modes mismatch), which is

98 another evidence for the fact that it's an universal (all scales) problem in quantum field theory.

For a more horrendous gauge transformation, the topological interpretation of the theory still exists, and the generalization was done in the 1980s. One of the most highlight is the use of it to restrict the possible super Yang - Mills gauge group allowed in ten-dimensional supergravity, and

(to some certain surprises) it agrees with string theory (with the conformal consistency requirements for particles and interactions, already) [13].

The Standard Model is a chiral gauge theory where the gauge fields couple to the right-hand and the left-hand fermions differently (experimental facts). This means the gauge group of the standard model should be constraint, so the theory is anomaly-free and no left-right mixing arise.

The Standard Model gauge group U(1) x SU(2) x SU(3) is indeed fit the bill. Along with it, there's other consistent choice (anomaly-free) of phenomenological interests - only for N < 3 [331:

U(1) x U(1) x U(1) , U(1) x SU(2) x SU(2) , (3.280)

U(1) x SU(3) x SU(3) , SU(2) x SU(2) x SU(2) (3.281)

For Minimal Supersymmetric Model, a single Higgsino should be anomalous, therefore another

Higgsino with the same helicity is required. Incidentally, the mixed gravity-gauge anomaly and

Witten's global anomaly (a theory with an odd number of SU(2) doublets is anomalous under large gauge transformations) are also cancelled with this choice [33].

In theories whose spectrum changes with the energy scale, the matching of anomalies between the high and low energy theories provided nontrivial nonperturbative information. The condition made by Gerard 't Hooft states that the calculation of any chiral anomaly by using the degrees of freedom of the theory at some energy scale, must not depend on what scale is chosen for the calculation [33]. It is also known as 't Hooft UV-IR anomaly matching condition, where UV stand for the high-energy limit (generator Ta), and IR for the low-energy limit (generator Ta) of the theory:

>ITr. (Tj{TTj}) - ETr (Ta{TI,Tj}) (3.282) R L

=Z Tr({Tj,7 } - Tr ( L"{JLTf}) (3.283) R L

99 Consider calculating the anomaly of a global gauge symmetry current by considering which fermionic degrees of freedom the theory has. One also may use either the degrees of freedom at the far IR (low energy limit) or the degrees of freedom at the far UV (high energy limit) in order to calculate the anomaly. In the former case one should only consider massless fermions which may be composite particles, while in the latter case one should only consider the elementary fermions of the underlying short-distance theory. In both cases, the answer must be the same.

Hence, one can think of this as the constraint on expected degrees of freedom at different energy scales of the theory.

To prove the anomaly matching condition, artificially add to the theory a gauge field which couples to the current related with this symmetry, as well as chiral fermions which couple only to this gauge field, and cancel the anomaly (so that the gauge symmetry will remain non-anomalous, as needed for consistency). In the limit where the coupling constants go to zero, one gets back to the original theory, plus the added fermions (which remain good degrees of freedom at every energy scale, as they are free fermions at this limit). The gauge symmetry anomaly can be computed at any energy scale, and must always vanishes, so that the theory is consistent. One may now get the anomaly of the symmetry in the original theory by subtracting the free fermions we have added, and the result is independent of the energy scale.

3.3.6 Wavefunctional

Let's make the connection back to quantum mechanics, by studying the very interesting object known as the wavefunctional with by using functional integration and canonical method together.

Consider a massive free scalar field quantum theory, described by the action:

2 2 S[D] - dtdKZ( - (6t@) + (0A) + m2 (3.284)

Let's change it to Euclidean signature by a Wick's rotation for time t -+ -it, then the Euclidean action and the equation of motion for the field:

3 2 2 SE[ (( 2 + + 2 2)42>- mfJ (3.285)

100 Spatial Fourier expanded the scalar field, one can rewrite the action:

d 3 k

t d 2 2 (3.287) 7-+m -* SE[ = J 2 k2 2 (27)3

Since the oscillatory behavior in Lorentzian time is replaced by the decay one, therefore the longest- lived state is the ground state, and by evolving from past infinity one can pull it out:

4 Of ;tf ![#5] = (#1 Q) ~ (#f = #; tf = 01 #i; ti = -oo) ~_ Do'e-sE[O' dO k (3.288) k

Of kj~b 2 2 f4e~Jdt(I0t(PI +wU Ik I ) '-e2 sinh (wptf-ti)) (3.289)

~NH e- bwIk = H e-2 2_ei= X k 1 k. (3.290) k k Therefore, the ground state wavefunctional:

(3.291)

To find the excited states, one canjustapplies the creation operator on the ground state:

a = dg6 (wi,

For example,for the first excited state:

-f[# =aF[(] = e- W4 ) - ao(2 ) 1Q[#] (3.293)

J d3 Feik#(-)e- f (3.294)

Note that, another way to find the ground state is to find the solution of:

d3= &FeWkj(w (+) -ifl(-)) -+ (3.295) Id 3 -ee(00) + 00(y) ) 101 a-'JQ [#] = d xeik (() + 0(y)) 'fQ[] 0 Vk (3.296)

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