His account of renormalization theory reflects the changes in our view of quantum field theory since the advent of effective field theories. The book's scope extends beyond quantum electrodynamics to elementary particle physics, and nuclear physics. It contains much original material, and is peppered with examples and insights drawn from the author's experience as a leader of elementary particle research.

Problems are included at the end of each chapter. This work will be an invaluable reference for all physicists and mathematicians who use quantum field theory, and it is also appropriate as a textbook for graduate students in this area. The development is. Comprehensive introduction to quantum field theory by Nobel Laureate Steven Weinberg, now available in paperback. Available for the first time in paperback, The Quantum Theory of Fields is a self-contained, comprehensive, and up-to-date introduction to quantum field theory from Nobel Laureate Steven Weinberg.

The development is fresh and logical throughout, with each step carefully motivated by. In this third volume of The Quantum Theory of Fields, available for the first time in paperback, Nobel Laureate Steven Weinberg continues his masterly exposition of quantum field theory.

This volume presents a self-contained, up-to-date and comprehensive introduction to supersymmetry, a highly active area of theoretical physics.

The text introduces. Based on a two-semester course held at the University of Heidelberg, Germany, this book provides an adequate resource for the lecturer and the student. The contents are primarily aimed at graduate students who wish to learn about the fundamental concepts behind constructing a Relativistic Quantum Theory of particles and fields.

This textbook covers a broad spectrum of developments in QFT, emphasizing those aspects that are now well consolidated and for which satisfactory theoretical descriptions have been provided.

The book is unique in that it offers a new approach to the subject and explores many topics merely touched upon, if covered. This book is a modern introduction to the ideas and techniques of quantum field theory.

After a brief overview of particle physics and a survey of relativistic wave equations and Lagrangian methods, the author develops the quantum theory of scalar and spinor fields, and then of gauge fields.

The emphasis. First Published in Quantum field theory provides the theoretical backbone to most modern physics. This book is designed to bring quantum field theory to a wider audience of physicists.

It is packed with worked examples, witty diagrams, and applications intended to introduce a new audience to this revolutionary theory.

A few years later Weisskopf repeated the calculation of the electron self-mass in the new hole theory, with all negative-energy states full.

The same calculation was being carried out at the suggestion of Bohr at that time by Carlson and Furry. After seeing Weisskopf's results, Furry realized that while Weisskopf had included an electrostatic term that he and Carlson had neglected, Weisskopf had made a new mistake in the calculation of the magnetic self-energy. However, despite this cancellation, an infinity remained : with a wave-number cutoff 1 f a, the self-mass was found to be63 Sac h flZ em M In 1.

An infinity of quite a different kind was encountered in 1 , apparently first by Dirac. Infinities also seemed to occur in a related problem, the scattering of light by light. Hans Euler, Bernard Kockel, and Heisenberg15 showed in w6 that these infinities could be eliminated by using amore-ar-less ar- bitrary prescription suggested earlier by DiraC66 and Heisenberg Soon after, Nicholas Kemmer and Weisskopf68 presented arguments that in this case the infinities are spuri- ous, and that Eq.

In 1 it was shown by Felix Bloch and Arne Nordsieck 68, that these infinities cancel provided one includes processes in which arbitrary numbers of low-energy photons are produced. This will be discussed in modern terms in Chapter Yet another infinity turned up in a calculation by Sidney Michael Dancoff69 in of the radiative corrections to the scattering of electrons by the static Coulomb field of an atom.

The calculation contained a mistake one of the terms was omitted , but this was not realized until later. Rather, they seemed to indicate a gap in the understanding of relativistic quantum field theory on the most fundamental level, an opinion reinforced by the problems with cosmic rays mentioned in the previous section. One of the symptoms of this uneasy pessimism was the continued exploration throughout the s and s of alternative formalisms.

As we shall see, the concept of the S-matrix has now become a vital part of modern quantum field theory, and for some theorists a pure S-matrix theory became an ideal, especially as a possible solution to the problems of the strong interactions.

They were able to show that a pure retarded or pure advanced potential could be obtained by taking into account the interaction not only between source and test charges, but also between these charges and all the other charges in the universe. A more conservative idea for dealing with the infinities was also in the air during the s. Also, Eq. Vacuum polarization gives finite results in lowest order if observables like scattering cross-sections are expressed in terms of eTOTAL rather than e.

The question was, whether all infinities in quantum field theory could be dealt with in this way. In VVeisslcopf 76 suggested that this is the case, and verified that known infinities could be eliminated by renormalization of physical parameters in a variety of sample calculations. However, it was impossible with the calculational techniques then available to show that infinities could always be eliminated in this way, and DancQff's calculation69 seemed to show that they could not. Another effect of the appearance of infinities was a tendency to believe that any effect which turned out to be infinite in quantum field theory was actually not there at all.

Later Bethego recalled that This shift comes out infinite in all existing theories, and has therefore always been ignored. The gloom surrounding quantum field theory began to lift soon after World War II. Quantum Mechanics at Shelter Island, NY brought theoretical physicists who had been working on the problems of quantum field theory through the s together with a younger generation of theorists who had started scientific work during the war, and of crucial importance a few experimental physicists.

A beam of hydrogen atoms from an oven, many in 2s and 2p states, was aimed at a detector sensitive only to atoms in excited states. The atoms in 2p states can decay very rapidly to the is ground state by one-photon Lyman a emission, while the 2s states decay only very slowly by two-photon emission, so in effect the detector was measuring the number of atoms in the metastable 2s state. The beam was also exposed to a microwave- frequency electromagnetic field, with a fixed frequency v - 10 GHz.

At a certain magnetic field strength the detector signal was observed to be quenched, indicating that the microwave field was producing resonant transitions from the metastable 2s state to the 2p state and thence b y a rapid Lyman a emission to the ground state. The total Zeeman plus intrinsic 2s-2p splitting at this value of the magnetic field strength would have to be just hv, from which the intrinsic splitting could be inferred.

A preliminary value of MHz was announced, in agreement with the earlier spectroscopic measurements. Kra- mers described his work on mass renormalization in the classical electro- dynamics of an extended electron, 79a which showed that the difficulties associated with the divergence of the self-energy in the limit of zero radius do not appear explicitly if the theory is reexpressed so that the mass pa- rameter in the formalism is identified with the experimental electron mass.

In fact, in , before he learned of Lamb's experiment, Weisskopf had already assigned this problem to a graduate student, Bruce French. He obtained the encouraging approximate value of MHz. Fully rela- tivistic calculations using the re-normalization idea to eliminate infinities were soon thereafter carried out by a number of other authors, 81 with excellent agreement with experiment.

Another exciting experimental result was reported at Shelter Island by Isidor I. Measurements in his laboratory of the hyperfine structure of hydrogen and deuterium had suggested 82 that the magnetic moment of the electron is larger than the Dirac value ehl2rrac by a factor of about 1. At Shel- ter both Breit and Schwinger described their efforts to calculate this correction. This, together with Beth's calculation of the Lamb shift, at last convinced physicists of the reality of radiative corrections.

The mathematical methods used in this period presented a bewilder- ing variety of concepts and formalisms. Tamonaga had grappled with infinities in Yukawa's meson theory in the s.

In he and his group were still out of the loop of scientific communication ; they learned about Lamb's experiment from an article in Newsweek. One result of great practical importance that came out of Feynman's work was a set of graphical rules for calculating 5'-matrix elements to any desired order of perturbation theory.

Unlike the old perturbation theory of the s and s, these Feynman rules automatically lumped together particle creation and antiparticle annihi- lation processes, and thereby gave results that were Lore ntz-in variant at every stage.

We have already seen in Weisskopf's early calculatian63 of the electron self-energy, that it is only in such calculations, including particles and antiparticles on the same footing, that the nature of the infinities becomes transparent. Dyson also carried out an analysis of the infinities in general Feynman diagrams, and outlined a proof that these are always precisely the sort which could be removed by renormalization.

In particular, an interaction like the Pauli term 1. With the publication of Dyson's papers, there was at last a general and systematic formalism that physicists could easily learn to use, and that would provide a common language for the subsequent applications of quantum field theory to the problems of physics.

I cannot leave the infinities without taking up a puzzling aspect of this story. Oppenheimer6l in had already noticed that most of the ultraviolet divergence in the self-energy of a bound electron cancels when one takes the difference between the shifts of two atomic energy levels, and Weisskopf 3 in had found that most of the divergence in the self-energy of a free electron cancels when one includes intermediate states containing positrons.

It would have been natural even in to guess that including positron intermediate states and subtracting the energy shifts of pairs of atomic states would eliminate the ultraviolet divergence in their relative energy shift.

In fact, this guess wou l d have been wro n g. As di scussed in Section So why did no one before attempt an numerical estimate of this energy difference? Strictly speaking, there was one such attempt8gQ in , but it focused on the wrong part of the problem, the charge radius of the proton, which has only a tiny effect on hydrogen energy levels.

The calculation gave a result in rough agreement with the early experiments. As long as one keeps all terms up to a given order, old-fashioned non-relativistic perturbation theory gives the same results as the manifestly relativistic formalisms of Feynman, Schwiner, and Tamonaga. In fact, after Be he's work, the first precise calculations8l of the Lamb shift in the USA by French and Weisskopf and Norman Kroll and Lamb were done in just this way, though Tomonoga's group 8 1 in Japan was already using covariant methods to solve this and other problems.

The one missing element was confidence in renormalization as a means of dealing with infinities. As we have seen, renormalization was widely discussed in the late s. But it had become accepted wisdom in the s, and a point of view especially urged by Oppenheimer,"' that quantum electrodynamics could not be taken seriously at energies of more than about Mel, and that the solution to its problems could be found only in really adventurous new ideas.

Several things happened at Shelter Island to change this expectation. One was news that the problems concerning cosmic rays discussed in the previous section were beginning to be resolved ; Robert Marshak presented the hypothesis58 that there were two types of "meson' with similar masses ; the muons that had actually been observed, and the pions responsible for nuclear forces. More important was the fact that now there were reliable experimental values for the Lamb shift and the anomalous magnetic moment that forced physicists to think carefully about radiative corrections.

Probably equally important was the fact that the conference brought together theorists who had in their own individual ways been thinking about renormalization as a solution to the problem of infinities. When the revolution came in the late s, it was made by physicists who though mostly young were playing a conservative role, turning away from the search by their predecessors for a radical solution.

Bibliography Beyer, ed. Brown and L. Hoddeson, eds. Cao and S. Dirac's Scientific work from to ,' in Aspects of Quantum Theory, ed. Salam and E. Wigner, Cambridge University Press, Cambridge, Pais, Inward Bound Clarendon Press, oxford, DeWitt and R. Stora North-Holland, Amsterdam, Weinberg, and E.

Section 1. Schwinger, ed. Hoddeson, op. Fierz and V. Weisskopf Interscience Publishers Inc. Eisasser, ,Nutu rwiss.

Davison and L. Heisenberg, A. Jordan , Z. Dirac, Proc. A, ; M. Born, W. Heisenberg, and P. Jordan, Z. These papers are reprinted in Sources of Quantum Mechanics, ed.

References Schrodinger, Ann. These papers are reprinted in English, unfortunately in a somewha t abridged form, in Were Mechanics, Ref. Also see Collected Papers on Wave Mechanics, trans. Schearer and W. Deans Blackie and Son, London, See, e.

Klein, Z. Also see V Fock, Z. Gordon, Z, f Whys. For the details of the calculation, see, e. Paschen, Ann. Ph ys. The fine structure of spectra l lines was first discovered interferametrically by A. Michelson, Phil. Pa ys.

Also s ee W. Wilson, Phil. Uhlenbeck and S. Goudsmit, Nat urwiss. The electron spin had been earlier suggested for other reasons by A. Compton, J. The general formula for Zeeman splitting in one-electron atoms ha d been discovered empirically by A. Lande, Z. Sammerfeld, Ann. It was only later that the extra angular momentum was recognized , as in Ref.

Heisenberg and P. Darwin, Proc. A , Darwin says that several authors did this work at about the same time, while Dirac quotes only Darwin. Thomas, Nature 11 7, Also see S. A, 1 Also see Dirac, ibid. For the probabilistic interpretation of non-relativistic quantum me- chanics, see M. Born Z. Wentzel, Z. Heisenberg, Z. Conversation between Dirac and J. Salem and E.

Darwin, P roc. A , ; ibid. G ord on, z. A, ; W Pauli, Z. Slater, Phys. Fermi, Z. Lineei 3, 14 5 Dirac, First R. Crane Lecture at the University of Michigan, April 17, , unpublished. Robertson Dover Publications, Inc. Also see P. Oppenheimer, Whys. Tatum, Z. Anderson, Science 76, ; Phy? The latter paper is reprinted in Foundations of Nu c lear Physics, ed.

Beyer Dover Publications, Inc. Born and Jordan, Ref. Unfortunately the relevant parts of this paper are not included in the reprint collection Sources of Quantum Mechan ics, cited ire Ref. Dirac, Froc. Al 12, For a more accessible derivation, see L. Schiff, Quantum Mechanics, 3rd eds. Einstein, Whys. A, ; reprinted in Quantum Electrody namics, ed. Schwinger Dover Publications, Inc. Lincei Rend. Jordan and W. Pauli, Z. Bohr and L. Rosenfeld, Kon. III, Na. Cohen and J. Stachel Reidel, Dordrecht, ; Whys.

Jordan and 0. Klein , Z. Jordan, Phys. Jordan and E. Wigner, Z. This article is reprinted in Quantu m Electrodynamics, Ref. Fierz, Hein Phys. Belinfante, Physica 7, Heisenberg and W Pauli, Z. A 1 36, ; P. Dirac W. Fork, and B. Padolsky, Phys. Dirac, Phys. The latter two art icles are reprinted in Quantum Electrodynamics, Ref. Also see L. Rosenfeld, Z. Fermi quotes unpublished work of Pauli for the proposition that an unobserved neutral particle is emitted along with the electron in beta decay.

This particle was called the neutrino to distinguish it from the recently discovered neutron. Furry and J.

Oppenheimer, Phys. This paper uses a density matrix formalism developed by P. Dirac, Prot. Also see R. Peierls, Proc. J: Pays. Pauli and V.

Weisskopf, HeIv. Acts 7, , reprinted in English translation in A. Also see W. Pauli, Ann. Henri Poincare 6, 13 7 6 -. Klein and Y. Nishina, Z. Nishina, ibid. Tamm, Z. Dirac, Prat. Miller, Ann. Bethe and W. Heitter, Proc. A, 83 ; also see G. Racah, Nuovo Cimento 11 , Na. Bhabha, Proc. A, Carlson and J. Ehrenfest and J. Heitler and G. Herzberg, Naturwiss. Rasetti, Z. Chadwick, Prot. This article is reprinted in the Foundations of Nuclear Physics, Ref.

Curie-Joliet and F. Joliot, Campt. For references, see L. Brown and H. Rechenberg, Hiss. Science, ZS, 1 Yukawa, Proc. Neddermeyer and C. Anderson, Phys. Street and E. Stevenson, Phys. Sakata and T. Inoue, Prog. Marshak and H. Bettie, Fhys. Latter, G.

Powell, Nature , , Rochester and C. Butler, Nature , Waller, Z. Weisskopf, Z. This last article is reprinted in Quantum Electrodynamics, Ref. For subsequent calculations based on less restrictive assumptions, see Heisenberg , Z. Serber, Whys. Pauli and M. Rosy, Phys. Also see Furry and Oppenheimer, Ref. Euler and B. Kockel, Naturw iss. Heisenberg and H. Bloch and A. Nordsieck, Whys. Lewis, Phys. Epstein, Phys. Also see J. Koba and S. Tarnonaga, Prog. Schwinger, in The Birth oj'Particle Physics, ed.

Hoddeson Cambridge University Press, Cambridge, : p. Heisenberg, Ann. Borand L. Infeld, Prac. Henri Pnincare 6, 1 93 6. Also see C. Miller, Kos. Benjamin, inc. New York, 1. Wheeler and R. Feynman, Rev. For further references and a discussion of the application of action-at-awdistance theories in cosmology, see S.

Weinberg G ravitation and Cosmology, Wiley, : Section A , 1 For a criticism, se e W. Pauli, Rev. For a review of classica l theories of this type, and of yet other attempts to solve the problem of infinities, see R. Stoops, Brussels, : p. Weisskopf, Kon. XJV, No. This a rticle is reprin ted in Quantum Electrodynamics, Ref. Fierz, Ref. Kramers, Ref. Pasternack, Phys. This suggestion was base d on experiments of W. Houston, Pays. Williams, Phys. For a report of contrary data , see J. Drinkwater, 0.

Richardson, and W. Williams, Proc. Lamb, Jr and R. Retherford, Phys. This article is reprinted in Qua ntum Electrodynamics, Ref.

Stoops, Brussels, 1 9 Bathe, Phys. This article is reprinted in Quantum Electrodynamics, Ref. French and V. Weisskopf; Phys. Kroll and W E. Lamb, ibid. Schwinger, Phys. Feynman, Rev, Alfod. Fukuda, Y. Miyarnoto, and S. Tomonaga, P rog. Nafe, E. Nelson, and I. Ra bi, Whys. Nagel, R. Julian, and J. Z acharias, Phys. Ku sch and H. Foley Phys. Breit, Phys. Schwinger in Ref. This article is reprinted in Qua n tum Electrodynamics, Ref.

All but the first two of these articles are reprinted in Qua ntum Electr odynamics, Ref, Tomonaga, Prog. Teti, and S. Tornonaga, ibid.

Kanesaw a and S. Tomonaga, ibid. Tomonaga, Phys. Ito, Z. Kona, and S. Tamonaga, ibid. The first and fourth of these articles are reprinted in Quantum Electrodynamics, Ref. All but the second and third of these articles are reprinted in Qua ntum Electrodynamics, Ref. Dyson, Phys. These articles are reprinted i n Quantum Electrodynamics, Ref.

Frohlich, W. Heitler, and B. Kahn, Prot. Roy, Soc. A1 7 1, ; Phys. Lamb, Jr, Phys. Quoted by R. Serbe r, in the Birth of Particle Physics, Ref.

Relativistic Quantum Mechanic s. The point of view of this book is that quantum field theory is the way it is because with certain qualifications this is the only way to reconcile quantum mechanics with special relativity. Therefore our first task is to study how symmetries like Lorentz invariance appear in a quantum setting. The reader is assumed to be already familiar with quantum mechanics ; this section provides only the briefest of summaries of quantum mechanics, in the generalized version of Dirac.

There are also certain technical assumptions that allow us to take limits of vectors within Hilbert space. A ray is a. A state represented by a ray. An elementary theorem tells us that for A Hermitian, x is real, and eigenvectors with different as are orthogonal.

A pair of rays is said to be orthogonal if the state-vectors from the two rays have vanishing scalar products. Another elementary theorem gives a total probability unity ; A symmetry transformation is a change in our point of view that does not change the results of possible experiments. If an observer 0 sees a system in a state represented by a ray ' or : fi r or R2. This is only a necessary condition for a ray transformation to b e a symmetry ; further conditions are discussed in the next chapter.

Wigner's proof omits some steps, A more complete proof is given at the end of this chapter in Appendix A. This condition cannot be satisfied for an antilinear operator, because in this case the right-hand side of Eq. Instead, the adjoint of an antilinear operator A is defined by. With this definition, the conditions for unitarily or antiuriitarity both take the form Ut - U - 1 2.

This operator is, of course, unitary and linear. Continuity then demands that any symmetry like a rotation or translation or Lorentz transformation that can be made trivial by a continuous change of some parameters like angles or distances or velocities must be represented b y a linear unitary operator U rather than one that is antilinear and antiunitary.

Symmetries represented by antiunitary antilinear operators are less prominent in physics ; they all involve a reversal in the direction of time's flow. See Section 2. For this to be unitary and linear, t must be Hermitian and linear, so it is a candidate for an observable.

Indeed, most and perhaps all of the observables of physics, such as angular momentum or momentum, arise in this way from symmetry transformations. The set of symmetry transformations has certain properties that define it as a group. If T, is a transformation that takes rays n into Win, and T2 is another transformation that takes n into ", then the result of per- forming both transformations is another symmetry transformation, which we write T2Tj, that takes.

The unitary or antiunitary operators U T corresponding to these sym- metry transformations have properties that mirror this group structure, but with a complication due to the fact that, unlike the symmetry trans- formations themselves, the operators U T act on vectors in the Hilbert space, rather than on rays. Consider any two different vectors TA,TB, which are not proportional to each other.

Then, applying Eq. Any unitary or antiunitary operator has an inverse its adjoint which is also unitary or antiunitary. Multiplying 2. The structure of the Lie group cannot by itself tell us whether physical state-vectors furnish an ordinary or a projective representation, but as we shall see, it can tell us whether the group has any intrinsically projective representations at all.

The exception to the argument that led to Eq. For instance, it is widely believed to be impossible to prepare a system in a superposition of two states whose total angular momenta are integers and half-integers, respectively. We will have more to say about these phases and projective representations in section 2. As we shall see there, any symmetry group with projective representations can always be enlarged without otherwise changing its physical implications in such a way that its representations can all be defined as non-projective, wit h 0.

Until section 2. There is a kind of group, known as a connected Lie group, of special importance in physics. These are groups of transformations T O that are described by a finite set of real continuous parameters, say 0a, with each element of the group connected to the identity by a path within the group.

The group multiplication law then takes the farm. Suppose that the U T B form an ordinary i. According to Eq. The presence of any terms of order 02 or would violate E q. This shows that if we are given the structure of the group, i. However, there is a consistency condition : the operator th, must be symmetric in b and c because it is the second derivative of U T 0 with respect to 6 and H ' so Eq. Such a set of commutation relations is known as a Lie a lgebra.

In Section 2. The extension to all 01' is discussed in Section 2. There is a special case of some importance, that we will encounter again and again. This is the case for instance for translations in spacetime, or for rotations about any one fixed axis though not for both together. Then the coefficients f a b, in Eq. The generators then all commut e. In this case, it is easy to calculate U T B for all 0'. From Eqs. It is distinguished from the Galilean principle of rela- tivity, obeyed by Newtonian mechanics, by the transformation connecting coordinate systems in different inertial frames.

Any coordinate transformation xP -f x'Y that satisfies Eq. These transformations form a group. If we first perform a Lorentz transformation 2. The bar is used here just to distinguis h. Conformal invariance in two dimensions has proved enormously important in string theory and statistical mechanics, but the physical relevance of these conformal transformations in four spacetime dimensions is not yet clear.

The transformations T A, a induced on physical states therefore satisfy the composition rule. Taking the determinant of Eq. The inverse of the transformation T , a is seen from Eq. The appropriate enlargement is described in Section 2. The whole group of transformations x' 11, a is properly known as the anhomogeneous L o r e n t z g r ou p , or Poirtcrxre group. It has a number of important subgroups. Also, we note from Eq. Further, from the components of Eqs. But Eq. Any Lorentz transformation is either proper and orthn chron o us, or may b e written as the product of an e l em ent of the proper orthochronous Lorentz group with one of the discrete transformations.

We will consider space inversion and time-reversal separately in Section 2. Until then, we will deal only with the homogeneous or inhomogeneous proper orthochronous Lorentz group.

As we saw in Section 2. The Lorentz condition 2. Keeping only the terms of first order in co in the Lorentz condition 2. Where superselection rules apply, it may be necessary to redefine U 1 70 by phase factors that depend on the sector on which it acts. The consistency of the choice in 2.

It follows then from 2. T " is a tensor and P P is a vector. In particular, the change of the space-space components of JP' under a spatial translation is just the usual change of the angular momentum under a change of the origin relative to which the angular momentum is calculated. Next, let's apply rules 2. Using Eq. In quantum mechanics a special role is played by those operators that are conserved, Le. Inspection of Eqs. These are not conserved, which is why we do not use the eigenvalues of K to label physical states.

In a three-dimensional notation, the commutation relations 2. The commutation relation 2,4. The pure translations T 1, Q form a subgroup of the inhomogeneous Lorentz group with a group multiplication rule given by 2. However, since we already have Eqs. For a system of particles of typical mass rn and typical velocity z', the momentum and the angular-momentum operators are expected to be of order J - 1, P - m v. Note that the product of a translation x In this respect, the mathematics of the Poincare group is simpler than that of the Galilean group.

However, there is nothing to prevent us from formally enlarging the Galilean group, by adding one mare generator to its Lie algebra, which commutes with all the other generators, and whose eigenvalues are the masses of the various states. In this case physical states provide an ordinary rather than a projective representation of the expanded symmetry group. The difference appears to be a mere matter of notation, except that with this reinterpre- tation of the Galilean group there is no need for a mass superselection rule.

We now consider the classification of one-particle states according to their transformation under the inhomogeneous Lorentz group.

Introducing a label a to denote all other degrees of freedom, we thus consider state-vectors 'Y ,. We take as part of the definition of a one-particle state, that the label cT is purely discrete, and will limit ourselves here to that case. However, a specific bound state of two or more particles, such as the lowest state of the hydrogen atom, is to be considered as a one- particle state.

It is not an e leme ntary particle, but the distinction between composite and elementary particles is of no relevance here. We must now consider how these states transform under homogeneous Lorentz transformations.

Using 2,4. In general, it may be possible by using suitable linear combinations of the 'Pp , , to choose the a labels in such a way that the matrix ,,, A, p is block-diagonal ; in other words, so that the 'Pp,, with a within any one block by themselves furnish a representation of the inhomogeneous Lorentz group.

It is natural to identify the states of a specific particle type with the components of a representation of the inhomogeneous Lorentz group which is irreducible, in the sense that it cannot be further decomposed in this ay. In some cases it may he convenient to define particle types as irreducible representations of larger groups that contain the inharnogeneous proper orthochronous Lorentz group as a sub- group ; for instance, as we shall sec, for massless particles whose interactions respect the symmetry of space inversion it is customary to treat all the components of an irreducible representation of the inhomogeneous Lorentz group including space inversion as a single particle type.

Up to this point, we have said nothing about how the ff labels are related for different momenta ; Eq. Operating on 2. I n particular, we may apply E q. This approach, of deriving representations of a group like the inhomogeneous Lorentz group from the representations of a little group, is called the method of induced representations.

Of these six classes of four-momenta, only a , c , and f have any known interpretations in terms of physical states. This is a good place to pause, and say something about the normal- ization of these states. By the usual orthonormalization procedure of quantum mechanics, we may choose the states with standard momentum k " to be orthonormal, in the sense tha t. This has the immediate consequence that the representation of the little group in Eqs.

Using the unitarity of the operator U A in Eqs. Table 2. Standard momenta and the corresponding little group for various classes of Four-rnornenta. Here K is an arbitrary positive energy, say I eV. The group IS O 2 is the group of Euclidean geometry, consisting of rotations and translations in two dimensions. It remain s to work out the proportionality fa ctor relating r5 3 k - k' and c53 p - p'. Since p' and p are related to k' and k respectively by a Lorentz transfor- mation, L p , we have the n.

In st e ad, I will here adopt the more usual con vention that. GS 2 Relativistic Q u antum Mechanics. The little group here is the three-dimensional rotation group. This is conveniently chosen as. To see this, note that the boost 2. Thus states of a moving massive particle and , by extension, multi-particle states have the same transformation un- der rotations as in non-relativistic quantum mechanics.

This is another piece of good news the whole apparatus of spherical harmonics, Clebsch-Gordan coefficients, etc. Mass sera First, we have to work out the structure of the little group. What group is this? Groups that do not have invariant Abelian subgroups have certain simple properties, and for this reason are called semi-si mple.

First, let's take a look at the Lie algebra of I SD 2. For O a, infinitesimal, the general group element is. Either from 2. Since A and B are commuting Hermitian operators they like the momen- tum generators of the inhomogeneous Lorentz group can be simultane- ously diagonalized by states Tk. The problem is that if we find one such set of non-zero eigenvalues of A, B, then we find a whole continuum. From Eq. Massless particles.

We are now in a position to calculate the Lorentz transformation properties of general massless particle states. First note that by use of the general arguments of Section 2. The Lorenz transformation rule for a massless particle of arbitrary helicity is now given by Eqs. We shall see in Section 5. At this point we have not yet encountered any reason that would forbid the helicity a of a massless particle from being an arbitrary real number.

As we shall see in Section 2. To calculate the little-group element 2. This may conveniently be chosen to have the for m. For instance, suppose we take p to have polar and azimuthal angles 9 and 0 :.

Since 2. Note that the helicity is Lorentz-invariant ; a massless particle of a given helicity a looks the same aside from its momentum in all inertial frames. Indeed, we would be justified in thinking of massless particles of each different helicity as different species of particles.

Even though the helicity of a massless particle is Lorentz-invariant, the state itself is not. In particular, because of the helicity-dependent phase factor exp i a H in Eq.

For instance, a general one-photon state of four-momenta may be writte n. Plane polarized gravitons can be defined in a similar way, and here Eq. We saw in Section 2. These transformation rules incorporate most of what is meant when we say that P or T are 'conserved'. In it became understaod8 that this is true for P only in the approximation in which one ignores the effects of weak interactions, such as those that produce nuclear beta decay.

Time-reversal survived for a while, but in there appeared indirect evidence9 that these properties of T are also only approximately satisfied. See Section 3. In what follows, we will make believe that operators P and T satisfying Sys. Let us apply Eqs. Using 2. There are no states of negative energy energy less than that of the vacuum , so we are forced to choose the other alternative- P is linear and unitary, and commutes rather than anticommutes with H.

To avoid this, we are forced here to conclude that T is antilinear and a n txura ita ry. Now that we have decided that P is linear and T is antilinear, we can conveniently rewrite Eqs. On the other hand, T reverses J, because after time-reversal an observer will see all bodies spinning in the opposite direction. Note by the way that Eq.