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Physics, B35 , 167–188. 218. ’t Hooft, G. (1973). Dimensional regularization and the renormalization group. Nuclear Physics, B61 , 455–468. 219. ’t Hooft, G. (1987). Can spacetime be probed below string size? Physics Letters, B198 , 61–63. 220. ’t Hooft, G. (1995). Black holes and the dimensional reduction of spacetime. In Lindström, U. (Ed.), The Oskar Klein Centenary, 19–21 Sept 1994, Stockholm, (1994) , World Scientiﬁc. See also: “Dimensional Reduction in Quantum gravity”, Utrecht preprint THU-93/26 (gr- qg/9310026). 221. ’t Hooft, G., & Veltman, M. J. G. (1972). Regularization and Renormalization of gauge ﬁelds. Nuclear Physics, B44 , 189–213. 222. Thirring, W. E. (Ed.). (2005). The stability of matter: From atoms to stars: Selecta of Elliott H. Lieb (4th ed.). Berlin: Springer. 223. Thorn, C. B. (1991). Reformulating string theory with the 1=N expansion. In International A. D. Sakharov Conference on Physics , Moscow (pp. 447–454). (hep-th/9405069).
224. Tomboulis, E. T. (1984). Unitarity in higher-derivative quantum gravity. Physical Review Letters, 52 , 1173–1176. 225. Tomonaga, S. (1972). Development of quantum electrodynamics: Personal recollections. In Nobel Lectures, Physics 1963–1970 , 6 May 1966, Amsterdam: Elsevier Publishing Company. 226. Tomonaga, S.-I. (1997). The story of spin . Chicago: University of Chicago Press. Translated by T. Oka. 227. Townsend, P. K. (1995). The eleven-dimensional supermembrane revisited. Physics Letters, B350, 184–187, hep–th/9501068. 228. Utiyama, R., & Sakamoto, J. (1976). Canonical quantization of non-abelian gauge ﬁelds. Progress of Theoretical Physics, 55 , 1631–1648. 229. Vanyashin, V. S., & Terentyev, M. V. (1965). Soviet Physics JETP , 21 , 375–380. Original Russian Version: Zhurnal Experimental’noi i Teoreticheskoi Fiziki, 48 , 565 (1965) 230. Veltman, M. J. G. (1981). The infrared-ultraviolet connection. Acta Physica Polonica, B12, 437.
231. Veneziano, G. (1968). Construction of a crossing-symmetric regge-behaved amplitude for linearly rising trajectories. Nuovo Cimento, 57A , 190–197. 232. Volkov, D. V., & Akulov, V. P. (1973). Is the neutrino a goldstone particle. Physics Letters, 46B, 109–110. Reprinted in Ferrara (1987). 233. Waller, I. (1930a). Die Streuung von Strahlung Durch Gebundene und Freie Elektronen Nach der Diracshen Relativistischen Mechanik. Zeitschrift fur Physik, 61 , 837–851. 234. Waller, I. (1930b). Bemerkung über die Rolle der Eigenenergie des Elektrons in der Quantentheorie der Strahlung. Zeitschrift fur Physik, 62 , 673–676. 235. Weinberg, S. (1960). High-energy behavior in quantum ﬁeld theory. Physical Review, 118 , 838–849. 236. Weinberg, S. (1967). A model of leptons. Physical Review Letters, 19 , 1264–1266.
42 1 Introduction 237. Weinberg, S. (1973). New approach to renormalization group. Physical Review, D8 , 3497– 3509. 238. Weinberg, S. (1980). Conceptual foundations of the uniﬁed theory of weak and electromag- netic interactions. Reviews of Modern Physics, 52 , 515–523. 239. Weisskopf, V. F. (1934). Über die Selbstenergie des Elektrons. Zeitschrift fur Physik, 89 , 27– 39. 240. Weisskopf, V. F. (1939). On the self-energy of the electromagnetic ﬁeld of the electron. Physical Review, 56 , 72–85. 241. Weisskopf, V. F. (1979). Personal impressions of recent trends in particle physics. CERN Ref.Th. 2732, CERN, Geneva, Aug 1979. [Also presented at the 17 International School of Subnuclear Physics “Ettore Majorana”, Erice, 31 July-11 August, 1979. pp.1–9, (ed. A.Zichichi), Plenum, New York,1982]. 242. Weisskopf, V. F. (1980). Growing up with ﬁeld theory, and recent trends in particle physics. “The 1979 Bernard Gregory Lectures at CERN”, 29 pages, CERN, Geneva.
243. Wess, J., & Zumino, B. (1974a). Supergauge transformations in four dimensions. Nuclear Physics, B70 , 39–50. Reprinted in Ferrara (1987). 244. Witte n, E. (1995). String theory dyna mics in various dimensions. Nuclear Physics, B443 , 85–126, hep–th/9503124. 245. Witte n, E. (1998). A nti-de-Sitter space and holography. Advances in Theoretical and Mathe- matical Physics, 2 , 253–291. (hep-th/9802150). 246. Wu, C. S. et al. (1957). Experimental tests of parity conservation in beta decay. Physical Review, 105 , 1413–1415. 247. Yang, C. N., & M ills, R. L. ( 1954). Conserva tion of isotopic spi n and isotopic gauge invariance. Physical Review, 96 , 191–195. 248. Yoneya, T. (1974). Connection of dual models to electrodynamics and gravidynamics. Progress of Theoretical Physics, 51 , 1907–1920. 249. Yukawa, H. (1935). On the interaction of elementary particles. I. Proceedings of the Physical
Mathematical Society Japan, 17, 48–57. Reprinted in “Foundations of Nuclear Physics”, (ed.R. T. Beyer), Dover Publications, Inc. New York (1949). 250. Zeidler, E. (2009). Quantum ﬁeld theory II: Quantum electrodynamics . Berlin: Springer. 251. Zimmermann, W. (1969). Convergence of Bogoliubov’s method of renormalization in momentum space. Communications in Mathematical Physics, 15 , 208–234. Recommended Reading 1. Becker, K, Becker, M., & Schwarz, J. H. (2006). String theory and M-theory: A modern introduction. Cambridge: Cambridge University Press. 2. Davies, P. (Ed.). (1989). The new physics. Cambridge: Cambridge University Press. 3. DeWitt, B. (2014). The global approach to quantum ﬁeld theory. Oxford: Oxford University Press. 4. Manoukian, E. B. (1983). Renormalization . New York/London/Paris: Academic Press. 5. Manoukian, E. B. (2006). Quantum theory: A wide spectrum . Dordrecht: Springer. 6. Manoukian, E. B. (2016). Quantum ﬁeld theory II: Introductions to quantum gravity, super-
symmetry, and string theory . Dordrecht: Springer. 7. Martin, P. C., & Glashow, S. L. (2008). Julian Schwinger 1918–1994”: A Bio-graphical memoir. National Academy of Sciences , Washington, DC, Copyright 2008. 8. Oriti, D. (E d.), (2009). Approaches to quantum gravity. Cambridge: Cambridge University Press. 9. Schweber, S. S. (2008). Quantum ﬁeld theory: From QED to the standard model In M. Jo Nye (Ed.), The Cambridge history of science. The modern physical and mathematical sciences (Vol. 5, pp. 375–393). Cambridge: Cambridge University Press.
References 43 10. Streater, R. F. (1985). Review of Renormalization by E. B. Manoukian. Bulletin of London Mathematical Society, 17 , 509–510. 11. Weinberg. S. (2000). The quantum theory of ﬁelds. III: supersymmetry . Cambridge: Cambridge University Press. 12. Weisskopf, V. F. (1980). Growing up with ﬁeld theory, and recent trends in particle physics. The 1979 Bernard Gregory Lectures at CERN, 29 pages. “Personal Impressions of Recent Trends in Particle Physics”. CERN Ref. Th. 2732 (1979). CERN, Geneva. 13. Zeidler, E. (2009). Quantum ﬁeld theory II: Quantum electrodynamics . Berlin: Springer. pp. 972–975.
Chapter 2 Preliminaries This preliminary chapter deals with basic tools needed for dealing with quantum ﬁeld theory. It begins with Wigner’s theory of symmetry transformations by showinghow symmetry is implemented in the quantum world via unitary or anti-unitaryoperators. This subject matter will be also important in deriving commutation(anti-commutation) rules between symmetr y generators such as in developing the Poincaré algebra of spacetime transformations including those in supersymmetricﬁeld theories which also involve fermion-boson exchanges. Some properties ofthe Dirac equation and related aspects are summarized in Appendix I,a tt h ee n d
of the book, for the convenience of the reader. Various representations of thegamma matrices are spelled out, however, in the present chapter which turn outto be important in modern ﬁeld theory, and the concept of a Majorana spinor isalso introduced. Special emphasis is put in the remaining sections on functionaldifferentiations and functional integrations involving Grassmann variables as well.Functional Fourier transforms are introduced which clearly show the intimateconnection that there exists between the functional differential formalism pioneeredby Julian Schwinger and the functional integral formalism pioneered by RichardFeynman. The last section on delta functionals, makes this connection even moretransparent and shows the simplicity of the functional formalism in general, whetherit is in differential or integral forms. It is, however, relatively easier to functionallydifferentiate than to deal with continual functional integrals. Both formalisms areused in this book for
greater ﬂexibility an d for a better unders tanding of quantum
ﬁeld theory. © Springer International Publishing Switzerland 2016 E.B. Manoukian, Quantum Field Theory I , Graduate Texts in Physics, DOI 10.1007/978-3-319-30939-2_245
46 2 Preliminaries 2.1 Wigner’s Symmetry Transformations in the Quantum World Invariance of physical laws under some given transformations leads to conservation laws and the underlying transformations are referred to as symmetry transforma-tions. For example, invariance under time translation (e.g., by setting one’s clocksback by a certain amount) or under space translation (by shifting the origin ofone’s coordinate system) lead, respectivel y, to energy and momentum conservations.
Invoking invariance properties in developing a dynamical theory, convenientlynarrows down one’s choices in providing the ﬁnal stages of a theory. Obviously, notall transformations of a given physical system are symmetry transformations. Butinvoking the invariance of a system under a set of given transformations may providethe starting point in describing a dynamical theory, and one may then consistentlymodify the theory to take into account any symmetry breaking. One may also havesymmetry breaking spontaneously which will be discussed later. A celebrated analysis of Eugene Wigner in the thirties 1originating on symmetry under rotations in space, spells out the nature of the transformations implemented onelements of a Hilbert space. This result is of central importance for doing quantum physics 2and quantum ﬁeld theory. The physical situation that presents itself is the following. One prepares a system in a statej i. The question then arises as to what is the probability of ﬁnding the
system in some state j/RSiifj iis what one initially has?.3The latter is given by jh/RSj ij2. Ifj 0i,j/RS0idenote the statesj i,j/RSi, resulting from a symmetry transforma- tion, then invariance of the above proba bility under such a transformation means that j˝ /RS0ˇˇ 0˛ j2Djh/RSj ij2; (2.1.1) andfj 0i;j/RS0igprovide an equivalent description as fj i;j/RSig. One may scale j 0i;j/RS0i;j i;j/RSiby arbitrary phase factors without changing the relevant probabilities given in ( 2.1.1 ). Although such overall phase factors are not important, the relative phases occurring when consid ering addition of such states are physically relevant with far reaching consequences.4 1See, e.g., his book Wigner [ 3]. See also Wigner [4 ]. 2For intricacies of Wigner’s Symmetry Transformations in quantum mechanics, see, e.g., Manoukian [ 1], pp. 55–65. 3More generally one may also have initially a mixture described by a density operator.
4One should also consider unit rays in this discussi on and in the subseque nt deﬁnitions but we will not go into these points here.
2.1 Wigner’s Symmetry Transformations in the Quantum World 47 To understand Wigner’s Theorem, one needs to deﬁne the following operators: 1. An operator Lis called linear or anti-linear, if for any j i;j/RSi;Œaj iCbj/RSi/c141, where a;bare, in general, complex numbers, then LŒaj iCbj/RSi/c141DaLj iCbLj/RSi; (2.1.2) or LŒaj iCbj/RSi/c141Da/ETXLj iCb/ETXLj/RSi; (2.1.3) respectively, with * denoting complex conjugation. 2. A linear or anti-linear operator Uis called unitary or anti-unitary, if hU/RSjU iDh/RSj i; (2.1.4) or hU/RSjU iDh/RSj i/ETXDh j/RSi; (2.1.5) respectively. 2.1.1 Wigner’s Symmetry Transformations Under a symmetry transformation, there exists a unitary or anti-unitary operator U such that ( 2.1.2 )/(2.1.4 )o r( 2.1.3 )/(2.1.5 ) hold true, respectively, with ˇˇ 0˛ DUj i;ˇˇ/RS0˛ DUj/RSi; (2.1.6) ˇˇŒa Cb/RS/c1410˛ DUŒaj iCbj/RSi/c141: (2.1.7) Continuous transformations, for which one may also consider inﬁnitesimal ones
close to the identity operation, such as space or time translations and rotations, are implemented by unitary operators since the identity operator Iitself is trivially a unitary one. An inﬁnitesimal change of a parameter •/CAN, under consideration, is implemented by a given operator G, referred to as the generator of symmetry transformation, and the corresponding unitary operator, for such an inﬁnitesimaltransformation, would take the form UDICi•/CANG: (2.1.8)
48 2 Preliminaries The unitarity condition U/c142UDIDUU/c142implies that •/CANGis a self-adjoint operator5: .•/CANG//c142D•/CANG: (2.1.9) For example, for an inﬁnitesimal space translation •x, via the momentum operator Passociated with the system in consideration, the corresponding unitary operator is given by UjspaceDICi•x/SOHP: (2.1.10) For inﬁnitesimal time translation •/FS, via the Hamiltonian Hof the system, one has UjtimeDI/NULi•/FSH: (2.1.11) Moreover for a rotation of a coordinate system by an inﬁnitesimal angle •/CANabout a unit vector n, via the angular operator J, UjrotationDICi•/CANn/SOHJ: (2.1.12) The latter may be more conveniently rewritten by introducing parameters •!ij,a n d rewrite Jias JiD1 2"ij kJjk;• !ijD"ij knk•/CAN; i;j;kD1;2;3; (2.1.13) where a summation over repeated indices is assumed, and "ij kis totally anti- symmetric, with "123DC1. We may then rewrite ( 2.1.12)a s UjrotationDICi 2•!ijJij: (2.1.14)
The generators H,P,a n d Jof the transformations are self-adjoint. In a relativistic setting, in which this book is based, these spacetime transfor- mations may be combined by i ntroducing, in the process, additional operators to this set of operators, denoted by J0i,iD1;2;3, which impart a frame with an inﬁnitesimal velocity change •v. The unitary operator corresponding to these inﬁnitesimal spacetime transformations then takes the elegant form UD1Ci/DLE •b/SYNP/SYNC1 2•!/SYN/ETBJ/SYN/ETB/DC1 ;• !/SYN/ETBD/NUL•!/ETB/SYN;/SYN ; /ETBD0;1;2;3; (2.1.15) 5One may conveniently, in general, absorb •/CANinG.
2.1 Wigner’s Symmetry Transformations in the Quantum World 49 with6P0DH, •b/SYND./NUL•/FS;•x/; •!ijD"ij knk•/CAN; •! 0iD•vi;• !00D0: (2.1.16) First we have to learn how the labels attached to an event in different, so- called, inertial frames are related. This will be taken up in the next section. The transformation rules relating this different labeling, are referred to as Lorentztransformations in Minkowski spacetime. On the other hand, the genertors P /SYN;J/SYN/ETB satisfy basic commutations relations and form an algebra in the sense that thecommutator of two generators is equal to a linear combination of the generators inquestion. The underlying algebra is called the Poincaré algebra and will be studiedin Sect. 4.2. In incorporating supersymmetry into physics, with an inherited symmetry exist- ing between fermions and bosons, one introduces a generator 7of fermionic type,
that is, it carries a spinor index, referred to as a supercharge operator, and thePoncaré algebra is extended to a larger one including anti-commutation relationsas well. This algebra is referre d to as Super-Poincaré algebra. 8 Typical discrete transformations, are provided by space P or time T reﬂections (parity and time reversal) or charge conj ugation C, for which a particle is replaced by its antiparticle. A key criterion for ﬁnding out if T, for example, should beunitary or anti-unitary is to eliminate the choice which would lead to the inconsistentresult that a Hamiltonian is unbounded from below. This is inferred as follows.Under an inﬁnitesimal time translation •/FS, via the Hamiltonian Hof the system under consideration, followed by a time-reversal applied to it is equivalent to a timereversal followed by a time translation, i.e., we have the equality Œ1/NULi•/FSH/c141TDTŒ1Ci•/FS.HC'//c141; (2.1.17) up to a phase factor '. If T were unitary, this gives HTD/NUL T.HC'/.F o r a
statej/DC1iwith positive and arbitrary large energy E, there would correspond a state Tj/DC1iwith energy/NUL.EC'/and the Hamiltonian would be unbounded from below. That is, T is to be implemented by an anti-unitary operator which, in the processof applying time reversal, it would complex conjugate the i factor on the right-hand side of ( 2.1.17). A similar analysis applied to P leads one to infer that it is to be implemented by a unitary operator. On the other hand, C, unlike P, and T, isnot involved with space and time reﬂections, and may be implemented by a unitaryoperator. Accordingly, the product CPT, in turn, is implemented by an anti-unitary 6With the Minkowski metric adopted in this book Œ/DC1/SYN/ETB/c141DdiagŒ/NUL1;1;1;1/c141 ,biDbi;b0D/NUL b0. 7One may also introduce more than one such operator. 8This will be studied in the accompanying book Manoukian [ 2]: Quantum Field Theory II: Introductions to Quantum Gravity, Supersymmetry, and string Theory. Minkowski spacetime is,
in turn, extended to what has been called superspace.
50 2 Preliminaries operator. The corresponding symmetry, embodied in the so-called CPT Theorem, will be dealt with in Chap. 4.9 2.2 Minkowski Spacetime: Common Arena of Elementary Particles A fundamental principle which goes to the heart of special relativity is theequivalence of inertial frames in describing physical laws. By this it is meant, inparticular, that dynamical equations take the same form in such frames up to a mere relabeling of the variables of the underlyi ng theories. Because of such relabeling of the variables, these equations are said to transform covariantly as one goes fromone inertial frame to another and the corresponding rules of transformations are, ingeneral, called Lorentz transformations. An event labeled by xD.x 0;x/,w h e r e x0Dt(time), in one frame, say, F will be labeled by x0D.x00;x0/in some other frame, say, F0. The transformation rules which connect this differen t labeling of the same event are the so-called Lorentz
transformations. One inertial frame F0may be moving with a uniform velocity with respect to another frame F, as determined in F, with a possible orientation ofthe cartesian space coordinate axes of F 0as also determined in F at some initial time x0D0.I f a n x/SYND0reading in F corresponds to an x0/SYND0reading in F0, then the Lorentz transformations are called homogeneous ones. Otherwise, they are called inhomogeneous. In the former case, the origins of the space coordinateaxes set up in F and F 0coincide at time readings x0D0,x00D0by observers located at the corresponding origins of these respective coordinate systems. We use the notation .x/SYN/D.x0;x//DC1x,/SYND0;1;2;3, and our Minkowski metric is deﬁned by Œ/DC1/SYN/ETB/c141DdiagŒ/NUL1;1;1;1/c141 ,Œ/DC1/SYN/ETB/c141DdiagŒ/NUL1;1;1;1/c141 .
Authors who have also used this signature for the metric include, Julian Schwinger,Steven Weinberg,. . . . Others may use the minus of ours. Surprisingly, Paul Dirachas used both signatures at different times of his career. It doesn’t matter whichone to use. It is easy to keep track of the relative minus sign. In our notation,x 0D/NUL x0,xiDxiforiD1;2;3, x/SYND/DC1/SYN/ETBx/ETB,x/SYND/DC1/SYN/ETBx/ETB,Œ/DC1/SYN/ETB/DC1/ETB/ESC/c141D Œ•/SYN/ESC/c141DdiagŒ1;1;1;1/c141 . Under a homogeneous Lorentz transformation, x0/SYND/ETX/SYN/ETBx/ETB; (2.2.1) 9The reader may wish to consult Manoukian [ 1], pp. 55–65, 112–115, where additional details, and proofs, are spelled out in quantum theory.
2.2 Minkowski Spacetime: Common Arena of Elementary Particles 51 where/ETX/SYN/ETBis independent of x,x0. Before we spell out the structure of the matrix ./ETX/SYN/ETB/, we note that @x0/SYN @x/ETBD/ETX/SYN/ETB; (2.2.2) and hence from the chain rule @/ETBD/ETX/SYN/ETB@0 /SYN;/NUL@/ETB/DC1@ @x/ETB/SOH: (2.2.3) Quite generally, we have the Lorentz invariant property .x0/SYN/NULy0/SYN//DC1/SYN/ETB.x0/ETB/NULy0/ETB/D.x/SYN/NULy/SYN//DC1/SYN/ETB.x/ETB/NULy/ETB/; (2.2.4) from which one may infer that /ETX/SYN/SUB/DC1/SYN/ETB/ETX/ETB/NAKD/DC1/SUB/NAK;. /ETX/NUL1//SUB/ETBD/ETX/ETB/SUB; (2.2.5) @0 /SYND/ETX/SYN/ETB@/ETB; /ETXD/ETX0;/NUL /ETX/DC1@/SYN@/SYN/SOH : (2.2.6) The Poincaré (inhomogeneous Lorentz) tr ansformations in Minkowski spacetime x!x0are deﬁned by x0/SYND/ETX/SYN/ETBx/ETB/NULb/SYN: (2.2.7) Consistency of such a transformation requires that its structure remains the same under subsequent transformations leading t o group properties spelled out below. For
a subsequent transformation x0!x00, w em a yu s e( 2.2.7 ) to write x00/NAKD/ETX0/NAK/SYNx0/SYN/NULb0/NAK; (2.2.8) or x00/NAKD/NUL /ETX0/NAK /SYN/ETX/SYN /ETB/SOH x/ETB/NUL/NUL /ETX0/NAK /SYNb/SYNCb0/NAK/SOH : (2.2.9) Thus a Poincaré transformation may be speciﬁed by a pair ./ETX;b/, with/ETXD ./ETX/SYN/ETB/,bD.b/ETB/, satisfying the group properties: 1. Group multiplication: ./ETX0;b0/./ETX; b/D./ETX0/ETX;/ETX0bCb0/. 2. Identity.I;0/:.I;0/./ETX; b/D./ETX;b/. 3. Inverse./ETX;b//NUL1D./ETX/NUL1;/NUL/ETX/NUL1b/:./ETX;b//NUL1./ETX;b/D.I;0/. 4. Associativity Rule: ./ETX3;b3/Œ./ETX2;b2/./ETX1;b1//c141DŒ./ETX3;b3/./ETX2;b2//c141./ETX1;b1/. To spell out the general structure of the matrices /ETX/SYN/ETB, we ﬁrst consider spatial rotations of coordinate systems. A point xspeciﬁed in a given 3D coordinate system
52 2 Preliminaries will be read as x0in a coordinate system obtained from the ﬁrst by a c.c.w rotation by an angle/CANabout a unit vector n, x0iDRijxj; RijRikD•jk; i;j;kD1;2;3; (2.2.10) with a summation over repeated indices understood. Here Rikare the matrix elements of the rotation matrix with RikD•ik/NUL"ij knjsin/CANC/NUL •ik/NULnink/.cos/CAN/NUL1/SOH ; (2.2.11) "ij kis totally anti-symmetric with "123DC1. If for a given unit vector n, we use the notation x0Dx./CAN/,t h e n from ( 2.2.10 ), (2.2.11), it is worth noting that d d/CANx./CAN/D/NUL n/STXx./CAN/; x.0/Dx: (2.2.12) The general structure of the matrices /ETX/SYN/ETB, involving such rotations and so- called Lorentz boosts is given by /ETXijD/ETXijDRijC.%/NUL1/Rikˇkˇj ˇ2; (2.2.13) /ETX0 0D/NUL/ETX00D%/DC1/NUL 1/NULˇ2/SOH/NUL1=2; (2.2.14) /ETX0iD/ETX0iD/NUL%ˇi; (2.2.15) /ETXi0D/NUL/ETXi0D/NUL%Rijˇj; (2.2.16) Using rather a standard notation for ˇ, its physical interpretation is as follows.
From the second identity in ( 2.2.5 ) it follows from the transformation law x0/SYND /ETX/SYN/ETBx/ETBin (2.2.1 )t h a t x/ETBD/ETX/SYN/ETBx0/SYN. The origin of the spacial coordinate system of the frame F0is speciﬁed by x0jD0;jD1;2;3. From this and ( 2.2.14), (2.2.15), we have x0D%x00,xiD%ˇix00.T h a t i s , ˇdenotes the velocity with which the origin of the spatial coordinate system of F0moves relative to the corresponding origin of F. Now put a (massive m¤0) particle at the origin of the spatial coordinate system of F, the so-called rest frame of the particle, i.e., with energy-momentum p/SYND.m;0/. Then if no initial relative rotation of the spatial coordinate systems of the frames is involved, the velocity of the particle in F0 (the “laboratory” frame) would be simply /NULˇ, i.e., of momentum p0D/NUL m%ˇ. With a relative rotation of the coordinate systems, the relation p0jD/ETXj0mgives p0jDRjk./NULm%ˇk/showing simply a c.w. rotation of the earlier expression.
Equally important are the transformations ( 2.2.7 ), (2.2.13), (2.2.14), (2.2.15), (2.2.16) for inﬁnitesimal changes •b,•/CAN,•ˇ. To ﬁrst order, •x/SYNDx/SYN/NULx0/SYND•b/SYN/NUL•!/SYN/ETBx/ETB; (2.2.17)
2.2 Minkowski Spacetime: Common Arena of Elementary Particles 53 where /ETX/SYN /ETBD•/SYN /ETBC•!/SYN /ETB;• !/SYN/ETBD/NUL•!/ETB/SYN; (2.2.18) •!ijD"ij knk•/CAN; (2.2.19) •!i0D/NUL•ˇi; (2.2.20) •!0 iD/NUL•ˇi; (2.2.21) •!00D0: (2.2.22) andi;jD1;2;3;/SYN;/ETBD0;1;2;3. When the parameters •b,•/CAN,•ˇare led continuously to go to zero, the induced transformations in the underlying vector space of particle states go over to the identity. The latter being unitary, we may refer to Wigner’s Theorem (Sect. 2.1) to infer that an induced transformation is represented by a unitary rather than byan anti-unitary operator. Such transformations, in turn, generate an algebra of thecorresponding generators referred to as t he Poincaré algebra. The corresponding analysis will be carried out in Sect. 4.2. Before closing this section, we note that the general structure in ( 2.2.5 ), rewritten in matrix form reads /ETX >/DC1/ETXD/DC1; (2.2.23)
and since det /DC1D/NUL1, we may infer that for the transformations in ( 2.2.1 )d e t/ETXD C1. It is not/NUL1as this transformation includes neither time nor space reﬂections. From this and ( 2.2.2 ) we learn that the Jacobian of the transformation x0!xis given by det /ETXD1, establishing the Lorentz invariance of the volume element in Minkowski spacetime: .dx/D.dx0/; where.dx//DC1dx0dx1dx2dx3: (2.2.24) This together with the deﬁnition of a Lorentz scalar ˚.x/by the condition ˚0.x0/D˚.x/; (2.2.25) under the above Lorentz transformations, guarantees the invariance of integrals of the form ADZ .dx/˚.x/; (2.2.26) such as the action integral, and lead to the d evelopment of Lorentz invariant theories.
54 2 Preliminaries 2.3 Representations of the Dirac Gamma Matrices; Majorana Spinors The Dirac formalism for the description of the relativistic electron in a quantum setting via the Dirac equation ./CR/SYN@/SYN=iCm/ D0, is summarized in Appendix I at the end of the book. The Dirac representation of the gamma /CR/SYNmatrices, in particular, are deﬁned by /CR0D/DC2I0 0/NULI/DC3 ;/CRD/DC20/ESC /NUL/ESC0/DC3 ;/CR5/DC1i/CR0/CR1/CR2/CR3D/DC20I I0/DC3 ; (2.3.1) satisfying the anti-commutation relations: f/CR/SYN;/CR/ETBgD/NUL2/DC1/SYN/ETB. Various representa- tions of the gamma matrices, satisfying the same anti-commutation relations, maybe similarly deﬁned. We here introduce two other representations which turn up tobe of quite importance in modern ﬁeld theory. They arise as follows. The unitary matrix GD1 p 2/DC2II /NULII/DC3 ;G/NUL1DG/c142D1p 2/DC2I/NULI II/DC3 ; (2.3.2) introduces via the transformation G/CR/SYNG/NUL1the following representation of gamma
matrices, known as the chiral representation, in which /CR5is diagonal /CR0D/DC20/NULI /NULI0/DC3 ;/CRD/DC20/ESC /NUL/ESC0/DC3 ;/CR5D/DC2I0 0/NULI/DC3 ; (2.3.3) where for simplicity we have used the same notation for the resulting /CR/SYNmatrices. Note that /CRcoincides with the one in the Dirac representation. The chiral representation is important in dealing with massless particles and in the study ofsupersymmetry. While the unitary matrix GDG /NUL1DG/c142D1p 2/DC2I/ESC2 /ESC2/NULI/DC3 ; (2.3.4) leads to the so-called Majorana representation /CR0D/DC20/ESC2 /ESC20/DC3 ;/CR5D/DC2/ESC20 0/NUL/ESC2/DC3 ; (2.3.5) /CR1D/DC2i/ESC30 0i/ESC3/DC3 ;/CR2D/DC20/NUL/ESC2 /ESC20/DC3 ;/CR3D/DC2/NULi/ESC10 0/NULi/ESC1/DC3 ; (2.3.6)
2.3 Representations of the Dirac Gamma Matrices; Majorana Spinors 55 in which/NUL /CR/SYN=i/SOH/ETXD/NUL /CR/SYN=i/SOH , making the Dirac operator/NUL /CR/SYN@/SYN=iCm/SOH real.T h i s representation is particularly convenient in analyzing general results in ﬁeld theory such as the Spin & Statistics Connection . If satisﬁes the Dirac equation in an external electromagnetic ﬁeld e A/SYN.x/ of charge e then CD C >,w h e r e D /c142/CR0,a n d Cis the charge conjugation matrix deﬁned by CDi/CR2/CR0, satisﬁes the Dirac equation with eA/SYN.x/replaced by/NULeA/SYN.x/i.e., with the sign of the charge e reversed. (See Eq. (I.3 ) in Appendix Iat the end of the book). In the Dirac and chiral representations the charge conjugation matrix are given by/NUL ./ESC2/>D/NUL/ESC2/SOH CDiracD/DC20/NULi/ESC2 /NULi/ESC20/DC3 ; CchiralD/DC2/NULi/ESC20 0i/ESC2/DC3 : (2.3.7) We note that /CR5C;/CR5/CR/SYNC, together with C, are anti-symmetric matrices.
Some properties of the charge conjugation matrix CDi/CR2/CR0, in general, are: (]stands for any of the operations .://c142;.://NUL1;.:/>) C\D/NUL C;C/NUL1/CR/SYNCD/NUL./CR/SYN/>;ŒC;/CR5/c141D0;C/NUL1/CR5/CR/SYNCD/NUL /CR5/CR/SYN/SOH>; (2.3.8) C/NUL1Œ/CR/SYN;/CR/ETB/c141CD/NUL.Œ/CR/SYN;/CR/ETB/c141/>; C/NUL1/CR5Œ/CR/SYN;/CR/ETB/c141CD/NUL/NUL /CR5Œ/CR/SYN;/CR/ETB/c141/SOH>: (2.3.9) Once a charge matrix Chas been deﬁned, one may deﬁne a Majorana spinor /DC2, not to be confused with the Majorana representation of the gamma matrices, by the condition /DC2D/DC2C,t h a ti s /DC2DC/DC2>;/DC2D/NUL/DC2>C/NUL1D/NUL.C/DC2/>: (2.3.10) From these deﬁnitions, one may infer the general structure of a Majorana spinor, for example, in the chiral representation, to be of the form: /DC2D0 BB@/DC21 /DC22 /DC23 /DC241 CCAD0 BB@/DC2/ETX 4 /NUL/DC2/ETX 3 /DC23 /DC241 CCA: (2.3.11)
56 2 Preliminaries 2.4 Differentiation and Integration with Respect to Grassmann Variables Consider nreal Grassmann variables /SUB1;:::/SUB n, i.e., variables that satisfy f/SUBi;/SUBkgD 0, pairwise, which, in particular, implies that for any k:/SUB2 kD0. The left-hand derivative with respect to a variable /SUBjof any product /SUBi1:::/SUB ikis deﬁned by @ @/SUBj/SUBi1:::/SUB ikD./NUL1/ıj/SUBi1:::b/SUBj:::/SUB ik; (2.4.1) whereıjdenotes the position of /SUBjfrom the left/NUL1, in the product /SUBi1:::/SUB ik,and /SUBjis omitted on the right-hand side of the equation as indicated by the “hat” sign on it. One may also deﬁne a right-hand derivat ive, taking a derivative from the right, with a corresponding rule. In particular note that, @ @/SUBk/SUBi/SUBjD•ik/SUBj/NUL•jk/SUBi; (2.4.2) and /SUB@ @/SUBj;/SUBk/ESC D•jk: (2.4.3) One may deﬁne the differential operator d DPn kD1d/SUBk@=@/SUB ksatisfying the rule d./SUBi/SUBj/Dd/SUBi/SUBjC/SUBid/SUBj: (2.4.4)
On the other hand by using the rule in ( 2.4.2 ) in applying the differential operator dg i v e s d./SUBi/SUBj/Dd/SUBi/SUBj/NULd/SUBj/SUBi; (2.4.5) which upon comparison with ( 2.4.4 )g i v e s d/SUBj/SUBiD/NUL/SUBid/SUBj: (2.4.6) To extend the above rules for differentiations with respect to complex variables, one may proceed as follows. Given two real Grassmann variables /SUBR;/SUBIsatisfying f/SUBR;/SUBIgD0, one may deﬁne a complex Grassmann variable /SUBD/SUBRCi/SUBI.U s i n g the notation/SUB/ETXfor the complex conjugate of /SUB, the following anti-commutation rules emerge f/SUB;/SUBgD0;f/SUB;/SUB/ETXgD0;f/SUB/ETX;/SUB/ETXgD0: (2.4.7)
2.4 Differentiation and Integration with Respect to Grassmann Variables 57 Imposing a reality restriction on the product /SUB/ETX/SUBD/NUL i/SUBI/SUBR; (2.4.8) i.e.,./SUB/ETX/SUB//ETXD./SUB/ETX/SUB/, leads to the following deﬁnition ./SUBI/SUBR//ETXD/NUL/SUBI/SUBR or./SUBI/SUBR//ETXD/SUBR/SUBI; (2.4.9) showing that complex conjugation of the product of two real Grassmann variables reverses their order. This in turn implies that for complex (or real) Grassmann variables/SUB1;:::;/SUB n. ./SUB1:::/SUB n//ETXD/SUB/ETX n:::/SUB/ETX 1: (2.4.10) One may also deﬁne two sets of derivatives with respect to variables /SUBi;/SUB/ETX i,a s before, and note that /SUB@ @/SUBi;@ @/SUBj/ESC D0;( @ @/SUBi;@ @/SUB/ETX j) D0: (2.4.11) We now consider integrations with respect to real Grassmann variables. Due to the property/SUB2 iD0, one has to investigate only the meanings of the following two integrals, for a given i, Z d/SUBi;Z d/SUBi/SUBi: (2.4.12)
Imposing translational invariance of these integrals under /SUBi!/SUBiC˛iD/SUB0 i, with˛ianti-commuting with /SUBi, simply means that Z d/SUB0 iDZ d/SUBi;Z d/SUB0i/SUB0iDZ d/SUBi/SUBi: (2.4.13) In particular, translation invariance gives Z d/SUB0i/SUB0iDZ d/SUBi/SUBiC/DC2Z d/SUBi/DC3 ˛i; (2.4.14) for arbitrary˛i. We may thus infer that Z d/SUBiD0: (2.4.15)
58 2 Preliminaries The second integral in ( 2.4.12) may be normalized as Z d/SUBi/SUBiD1; (2.4.16) for any i, a normalization condition that will be adopted in this book. For any function f./SUBi/D˛0Cc1/SUBi,w h e r e c1is a c-number, and ˛0anti- commutes with /SUBi, one clearly has Z d/SUBi./SUBi/NULˇi/f./SUBi/Df.ˇi/: (2.4.17) This allows one to introduce a Dirac delta given by •./SUB i/NULˇi/D./SUBi/NULˇi/; (2.4.18) being a Grassmann variable itself. That is d /SUBi•./SUB i/NULˇi/commutes with Grassmann variables. Before introducing integrations over complex Grassmann variables, consider making a change of integration variables /SUBiDCij/CANj; (2.4.19) in an integral, where a summation over junderstood, and the Cijare c-numbers, i;jD1;:::; n. We are thus led to consider, in general, the integral Z d/SUB1:::d/SUBn/SUB1:::/SUB nDZ d/CAN1:::d/CANnJC1i1/CANi1:::Cnin/CANin; (2.4.20) where Jis the Jacobian of the transformation to be determined. Using the anti-
commutativity property of the /CANj, we may write /CANi1:::/CAN inD"i1:::in/CAN1:::/CAN n; (2.4.21) where"i1:::inis the Levi-Civita symbol totally anti-symmetric with "1:::nDC1. Using the deﬁnition of a determinant of the matrix ŒCij/c141 C1i1:::Cnin"i1:::inDdetC; (2.4.22) we immediately obtain JD1=det C; (2.4.23) being the inverse of det C.
2.4 Differentiation and Integration with Respect to Grassmann Variables 59 As before, out of two real Grassmann variables /SUBR;/SUBI, we may deﬁne a complex one/SUBD/SUBRCi/SUBI. Now consider the transformation /SUBR;/SUBI!/SUB;/SUB/ETXdeﬁned by /DC2/SUBR /SUBI/DC3 D1 2/DC211 i/NULi/DC3/DC2/SUB/ETX /SUB/DC3 ; (2.4.24) with the Jacobian of the transformation given by JD/DC2 det1 2/DC211 i/NULi/DC3/DC3/NUL1 D2i: (2.4.25) Using the fact that /SUBI/SUBRD/SUB/SUB/ETX=2i, we may infer that 1DZ d/SUBRd/SUBI/SUBI/SUBRDZ d/SUB/ETXd/SUB.2i/./SUB/SUB/ETX=2i/; (2.4.26) or Z d/SUB/ETXd/SUB/SUB/SUB/ETXD1: (2.4.27) Hence for consistency with the above integral we may set Z d/SUB/SUBD1;Z d/SUB/ETX/SUB/ETXD1: (2.4.28) For integrations over ncomplex Grassmann variables /SUB1;:::;/SUB n, consider the integral IDZd/SUB/ETX 1d/SUB1 i/SOH/SOH/SOHd/SUB/ETX nd/SUBn iexpŒ/NULi/SUB/ETX iAij/SUBj/c141 (2.4.29) where the Aijare c-numbers. The commutativity of /SUB/ETX iAi1/SUB1, for example, with all
the Grassmann variables, and so on, allows us to rewrite the integral Ias (since ./SUB/ETX 1/2D0;:::;./SUB/ETX n/2D0) IDZ d/SUB/ETX 1/NUL /NUL/SUB/ETX 1A1j1/SUBj1/SOH d/SUB1/SOH/SOH/SOHd/SUB/ETX n/NUL /NUL/SUB/ETX nAnjn/SUBjn/SOH d/SUBn DZ d/SUB1/SUBj1/SOH/SOH/SOHd/SUBn/SUBjnA1j1/SOH/SOH/SOHAnjn DZ d/SUB1/SUB1/SOH/SOH/SOHd/SUBn/SUBn"j1:::jnA1j1/SOH/SOH/SOHAnjn; (2.4.30)
60 2 Preliminaries where we have used ( 2.4.21). From ( 2.4.28) and the deﬁnition of a determi- nant ( 2.4.22), the following expression for the integral emerges Zd/SUB/ETX 1d/SUB1 i/SOH/SOH/SOHd/SUB/ETX nd/SUBn iexpŒ/NULi/SUB/ETX iAij/SUBj/c141DdetA: (2.4.31) For det A¤0, the above integral may be used to give another useful integral. To this end, we note the following. We may call the variables /SUB/ETX iin (2.4.31) anything we like and use any notation for them. In the language of calculus they are dummy variables. We may go even further by noting that we are free to make completelydifferent change of variables for the /SUB jand the former variables. Taking these two points into account, we m ake the substitutions /SUBj!/SUBj/NUL.A/NUL1/DC1/jand/SUB/ETX i!/SUBi/NUL ./DC1A/NUL1/ito obtain directly from ( 2.4.31) Zd/SUB1d/SUB1 i/SOH/SOH/SOHd/SUBnd/SUBn iexpŒi./NUL/SUBiAij/SUBjC/SUBi/DC1iC/DC1i/SUBi//c141D.det A/expŒi/DC1iA/NUL1 ij/DC1j/c141: (2.4.32)
This integral is to be compared with the corresponding one for integrations over commuting complex variables Zdz/ETX1dz1 2/EM=i/SOH/SOH/SOHdz/ETXndzn 2/EM=iexpŒi./NULz/ETXiAijzjCz/ETXiKiCK/ETX izi//c141D/DC21 detA/DC3 exp/NUL iK/ETX iA/NUL1 ijKj/SOH : (2.4.33) For integrations over real commuting variables one also has Zdy1p 2/EM=i/SOH/SOH/SOHdynp 2/EM=iexph i/DLE /NUL1 2yiAijyjCKiyi/DC1i D/DC21p detA/DC3 exphi 2KiA/NUL1 ijKji : (2.4.34) 2.5 Fourier Transforms Invol ving Grassmann Variables The study of Fourier transform theory of Grassmann variables rests on the followingtwo integrals. The ﬁrst one is Z d /DC11d/DC11:::d/DC1nd/DC1nexp iŒ/DC1./SUB/NUL˛/C./SUB/NUL˛//DC1/c141 D./SUB1/NUL˛1/./SUB1/NUL˛1/:::./SUB n/NUL˛n/./SUBn/NUL˛n/; (2.5.1) which, by now, should be straightforward to verify (see Problem 2.6). In the above equation, we have used the standard notation /DC1./SUB/NUL˛/DP j/DC1j./SUBj/NUL˛j/. The second
2.5 Fourier Transforms Involving Grassmann Variables 61 one is Z d/SUB1d/SUB1:::d/SUBnd/SUBn./SUB1//DC41./SUB1/"1:::./SUBn//DC4n./SUBn/"n./SUB1/NUL˛1//STX /STX./SUB1/NUL˛1/:::./SUB n/NUL˛n/./SUBn/NUL˛n/D.˛1//DC41.˛1/"1:::.˛n//DC4n.˛n/"n; (2.5.2) where the/DC4j;"jare either0or1, and a factor such as .0/0, (e.g., when ˛jD 0;" jD0), is to be replaced by 1 on the ri ght-hand side of the equation. The second one seems more difﬁcult to verify but easily follows by ﬁrst writing ./SUB//DC4Dı/DC40C/SUBı/DC41;. /SUB /"Dı"0C/SUBı"1: (2.5.3) The integral on the left-hand side of (2.5.2 ) then becomes Z d/SUB1.ı/DC410/NUL/SUB1ı/DC411/./SUB1/NUL˛1/./SUB1/NUL˛1/d/SUB1.ı"10C/SUBı"11//STX /SOH/SOH/SOH/STX d/SUBn.ı/DC4n0/NUL/SUBnı/DC4n1/./SUBn/NUL˛n/./SUBn/NUL˛n/d/SUBn.ı"n0C/SUBı"n1/: (2.5.4) Therefore it remains to face the following integral which works out as follows: Z d/SUB.ı/DC40/NUL/SUBı/DC41/./SUB/NUL˛/./SUB/NUL˛/d/SUB.ı"0C/SUBı"1/ DZ
d/SUB.ı/DC40/NUL/SUBı/DC41/./SUB/NUL˛/d/SUB./SUB/NUL˛/.ı"0C/SUBı"1/ DZ d/SUB/SUB.ı/DC40C˛ı/DC41/d/SUB/SUB.ı"0C˛ı"1/D.˛//DC4.˛/"; (2.5.5) which leads from ( 2.5.4 ) to the expression on the right-hand side of ( 2.5.2 ). Now we are ready to introduce Fourier transforms involving Grassmann variables. Given a function FŒ/SUB;/SUB/c141 which is a linear combination of terms as Œ./SUB1//DC41./SUB1/"1:::./SUBn//DC4n./SUBn/"n/c141, we deﬁne its Fourier transform: ./SUB;/SUB/!./DC1;/DC1/ by QFŒ/DC1;/DC1/c141DZd/SUB1d/SUB1 i:::d/SUBnd/SUBn iFŒ/SUB;/SUB/c141 expŒi./SUB/DC1C/DC1/SUB//c141: (2.5.6) To ﬁnd the inverse Fourier transform: ./DC1;/DC1/!./SUB;/SUB/ , we multiply the above integral by id/DC11d/DC11:::id/DC1nd/DC1nexpŒ/NULi./DC1˛C˛/DC1//c141;
62 2 Preliminaries which commutes with everything, and integrate using ( 2.5.1 )a n dt h e n( 2.5.2 ). The right-hand becomes10 Z d/SUB1d/SUB1:::d/SUBnd/SUBnFŒ/SUB;/SUB/c141Z d/DC11d/DC11:::d/DC1nd/DC1nexp iŒ/DC1./SUB/NUL˛/C./SUB/NUL˛//DC1/c141 DZ d/SUB1d/SUB1:::d/SUBnd/SUBnFŒ/SUB;/SUB/c141./SUB1/NUL˛1/./SUB1/NUL˛1/:::./SUB n/NUL˛n/./SUBn/NUL˛n/DFŒ˛;˛/c141: (2.5.7) By equating this expression with the resulting one on the left-hand side of the equation in question, the following expression emerges for the inverse Fouriertransform FŒ /SUB;/SUB/c141DZ id/DC11d/DC11:::id/DC1nd/DC1nQFŒ/DC1;/DC1/c141 expŒ/NULi./DC1/SUBC/SUB/DC1//c141; (2.5.8) written in terms of variables ./SUB;/SUB/ . For integration over commuting complex variables, with zD.z1;:::; zn/>,w e deﬁne a Fourier transform by QFŒK/c142;K/c141DZ/DLEnY jdz/ETX jdzj 2/EM=i/DC1 FŒz/c142;z/c141expŒi.z/c142KCK/c142z//c141; (2.5.9) then for the inverse Fourier transform, we have FŒz/c142;z/c141DZ/DLEnY jdK/ETX jdKj 2/EMi/DC1
QFŒK/c142;K/c141expŒ/NULi.z/c142KCK/c142z//c141: (2.5.10) For integrations over commuting real variables, we deﬁne a Fourier transform as QFŒK/c141DZ/DLEnY jd/RSjp 2/EM=i/DC1 FŒ/RS/c141expŒiK>/RS/c141; (2.5.11) and the inverse Fourier transform is then given by FŒ/RS/c141DZ/DLEnY jdKjp 2/EMi/DC1 QFŒK/c141expŒ/NULiK>/RS/c141: (2.5.12) 10Note that the factors ./SUBj/NUL˛j/./SUBj/NUL˛j/commute with everything.
2.6 Functional Differentiation and Integration; Functional Fourier Transforms 63 2.6 Functional Differentiation and Integration; Functional Fourier Transforms We begin by extending the concept of differentiation with respect to Grassmann variables/DC11;:::;/DC1 n;/DC11;:::;/DC1nto a continuum limit as n!1 . Such an extension for commuting variables will then beco me straightforward to carry out. By a continuum limit as n!1 , it is meant that one introduces Grassmann type functions/DC1a.x/;/DC1a.x/of a continuous variable x, such as specifying spacetime points, moreover they may also depend, in ge neral, on discrete parameters, such as a spinor index. By Grassmann type functions, one also means that these functionsall anti-commute for all xand all a. The simple rules of differentiations taken from the left with respect to a discrete set of Grassmann variables, deﬁned earlier in Sect. 2.4,s u c ha s @ @/DC1i/DC1jD•ji;@ @/DC1i/DC1jD•ji;@ @/DC1i/DC1jD0;@ @/DC1i/DC1jD0; (2.6.1) @
@/DC1i/DC1j/DC1k/DC1m/DC1sDıji/DC1k/DC1m/DC1s/NUL/DC1j/DC1k/DC1mısi; (2.6.2) become replaced, in this limit, by11 • •/DC1a.x//DC1b.x0/Dıbaı.x0/NULx/;• •/DC1a.x//DC1b.x0/Dıbaı.x0/NULx/; (2.6.3) • •/DC1a.x//DC1b.x0//DC1c.y0//DC1d.y00//DC1e.x00/Dıbaı.x0/NULx//DC1c.y0//DC1d.y00//DC1e.x00/ /NUL/DC1b.x0//DC1c.y0//DC1d.y00/ıeaı.x00/NULx/; (2.6.4) involving Dirac deltas instead of just Kronecker deltas. One uses the notation•=•/DC1 a.x/etc, for the functional derivatives , i.e., for derivatives with respect to functions. These functions /DC1a.x/;/DC1a.x/together with the functional derivatives with respect to them satisfy the anti-commutation rules: f/DC1a.x/;/DC1 b.x0/gD0;f/DC1a.x/;/DC1b.x0/gD0;f/DC1a.x/;/DC1b.x0/gD0; (2.6.5) /SUB• •/DC1a.x/;• •/DC1b.x0//ESC D0;/SUB• •/DC1a.x/;• •/DC1b.x0//ESC D0;/SUB• •/DC1a.x/;• •/DC1b.x0//ESC D0; (2.6.6) 11We interchangeably use the notations ı.x/NULx0/andı.D/.x/NULx0/where Dis the dimensionality of spacetime.
64 2 Preliminaries /SUB• •/DC1a.x/;/DC1b.x0//ESC Dıbaı.x0/NULx/;/SUB• •/DC1a.x/;/DC1b.x0//ESC Dıbaı.x0/NULx/: (2.6.7) /SUB• •/DC1a.x/;/DC1b.x0//ESC D0;/SUB• •/DC1a.x/;/DC1b.x0//ESC D0: (2.6.8) Important functionals, i.e., functions of functions, that occur repeatedly in ﬁeld theory are the exponentials of lin ear and/or bilinear forms such as: expŒi./DC1/SUBC/SUB/DC1//c141; where/DC1/SUB/DC1X aZ .dx//DC1a.x//SUBa.x/; (2.6.9) /SUBa.x/;/SUBa.x/anti-commute with /DC1b.x0/;/DC1b.x0/as well as with the functional derivatives with respect to the latter, and expŒi/DC1A/DC1/c141; where/DC1A/DC1/DC1X a;bZ .dx/.dx0//DC1a.x/Aab.x;x0//DC1b.x0/; (2.6.10) /DC1a.x/;/DC1a.x/commute with Abc.x0;x00/. Note that the above two functionals being even in anti-commuting functions commute with everything. Typical functional derivatives of such functionals are • •/DC1a.x/expŒi./DC1/SUBC/SUB/DC1//c141D/NUL i/SUBa.x/expŒi./DC1/SUBC/SUB/DC1//c141; (2.6.11) •
•/DC1a.x/expŒi./DC1/SUBC/SUB/DC1//c141DC i/SUBa.x/expŒi./DC1/SUBC/SUB/DC1//c141; (2.6.12) note the minus sign on the right-hand side of the ﬁrst equation, • •/DC1a.x/expŒi/DC1A/DC1/c141DiX bZ .dx0/Aab.x;x0//DC1b.x0/expŒi/DC1A/DC1/c141; (2.6.13) • •/DC1a.x/expŒi/DC1A/DC1/c141D/NUL iX cZ .dx0//DC1c.x0/Aca.x0;x/expŒi/DC1A/DC1/c141; (2.6.14) • •/DC1c.y/• •/DC1a.x/expŒi/DC1A/DC1/c141DiA.x;y/acexpŒi/DC1A/DC1/c141CexpŒi/DC1A/DC1/c141/STX /STX X dZ .dx00//DC1d.x00/Adc.x00;y/! X bZ .dx0/Aab.x;x0//DC1b.x0/! ; (2.6.15) • •/DC1a.x/• •/DC1b.x0/expŒi/DC1A/DC1/c141D/NUL• •/DC1b.x0/• •/DC1a.x/expŒi/DC1A/DC1/c141: (2.6.16)
2.6 Functional Differentiation and Integration; Functional Fourier Transforms 65 A point which is perhaps not sufﬁciently emphasized is that a functional derivative may be also applied in a simple manner to a derivative such as shown below @/SYN/DC1a.x/D@/SYNZ .dx0/ı.x/NULx0//DC1a.x0/DZ .dx0//DC1a.x0/@/SYNı.x/NULx0/; • •/DC1b.x00/@/SYN/DC1a.x/DZ .dx0/ıabı.x0/NULx00/@/SYNı.x/NULx0/Dıab@/SYNı.x/NULx00/: (2.6.17) We may take over the integral in ( 2.4.32) in a continuum limit, and use the notationQ xa.d/SUBa.x/d/SUBa.x/=i//DC1.D/SUBD/SUB/, which is even in anti-commuting variables, to obtain for a matrix M=ŒMab.x;x0//c141: expŒi/DC1M/NUL1/DC1/c141D.detM//NUL1Z .D/SUBD/SUB/expŒi./NUL/SUBM/SUBC/SUB/DC1C/DC1/SUB//c141; (2.6.18) where we note that detMDZ .D/SUBD/SUB/expŒi./NUL/SUBM/SUBC/SUB/DC1C/DC1/SUB//c141ˇˇˇ /DC1D0;/DC1D0: (2.6.19) Here one may use a matrix notation for Mab.x;x0//DC1hxjMabjx0i,hxjIabjx0i Dıabı.x/NULx0/, and note that X bZ hxjM/NUL1
abjx0i.dx0/hx0jMbcjx00iDıacı.x/NULx00/: (2.6.20) To see how these integrals work, consider, for example, the Dirac operator in the presence of an external electromagnetic ﬁeld A/SYN.x/(see Eq. ( I.2) in Appendix Iat the end of the book) MDh /CR/SYN/DLE@/SYN i/NULeA/SYN/DC1 Cmi /DC1M.e/; (2.6.21) hxjM.e/jx0iDh /CR/SYN/DLE@/SYN i/NULeA/SYN.x//DC1 Cmi ı.x/NULx0/; (2.6.22) corresponding to the interaction of an electron with an external electromagnetic potential. The following notation is often used for the matrix M.0/D/DLE /CR/SYN@/SYN iCm/DC1 /DC1S/NUL1 C: (2.6.23)
66 2 Preliminaries Subject to an appropriate boundary c ondition, we will learn in Sect. 3.1how to invert the matrix S/NUL1 C, thus deﬁning the matrix SC. We may conveniently write M.e/D .S/NUL1 C/NULe/CRA/, and note that detM.e/DexpŒTr ln M.e//c141DexpŒTr ln.S/NUL1 C/NULe/CRA//c141: (2.6.24) In particular, detM.e/ detM.0/DexpŒTr ln.I/NULeSC/CRA//c141; (2.6.25) and the expression in the exponential, on the right-hand side may be formally expanded as follows Tr ln.I/NULeSC/CRA/D/NULX n>1.e/n nZ .dx1/:::. dxn/A/SYN1.x1/:::A/SYNn.xn//STX /STXTrh SC.x1;x2//CR/SYN2SC.x2;x3//CR/SYN3:::SC.xn;x1//CR/SYN1i ;.dx//DC1dx0dx1dx2dx3; (2.6.26) where the trace operation is over the gamma matrices. Such expressions will bestudied in detail later and applications will be given. Using the notation M.e//DC1ŒS A C/c141/NUL1, for the matrix depending on the elec- tromagnetic potential, and taking into account the expression in ( 2.6.19), we may equivalently rewrite ( 2.6.18)a s expŒi/DC1SA
C/DC1/c141DZŒ/DC1;/DC1/c141 ZŒ0;0/c141; (2.6.27) ZŒ/DC1;/DC1/c141DZ D/SUBD/SUBexp ih /NUL/SUB/DLE /CR/SYN/DLE@/SYN i/NULeA/SYN/DC1 Cm/DC1 /SUBC/SUB/DC1C/DC1/SUBi : (2.6.28) We note that this deﬁnes a functional Fourier transform of exph /NULi/SUB/DLE /CR/SYN/DLE@/SYN i/NULeA/SYN/DC1 Cm/DC1 /SUBi ; (2.6.29) giving the Fourier transform from ./SUB;/SUB/ to./DC1;/DC1/ variables We will recognize the expression in the exponential in ( 2.6.29), multiplying the i factor, as the Dirac Lagrangian in the presence of an external electromagnetic ﬁeld, with ./DC1;/DC1/ in (2.6.28) representing external sources coupled to the Dirac ﬁeld. ZŒ/DC1;/DC1/c141= ZŒ0;0/c141
2.6 Functional Differentiation and Integration; Functional Fourier Transforms 67 acts as a generating functional for determining basic components of an underlying theory as we will see this on several occasions later on in the book. In particularby functional differentiating ( 2.6.27) with respect to /DC1a n dt h e nb y/DC1, and setting them equal to zero, gives12iSA C, with SA Cinterpreted as the propagator of a charged particle of charge e in the presence of an external electromagnetic ﬁeld A/SYN.x/.SA C may be then analyzed depending on how complicated the external ﬁeld is. This willbe dealt with in the next chapter. By using the notationQ xd/RS/ETX.x/d/RS.x/=.2/EM= i//DC1.D/RS/ETXD/RS/,f o r c o m p l e x functions/RS/ETX.x/;/RS.x/, of a real variable, the integral ( 2.4.33) in the continuum limit may be written for a matrix M 1 detMexpŒiK/ETXM/NUL1K/c141DZ .D/RS/ETXD/RS/expŒi./NUL/RS/ETXM/RSC/RS/ETXKCK/ETX/RS//c141: (2.6.30)
On the other-hand for a real function /RS.x/of a real variable, the integral in (2.4.34) in the continuum limit, with the notationQ xd/RS.x/=p 2/EM=i/DC1.D/RS/, takes the form 1p detMexphi 2KM/NUL1Ki DZ .D/RS/exph i/DLE /NUL1 2/RSM/RSC/RSK/DC1i : (2.6.31) Functionals FŒ•=•K;•=•K/ETX/c141and FŒ•=•K/c141may be applied, respectively, to (2.6.30)a n d( 2.6.31) to generate more complicated functional integrals. The Functional Fourier transforms in the continuum case may be taken over from ( 2.5.6 ), (2.5.8 ), (2.5.9 ), (2.5.10), (2.5.11), (2.5.12). For anti-commuting ﬁelds we deﬁne QFŒ/DC1;/DC1/c141DZ .D/SUBD/SUB/FŒ/SUB;/SUB/c141 expŒi./SUB/DC1C/DC1/SUB//c141; (2.6.32) and the inverse Functional Fourier transform of QFŒ/DC1;/DC1/c141 is then given by FŒ/SUB;/SUB/c141DZ .QD/DC1QD/DC1/QFŒ/DC1;/DC1/c141 expŒ/NULi./SUB/DC1C/DC1/SUB//c141; (2.6.33) using the notationQ xa/DLE id/DC1a.x/d/DC1a.x///DC1 D.QD/DC1QD/DC1/.
For complex ﬁelds, the Functional Fourier transform reads (see ( 2.5.9 ), (2.5.10)) QFŒK/ETX;K/c141DZ .D/RS/ETXD/RS/QFŒ/RS/ETX;/RS/c141expŒi./RS/ETXKCK/ETX/RS//c141; (2.6.34) 12See Problem 2.8.
68 2 Preliminaries whereQ x/DLE .d/RS/ETX.x/d/RS.x//=.2/EM= i//DC1 =.D/RS/ETXD/RS/, and for the inverse Fourier trans- form one has FŒ/RS/ETX;/RS/c141DZ .QDK/ETXQDK/QFŒK/ETX;K/c141expŒ/NULi.K/ETX/RSC/RS/ETXK//c141; (2.6.35) where nowQ x/DLE .dK/ETX.x/dK.x/.=.2/EM i//DC1 =.QDK/ETXQDK/. For real ﬁelds, the Functional Fourier transform reads (see (2.5.11), (2.5.12)) QFŒK/c141DZ .D/RS/FŒ/RS/c141expŒiK/RS/c141; (2.6.36) whereQ x/DLE d/RS.x/=p .2/EM= i//DC1 =.D/RS/, and the Fourier transform of QFŒK/c141is given by FŒ/RS/c141DZ .QDK/FŒK/c141expŒ/NULiK/RS/c141; (2.6.37) whereQ x/DLE dK.x/=p .2/EMi//DC1 =.QDK/. 2.7 Delta Functionals Consider a real function K.x/and deﬁne the functional GD/NUL1 2Z .dx0/.dx00/./NULi/• •K.x0/M.x0;x00/./NULi/• •K.x00/; (2.7.1) where M.x0;x00/DM.x00;x0/. Then we have the commutation relation iŒG;K.x//c141DiZ .dx0/M.x;x0/• •K.x0/: (2.7.2) By using the relation MM/NUL1DI, we obtain /DLE iŒG;K.x//c141CK.x//DC1 exphi 2KM/NUL1Ki D0: (2.7.3)
SinceŒG;ŒG;K.x//c141/c141D0, we may use the identity eABe/NULADŒA;B/c141CB; forŒA;ŒA;B/c141/c141D0; (2.7.4)
2.7 Delta Functionals 69 with BDK,ADiG, to infer, upon multiplying ( 2.7.3 ), in the process, by exp./NULiG/,t h a t K.x//DLE exp./NULiG/exphi 2KM/NUL1Ki/DC1 D0; (2.7.5) for all x. Therefore for every point xwe may introduce a delta function ı.K.x//and introduce the delta functional ı.K/DY xı/NUL K.x//SOH ; (2.7.6) and use the fact that xf.x/D0implies that f.x/is a delta function ı.x/,u p t o a multiplicative constant, to infer that the expression between the round brackets in (2.7.5 ) is equal toı.K/, up to a proportionality constant, since it is valid for all x, i.e., exphi 2KM/NUL1Ki DCexp.iG/SOH ı.K/; (2.7.7) where Cis aK-independent multiplicative factor, and Gis deﬁned in ( 2.7.1 ). Equivalence of the above formulation with the Functional Fourier transform method developed in the previous section may be established by writing ı.K/DY xZ1 /NUL1d/RS.x/ 2/EMexpŒiK.x//RS.x//c141; (2.7.8) which leads from ( 2.7.7 )t o exphi 2KM/NUL1Ki DC/DC2Y xZ1 /NUL1d/RS.x/ 2/EM/DC3 exph i/DLE /NUL1
2/RSM/RSCK.x//RS.x//DC1i : (2.7.9) Since the left-hand side is unity for KD0, we obtain for 1=C the functional integral 1 CD/DC2Y xZ1 /NUL1d/RS.x/ 2/EM/DC3 exph /NULi 2/RSM/RSi : (2.7.10) Obviously ( 2.7.9 ) coincides with that given in ( 2.6.31) since the .2/EM/ factors in (2.7.9 ) cancel out anyway in determining exp ŒiKM/NUL1K=2//c141. The interest in the expression in ( 2.7.7 ) is that one may also apply to it an arbitrary functional FŒı=ıK/c141 which, as we will see later, essentially provides the solution of ﬁeld theory.
70 2 Preliminaries For complex functions K.x/;K/ETX.x/of a real variable x, the same analysis as above involving both functions, with GD/NULZ .dx0/.dx00/./NULi/• •K.x/M.x0;x00/./NULi/• •K/ETX.x/; (2.7.11) gives expŒiK/ETXM/NUL1K/c141DCexp/NUL iG/SOH ı.K/ETX;K/; (2.7.12) up to a ( K/ETX;K)-independent multiplicative factor C,a n dı.K/ETX;K/may be taken as the productQ xı.KR.x//ı.K I.x//or equivalently as ı.K/ETX;K/DY xZd/RS/ETX.x/d/RS.x/ i.2/EM/2exp/STX i/NUL K/ETX.x//RS.x/C/RS/ETX.x/K.x//SOH/ETX : (2.7.13) For anti-commuting /DC1;/DC1, paying special attention to this property, the same analysis as above gives expŒi/DC1M/NUL1/DC1/c141DCexp.iG/ı./DC1;/DC1/; (2.7.14) up to a (/DC1;/DC1)-independent multiplicative factor C, with GD/NULZ .dx0/.dx00/.i/• •/DC1a.x0/Mab.x0;x00/./NULi/• •/DC1b.x00/; (2.7.15) andı./DC1;/DC1/, absorbing any /EMfactors in C,m a y b e t a k e n a s ı./DC1;/DC1/DY xZ d/SUB.x/d/SUB.x/exp iŒ/DC1.x//SUB.x/C/SUB.x//DC1.x//c141; (2.7.16)
with the functional differentiations on the right-hand side of ( 2.7.14) to be carried out before carrying out the integrations in ( 2.7.16 ). Problems 2.1Consider the matrix GDIcos/DC2C/CR/SOHnsin/DC2,G/c142DG/NUL1DIcos/DC2/NUL/CR/SOHnsin/DC2, where nDa=jaj, for some non-zero dimensionless 3-vector a,a n d cos/DC2Dh/NULp a2C1C1/SOHı/NUL 2p a2C1/SOHi1=2 : Show that G/CR0./CR/SOHaC1/G/NUL1D/CR0p a2C1.
Recommended Reading 71 2.2The Dirac Hamiltonian is given by HD/CR0Œ/CR/SOHpCm/c141in the momentum description. Use the result in Problem 2.1to diagonalize the Hamiltonian in the Dirac representation. 2.3Evaluate the integral over the Grassmann variable /SUBjin (2.4.17). 2.4As a curious property of integration over a Grassmann variable, consider again, as in the previous problem, the function: f./SUB/D˛0Cc1/SUBof a single variable. Show thatR d/SUBf./SUB/D.@=@/SUB/f./SUB/implying an equivalence relationship between integration and differentiation.2.5Use the property of complex conjugation of the product d /SUB R/SUBRof a real Grassmann variable /SUBRto infer that (i).d/SUBR//ETXD/NUL d/SUBR. From this conclude that for a complex Grassmann variable /SUB,.d/SUB//ETXD/NUL d/SUB/ETX. (ii)/DLER d/SUB/SUB/DC1/ETX DR d/SUB/ETX/SUB/ETX. This justiﬁes of assignining the same real numerical values for both integrals:Rd/SUB/SUB,a n dRd/SUB/ETX/SUB/ETXover complex Grassmann variables.
2.6Verify the validity of the integral in ( 2.5.1 ). 2.7Verify the functional differentiations carried out in ( 2.6.15). 2.8Evaluate the functional integral in ( 2.6.27) to the leading order in the external potential as a functional of SC. Then use this expression to determine SA Cto this leading order. References 1. Manoukian, E. B. (2006). Quantum theory: A wide spectrum . Dordrecht: Springer. 2. Manoukian, E. B. (2016). Quantum ﬁeld theory II: Introductions to quantum gravity, supersym- metry, and string theory . Dordrecht: Springer. 3. Wigner, E. P. (1959). Group theory, and its applications to the quantum mechanics of atomic spectra . New York: Academic. 4. Wigner, E. P. (1963). Invariant quantum mechanical equations of motion. In: Theoretical physics (p. 64). Vienna: International Atomic Energy Agency. Recommended Reading 1. Manoukian, E. B. (2006). Quantum theory: A wide spectrum . Dordrecht: Springer.
Chapter 3 Quantum Field Theory Methods of Spin 1=2 With the many-particle aspect encountered by emerging relativity and quantum physics and the necessity of describing thei r creation and annihilation in arbitrary numbers, one introduces the very rich concept of a quantum ﬁeld. As a ﬁrstencounter with this concept, we investigate, in this chapter, the very signiﬁcanceof a quantum ﬁeld for the very simple system described by the celebrated and wellknown Dirac equation as given in Eq. (I.1 ) in Appendix I,a sw e l la so ft h eD i r a c equation in the presence of a general ex ternal electromagnetic potential in Eq. ( I.2). The quantum ﬁeld inevitably leads to the concept of a propagator as describing
the transfer of energy and momentum between emission sources and particledetectors or between particles, in general. In this particular situation involved withthe Dirac ﬁeld, this necessarily leads one to investigate and understand how to invertthe Dirac operator ./CR@= iCm/, based on physical grounds, and deﬁne carefully the boundary conditions involv ed in doing this (Sect. 3.1). The experimental set-up where a particle may be emitted or detected before or after having participated in some physical process, respectively, as well as of theunderlying physical process itself are quite appropriately described by introducingthe so-called vacuum-to-vacuum transition amplitude (Sects. 3.1,3.2,3.3,3.4,3.5, 3.6,3.7,a n d 3.8). The latter amplitude leads quite naturally to the determination of transition amplitudes of physical processes. In particular, the positron is readily re-
discovered by extracting the amplitude by such a method for the system describedby the Dirac equation in an external electromagnetic potential probe (Sect. 3.3). Various applications will be given including to the Coulomb scattering of relativistic electrons, derivation of the Eu ler-Heisenberg effective Lagrangian, as an example of a modiﬁcation of Maxwell’ s equations, as well of the decay of the vacuum and of the underlying Schwinger effect of pair productions by a constantelectric ﬁeld (Sect. 3.8). Particular emphasis is put on deriving the spin & statistics connection of the Dirac quantum ﬁeld, and of the generation of gauge invariantcurrents (Sects. 3.6,3.9,a n d 3.10) based on Schwinger’s elegant point splitting method. Finally it is noted that a conservation law that may hold classically does © Springer International Publishing Switzerland 2016 E.B. Manoukian, Quantum Field Theory I , Graduate Texts in Physics, DOI 10.1007/978-3-319-30939-2_373
74 3 Quantum Field Theory Methods of Spin 1=2 not necessarily hold quantum mechanically and lead to modiﬁcations of the theory in the quantum world. The breakdown of such conservation laws lead to calculableanomalies and are treated in Sect. 3.9and3.10, and we provide as well of a concrete physical example where such an anomaly is actually veriﬁed experimentally. 3.1 Dirac Quantum Field, Propagator and Energy-Momentum Transfer: Schwinger-Feynman Boundary Condition In this section, we investigate the very signiﬁcance of a quantum ﬁeld for the very simple system described by the Dirac equation in (I.1 ). To this end, one may consider the matrix element of the Dirac ﬁeld .x/, treated as a quantum ﬁeld, between the no-particle state (vacuum) and a single particle state of a particle ofmomentum-spin ( p;/ESC) as followshvacj .x/jp/ESCi. The particle may have been produced by some external emission source, and a
much more transparent treatment for understanding the meaning of such a matrixelement is to introduce a source term /DC1.x/on the right-hand side of the Dirac equation to reﬂect this fact: /DC2/CR@ iCm/DC3 .x/D/DC1.x/: (3.1.1) The presence of the external source /DC1.x/makes the physical interpretation of the problem at hand clearer, as it plays the role of an emitter and detector ofparticles. The source is assumed to be of compact support in time, i.e., it vanishesexcept within a given interval or given sets of intervals in time. Physically it meansoutside the interval(s) the source ceases to operate, i.e., it is switched off, and is setequal to zero. 1The explicit structure of /DC1.x/isunimportant . It does not appear in the expressions of transition amplitudes of quantum particles. The introduction ofsuch sources facilitate tremendously the c omputation of transition amplitudes, and
hence their inclusion in the analysis of the underlying physical problems is of quitepractical value. /DC1.x/, by deﬁnition of an external source, is not a quantum ﬁeld. It is not, however, a classical one either as such, but a Grassmann variable. This latterpoint does not concern us yet at this stage and we will come back to it later. Letj0 /NULidesignates the vacuum state before the source /DC1.x0/is switched on. After the source is eventually switched off, the latter may have created someparticles. This means that if j0 Cidesignates the vacuum state after the source is switched-off, then quantum mechanics says that h0Cj0/NULiis not necessarily one or just a phase factor as the system may be in some other state involving some 1It is important to emphasize, that the source need not vanish abruptly at any point and may be taken to be continuous at a point where it vanishes (see Problem 3.1).
3.1 Dirac Quantum Field, Propagator and Energy-Momentum. . . 75 particles. This amplitude corresponds to the case where we begin with the vacuum before the source /DC1.x0/is switched on, and end up with the vacuum after the source is switched off. Here we have conveniently used the argument x0for the source. We consider the matrix element h0Cj .x/j0/NULi,a n d s o l v e ( 3.1.1 ) for the latter. We will then investigate the meaning of this matrix element later. By carrying out a 4D Fourier transform of ( 3.1.1 ) and inverting, in the process, the matrix./CR@= iCm/, we obtain for the matrix element in question h0Cj .x/j0/NULi h0Cj0/NULiDZ .dx0/Z.dp/ .2/EM/4eip.x/NULx0/./NUL/CRpCm/ .p2Cm2//DC1.x0/; (3.1.2) .dx/Ddx1dx2dx3dx4;p.x/NULx0/Dp/SOH.x/NULx0//NULp0.x0/NULx0/; (3.1.3) where we have used the relation ./NUL/CRpCm/./CRpCm/D.p2Cm2/. Regarding ( 3.1.2 ), we are faced with the interpretation of the pole that may arise
from the vanishing of the denominator .p2Cm2/, whose physical signiﬁcance and the evaluation of the expression on the right-hand side of ( 3.1.2 ) provides the ﬁrst step for carrying out computations in quantum ﬁeld theory. The denominator in ( 3.1.2 ) may be rewritten as .p2Cm2/D/NUL.p0/NULE/.p0CE/; EDp p2Cm2: (3.1.4) Ifx0is such that x0Dx0 2>x00, as shown in Fig. 3.1,f o r all x00contributing to thex00-integral in ( 3.1.2 ), that is, if the source is switched off before the time x0, we have the physical situation of a causal arrangement ( x0>x00), where energy is being transferred from the source to the point x2, with the source playing the role of an emitter. Thus we have to pick up the pole p0DC Ein (3.1.2 ) corresponding to an energy gain at the point in question. To do this, we make a transition to a complex p0-plane. Let Im p0denote the imaginary part of p0. For the case considered above, x0/NULx00/DC1T>0, and the real part of ./NULip0T/is given by Re ./NULip0T/D.Imp0/T
(Fig. 3.2). SOURCE x20x0 x10 Fig. 3.1 x10;x20are, respectively, two arbitrary points before the source is switched on and after it is switched off at which h0Cj .x/j0/NULiis examined
76 3 Quantum Field Theory Methods of Spin 1=2 −E+i E−ip0−plane x0>x0x0<x0 Fig. 3.2 Figure showing the direction of the contour that is to be taken in the complex p0/NULplane when applying the residue theorem, taken c.w. below the horizontal axis for x0>x00, and c.c.w. above it for x0<x00, generating, in turn, two semi-circles of inﬁnite radii, one below and one above the horizontal axis , respectively To pick up the pole at p0DC Ewe displace Eby/NULi/SI,/SI!0,i n ( 3.1.4 ), i.e., we make the replacement .p2Cm2/!/NULŒp0/NUL.E/NULi/SI//c141Œp0C.E/NULi/SI//c141D.p2Cm2/NULi/SI/: (3.1.5) We close the p0/NULcontour c.w. from below the horizontal axis, for which Im p0< 0, ensuring that the inﬁnite semi-circle part of the contour in the lower plane gives no contribution. The latter is as a consequence of the fact that exp Œ.Imp0/T/c141!0 for.Imp0/T!/NUL1 ,T>0, and, in the process pick up the pole p0DC E,b y the application of the Cauchy Theorem. We note that for any real number a>0,
a/SIis equivalent to /SIin complex integration for /SI!C0. An application of the residue theorem gives h0Cj .x/j0/NULi h0Cj0/NULiDiZ .dx0/Zd3p .2/EM/32p0eip.x/NULx0/./NUL/CRpCm//DC1.x0/; (3.1.6) where x0>x00, and now p0DCp p2Cm2. Here we recall the source /DC1.x0/ acts as an emitter. On the other hand for x0Dx10in Fig. 3.1, i.e., x0/NULx00/DC1T0<0, meaning that the source is switched on after the time x0in question, energy is absorbed by the source, now acting as a detector, with energy loss from point x1. That is, in this case, we have to pick up the pole at p0D/NUL E. We may then close the p0/NULcontour c.c.w. from above the horizontal axis in the complex p0/NULplane by noting, in the process, that exp Œ.Imp0/T0/c141!0,f o rI m p0!C1 ,T0<0, t oo b t a i nb yt h e residue theorem h0Cj .x/j0/NULi h0Cj0/NULiDiZ .dx0/Zd3p .2/EM/32p0e/NULip.x/NULx0/./CRpCm//DC1.x0/; (3.1.7)
3.1 Dirac Quantum Field, Propagator and Energy-Momentum. . . 77 where x0<x00,p0DCp p2Cm2(not minus), and we made a change of variables of integration p!/NUL pto ﬁnally write down the above integral. Accordingly, and quite generally, from ( 3.1.5 ), (3.1.6 ), (3.1.7 ) one may rewrite the expression in (3.1.2 ) in a uniﬁed manner as h0Cj .x/j0/NULi h0Cj0/NULiDZ .dx0/SC.x/NULx0//DC1.x0/; (3.1.8) where SC.x/NULx0/is referred to as the Dirac (fermion) propagator given by SC.x/NULx0/DZ.dp/ .2/EM/4eip.x/NULx0/./NUL/CRpCm/ .p2Cm2/NULi/SI/; (3.1.9) responsible for the transfer of energy (and momentum) as discussed above. The /NULi/SI term, with/SI!C0, in the denominator, dictated by the physics of the situation, speciﬁes the (Schwinger-Feynman) boundary condition for the proper integration in the complex p0/NULplane. Also with p0DCp p2Cm2, SC.x/NULx0/DiZd3p .2/EM/32p0eip.x/NULx0/./NUL/CRpCm/;forx0>x00; (3.1.10) SC.x/NULx0/DiZd3p .2/EM/32p0e/NULip.x/NULx0/./CRpCm/;forx0<x00; (3.1.11) /DC2/CR@
iCm/DC3 SC.x/NULx0/Dı.4/.x/NULx0/; (3.1.12) as follows from ( 3.1.9 ). The propagator also satisﬁes SC.x0/NULx/ /NUL/CR /NUL@ iCm! Dı.4/.x0/NULx/: (3.1.13) The ﬁeld equation for .x//DC1 /c142.x//CR0, in the presence of the external source, is given by .x/ /NUL/CR /NUL@ iCm! D/DC1.x/: (3.1.14) The vacuum expectation value of .x/is then easily worked out to be h0Cj .x/j0/NULi h0Cj0/NULiDZ .dx0//DC1.x0/SC.x0/NULx/; (3.1.15)
78 3 Quantum Field Theory Methods of Spin 1=2 and from (3.1.10), (3.1.11), h0Cj .x/j0/NULi h0Cj0/NULiDiZ .dx0/Zd3p .2/EM/32p0eip.x0/NULx//DC1.x0/./NUL/CRpCm/; (3.1.16) ifx0<x00, i.e., if the source /DC1.x0/is switched on after the time x0, h0Cj .x/j0/NULi h0Cj0/NULiDiZ .dx0/Zd3p .2/EM/32p0e/NULip.x0/NULx//DC1.x0/./CRpCm/; (3.1.17) ifx0>x00, i.e., if the source /DC1.x0/is switched off before the time x0. Thus the Schwinger-Feynman boundary c ondition, based on physical grounds, amounts in replacing the mass m by m/NULi/SIin the denominator in deﬁning the propagator. Needless to say for mD0the/NULi/SIshould survive. We note that due to the equality .dp/ı.p2Cm2/™.p0/Dd3p 2p0dp0ı.p0/NULp p2Cm2/; (3.1.18) where™.p0/D1forp0>0,™.p0/D0forp0<0, and due to the fact that a Lorentz transformation does not change the sign of p0, we may infer the Lorentz invariance of the measure on the right-hand side of ( 3.1.18) as well. We are now ready to investigate, in the next section, the role of .x/as a
quantum ﬁeld and the particle content of the theory. 3.2 The Dirac Quantum Field Concept, Particle Content, a n dC ,P ,TT a n s f o r m a t i o n s For the interpretation of the role played by the quantum ﬁeld .x/for the simple system described by the Dirac equation, we consider the expression for the matrix elementh0Cj .x/j0/NULi=h0Cj0/NULiin (3.1.6 ), that is for x0>x00in reference to Fig. 3.1, where the source acts as an emitter. With the functional time dependence of /DC1.x0/being of compact support in time explicitly absorbed in/DC1.x0/, we may carry out the x0-integral in ( 3.1.6 )o v e ra l l x0 to obtain h0Cj .x/j0/NULi h0Cj0/NULiDZd3p .2/EM/32p0eipx./NUL/CRpCm/i/DC1.p/; (3.2.1) /DC1.p/DZ .dx0/e/NULipx0/DC1.x0/: (3.2.2)
3.2 The Dirac Quantum Field Concept, Particle Content, and C, P, T. . . 79 Using the expression for the projection operator in Eq. ( I.21) in Appendix Iover spin states, we rewrite ( 3.2.1 )a s h0Cj .x/j0/NULi h0Cj0/NULiDZX /ESCm p0d3p .2/EM/3eipxu.p;/ESC/hp/ESCj0/NULi: (3.2.3) where hp/ESCj0/NULi/DC1Œiu.p;/ESC//DC1. p//c141; (3.2.4) and as it will be evident below, and as shown in the next section, hp/ESCj0/NULi represents an amplitude that the source, as an emitter, emits a particle of momentum- spin ( p;/ESC) which persists after the source is switched off. Since we have the vacuum state on the left-hand side of ( 3.2.3 ), we reach the inevitable conclusion that .x/has annihilated the particle in question to end up in the vacuum. At this stage, it is instructive to intr oduce the normalization condition of a particle state,2as well as of the resolution of the identity of the momenta and spin of such a particle deﬁned, in turn, by hp0/ESC0jp/ESCiD.2/EM/3p0
mı/ESC0/ESCı3.p0/NULp/;IDZX /ESCm p0d3p .2/EM/3jp/ESCihp/ESCj; (3.2.5) to rewrite ( 3.2.3 )a s h0Cj .x/j0/NULi h0Cj0/NULiDZX /ESCh0Cj .x/jp/ESCi h0Cj0/NULim p0d3p .2/EM/3hp/ESCj0/NULi: (3.2.6) and infer from ( 3.2.3 )t h a t h0Cj .x/jp/ESCi h0Cj0/NULiDeipxu.p;/ESC/: (3.2.7) A similar analysis as above dealing with h0Cj .x/j0/NULiin (3.1.16)f o rt h e causal arrangement, now, with x0<x00, relative to the source /DC1.x0/, in reference to Fig. 3.1, leads to the identiﬁcation Œi/DC1.p/u.p;/ESC//c141/DC1h0Cjp/ESCi; (3.2.8) 2Particle states and the Poincaré Algebra will be considered in Sect. 4.2.
80 3 Quantum Field Theory Methods of Spin 1=2 with the latter representing an amplitude that the source /DC1, as a detector, absorbs a particle, and ﬁnally obtain hp/ESCj .x/j0/NULi h0Cj0/NULiDe/NULipxu.p;/ESC/: (3.2.9) Thus we reach the inevitable conclusion that .x/has created a particle since we began with the vacuum and ended up having a particle detected by the source. It remains to consider the situation for h0Cj .x/j0/NULiwith x0<x00relative to/DC1.x0/,a n dh0Cj .x/j0/NULiwith x0>x00, relative to/DC1.x0/. This will be done in the next section. In the remaining part of this section and in view of applications in the following sections, we consider the Dirac equation in an external electromagnetic ﬁeld A/SYNin the presence of an external source /DC1. This is given by /DC4 /CR/SYN/DC2@/SYN i/NULeA/SYN.x//DC3 Cm/NAK .x/D/DC1.x/; (3.2.10) with e denoting the charge. We deﬁne it to carry its own sign, i.e., e D/NULj ejfor an electron.
In analogy to (3.1.12), we introduce the propagator SA C.x;x0/, now in the presence of an external elect romagnetic ﬁeld, satisfying /DC4 /CR/SYN/DC2@/SYN i/NULeA/SYN.x//DC3 Cm/NAK SA C.x;x0/Dı.4/.x/NULx0/: (3.2.11) The reason as to why we have written the functional dependence of SA Cas.x;x0/ rather than.x/NULx0/is that for A/SYN.x/¤0translational invariance is generally broken. One may convert ( 3.2.11) to the following integral equation SAC.x;x0/DSC.x/NULx0/CeZ .dx00/SC.x/NULx00//CR/SYNA/SYN.x00/SAC.x00;x0/: (3.2.12) This is easily veriﬁed by applying the uperator ./CR@= iCm/to (3.2.12), use ( 3.1.12) and ﬁnally integrate over a delta function to see that ( 3.2.11) is indeed satisﬁed. The expression for h0Cj .x/j0/NULiA=h0Cj0/NULiA, now in the presence of the external electromagnetic ﬁeld A/SYN.x/, follows from ( 3.2.10) and takes the form h .x/iA/DC1h0Cj .x/j0/NULiA h0Cj0/NULiADZ .dx0/SA C.x;x0//DC1.x0/: (3.2.13)
We consider the following physical situation where /DC1.x0/is so chosen that x00 lies sufﬁciently in the remote past and x0lies sufﬁciently forward in the future, with A/SYN.x00/in (3.2.12) being effective only in the region, x00<x000<x0.I n
3.2 The Dirac Quantum Field Concept, Particle Content, and C, P, T. . . 81 x0x0x0 Fig. 3.3/DC1.x0/is so chosen that x00lies sufﬁciently in the remote past and x0lies sufﬁciently forward in the future, with A/SYN.x00/being effective only in the region x00<x000<x0 practice, one may choose x00/FSx000/FSx0, if necessary, depending on A/SYN.x00/, in a limiting sense for the time variables x00;x0(Fig. 3.3). For a sufﬁciently weak external ﬁeld A/SYN,(3.2.12) becomes SA C.x;x0/'SC.x/NULx0/CeZ .dx00/SC.x/NULx00//CR/SYNA/SYN.x00/SC.x00/NULx0/: (3.2.14) The condition of a weak external ﬁeld will be relaxed in Sect. 3.6ı Appendix IVand an all order treatment will be given. We may carry out the integrations in ( 3.2.13), (3.2.14) to obtain3 h .x/iA'ZX /ESC/ESC0m p0d3p .2/EM/3m p00d3p0 .2/EM/3eip0xu.p0;/ESC0/hp0/ESC0jp/ESCiAŒiu.p;/ESC//DC1. p//c141; (3.2.15) where hp0/ESC0jp/ESCiADh .2/EM/3p0 mı/ESC0/ESCı3.p0/NULp/Cieu.p0;/ESC0//CR/SYNA/SYN.p0/NULp/u.p;/ESC/i ; (3.2.16)
which coincides with ( 3.2.5 )f o r eD0. In the process of deriving (3.2.15), (3.2.16), we have, by using ( 3.1.10), (I.21), conveniently rewritten the ﬁrst term SC.x/NULx0/ on the right-hand side of ( 3.2.14)a s SC.x/NULx0/DiZX /ESC/ESC0m p0d3p .2/EM/3m p00d3p0 .2/EM/3ei.p0x/NULpx0//STX /STXu.p0;/ESC0/Œ.2/EM/3p0 mı/ESC/ESC0ı3.p0/NULp//c141u.p;/ESC/; (3.2.17) 3See Problem 3.3.
82 3 Quantum Field Theory Methods of Spin 1=2 forx0>x00, and deﬁned the Fourier transform A/SYN.x/DZ.dQ/ .2/EM/4eiQxA/SYN.Q/: (3.2.18) We note that ( 3.2.16) is also valid for a time-independent external ﬁeld A/SYN.x/as long as x0;/NULx00are taken large in a limiting sense. In this case A/SYN.p0/NULp/ in (3.2.16) will be proportional to ı.p00/NULp0/. In the causal arrangement discussed above, the quantum ﬁeld .x/, in a source free region, destructs the particle after it has scattered off the external ﬁeld A/SYN, with the transition amplitude for the process corresponding to p/ESC!p0/ESC0,s a y , forp¤p0, given from ( 3.2.16)t ob e hp0/ESC0jp/ESCiADŒieu.p0;/ESC0//CR/SYNA/SYN.p0/NULp/u.p;/ESC//c141: (3.2.19) The external electromagnetic ﬁeld A/SYN.x/may in turn create particles. This will be taken up in Sect. 3.8. The structure given in ( 3.2.16) is interesting as, in particular, it provides us with the value of the magnetic moment of the charged particle (electron). To this end,
we may use the equation ./CRpCm/u.p;/ESC/D0(see ( I.10)), to write u.p;/ESC/D /NUL./CRp=m/u.p;/ESC/, multiply the latter from the left by /CR/SYNto derive the identity /CR/SYNu.p;/ESC/Dp/SYN mu.p;/ESC//NULp/ETBŒ/CR/SYN;/CR/ETB/c141 2mu.p;/ESC/; (3.2.20) and similarly u.p0;/ESC0//CR/SYNDu.p0;/ESC0/p0/SYN mCu.p0;/ESC0/p0 /ETBŒ/CR/SYN;/CR/ETB/c141 2m: (3.2.21) By multiplying ( 3.2.20) from the left by u.p0;/ESC0/and (3.2.21) from the right by u.p;/ESC/, and adding the resulting two equations lead to the following decomposition, referred to as the Gordon decomposition, eŒu.p0;/ESC0//CR/SYNu.p;/ESC//c141 A/SYN.Q/ Deu.p0;/ESC0/h.p/SYNCp0/SYN/ 2mCŒ/CR/SYN;/CR/ETB/c141 4mQ/ETBi u.p;/ESC/A/SYN.Q/ Du.p0;/ESC0/h e.p/SYNCp0/SYN/ 2mA/SYN.Q/Cie 2m/NUL Q/SYNA/ETB.Q//NULQ/ETBA/SYN.Q//SOHi 4Œ/CR/SYN;/CR/ETB/c141i u.p;/ESC/; (3.2.22) and Q/ETBDp0/ETB/NULp/ETBis the (four) momentum transfer. In reference to the second term in the second equality on the right-hand side of ( 3.2.22), note that the speciﬁc
3.2 The Dirac Quantum Field Concept, Particle Content, and C, P, T. . . 83 contribution over the sum of the indices /SYN;/ETB,f o r/SYNDi,/ETBDj,i s g i v e n b y ie 2m"ij k/NUL QiAj.Q//NULQjAi.Q//SOH/STX u.p0;/ESC0/Sku.p;/ESC//ETX ;i;j;kD1;2;3; (3.2.23) with a sum over repeated indices understood, where SkDi"s`kŒ/CRs;/CR`/c141=8is the spin matrix in ( I.15), and in writing ( 3.2.23), we have used, in the process, the identity "ij k"s`kDıisıj`/NULıi`ıjs: (3.2.24) We note that i "ij k/NUL QiAj.q//NULQjAi.Q//SOH =2denotes the Fourier transform of the magnetic ﬁeld Bk.Q/Di 2"ij k/NUL QiAj.Q//NULQjAi.Q//SOH (3.2.25) and we may rewrite ( 3.2.23)a s Œu.p0;/ESC0//SYNu.p;/ESC//c141/SOHB.Q/; /SYNDge 2mS; (3.2.26) where /SYNis the magnetic dipole moment of the charged particle (electron) with theg-factor equal to 2. The modiﬁcation of this value due to so-called radiative corrections in QED will be considered later in Sect. 5.11.2 . 3.2.1 Charge Conjugation (C), Parity Transformation (T),
and Time Reversal (T) of the Dirac Quantum Field In the remaining part of this section, we investigate the transformation rules of the Dirac quantum ﬁeld under charge conjugation .C/, parity transformation .P/,a n d time reversal .T/. In the presence of an external electromagnetic ﬁeld, the Dirac equation reads from ( 3.2.10 ), with/DC1D0, /DC4 /CR/SYN/DC2@/SYN i/NULeA/SYN.x//DC3 Cm/NAK .x/D0; (3.2.27) while from Eq. ( I.3),C >/DC1 Csatisﬁes the same equation with sign of the charge e reversed, i.e., h /CR/SYN/DC2@/SYN iCeA/SYN.x//DC3 Cmi C.x/D0; (3.2.28)
84 3 Quantum Field Theory Methods of Spin 1=2 where CDi/CR2/CR0is the charge conjugation matrix deﬁned in Eq. ( I.3). Hence we may deﬁne the charge conjugation of a Dirac quantum ﬁeld by C .x/C/NUL1DC >.x/; (3.2.29) up to a phase factor. On the other hand, according to Eq. (I.18), /CR0 .x0/, with xD.x0;/NULx/, satisﬁes the same Dirac equation ./CR/SYN@/SYN=iCm/ .x/D0,i . e . , /DLE/CR/SYN@/SYN iCm/DC1 /CR0 .x0/D0; x0D.x0;/NULx/: (3.2.30) Hence we may infer that the parity transformation of the Dirac quantum ﬁeld may be deﬁned by P .x/P/NUL1D/CR0 .x0/; (3.2.31) up to a phase factor. For time reversal, we note that by setting x00D./NULx0;x/, that we may rewrite the Dirac equation ./CR/SYN@/SYN=iCm/ .x/D0as h /NUL/CR0@000 iC/CR/SOH@ iCmi .x/D0: (3.2.32) Upon multiplying the latter equation by /CR5C, making the substitution x00$x, and using the facts that Œ/CR5C;/CR0/c141D0; Œ/CR5C;/CR2/c141D0;f/CR5C;/CR1gD0;f/CR5C;/CR3gD0; (3.2.33) as is easily veriﬁed, we may rewrite ( 3.2.32)a s h
C/CR2@2 i/NUL/CR0@0 i/NUL/CR1@1 i/NUL/CR3@3 iCmi /CR5C .x00/D0: (3.2.34) Finally we make use of the property of T, as an anti-unitary operator, that it complexconjugates, and note that, e.g., in the Dirac representation, ./CR 2//ETXD/NUL/CR2,a n d /CR0;/CR1;/CR3are real, to obtain upon multiplying ( 3.2.34) from the left by T/NUL1,a n d from the right by T, the equation /DLE/CR/SYN@/SYN iCm/DC1 T/NUL1/CR5C .x00/TD0; (3.2.35)
3.3 Re-Discovering the Positron and Eventual Discovery of Anti-Matter 85 which by comparing it with the Dirac equation, gives the following transformation rule for time reversal of the Dirac quantum ﬁeld T .x/T/NUL1D/CR5C .x00/; x00D./NULx0;x/; (3.2.36) up to a phase factor. 3.3 Re-Discovering the Positron and Eventual Discovery of Anti-Matter We have seen in Sect. 3.2,(3.2.1 ),(3.2.6 ), that the vacuum expectation value of the Dirac quantum ﬁeld .x/, in the presence of an external source /DC1.x/, i.e., satisfying ( 3.1.1 ), is given by h0Cj .x/j0/NULi h0Cj0/NULiDiZ .dx0/Zd3p .2/EM/32p0eip.x/NULx0/./NUL/CRpCm//DC1.x0/ DZX /ESCm p0d3p .2/EM/3h0Cj .x/jp/ESCi h0Cj0/NULihp/ESCj0/NULi; (3.3.1) when the source /DC1.x0/is in operation, i.e., non-zero, and then switched off sometime before the time x0, i.e., x0>x00in reference to Fig. 3.1,f o ra l l x00 contributing to the above integral. Here hp/ESCj0/NULidenotes an amplitude that

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