NSCASCPPR PERSPECTIVE PAPER: CLASSIC PNAS

The sources of Schwinger’s Green’s functions

Silvan S. Schweber* Dibner Institute, Massachusetts Institute of Technology, Cambridge, MA 02139; and Department of Physics, Brandeis University, Waltham, MA 02454-9110

Julian Schwinger’s development of his Green’s functions methods in quantum field theory is placed in historical context. The relation of Schwinger’s quantum action principle to ’s path-integral formulation of quantum mechanics is reviewed. The nonperturbative character of Schwinger’s approach is stressed as well as the ease with which it can be extended to finite tempera- ture situations.

n his introduction to the volume containing the addresses essentially infinite. , protons, and nuclei could be speci- that were made at the three memorial symposia held after fied by their mass, spin, number, and electromagnetic properties Julian Schwinger’s death in 1994 (1), Ng observed that few such as charge and magnetic moment. In addition, the formalism have matched Schwinger’s contribution to, and could readily incorporate the consequence of the strict identity Iinfluence on, the development of physics in the 20th century. As and indistinguishabilty of these ‘‘elementary’’ entities. Their in- Paul C. Martin and Sheldon L. Glashow, two of Schwinger’s distinguishability implied that an assembly of them obeyed char- most outstanding students, noted: ‘‘[Schwinger] set standards acteristic statistics depending on whether their spin is an integer and priorities single-handedly...Hisideas, discoveries, and tech- or half odd integer multiple of Planck’s constant, h. niques pervade all areas of ’’ (1). Indeed in In 1927 Dirac extended the formalism to include the interac- the post-World War II period Schwinger set the standards and tions of charged particles with the electromagnetic field by de- priorities of theoretical physics. Schwinger was the first to make scribing the electromagnetic field as an assembly of photons. For use of the ideas that had been advanced by Dirac, particles were the ‘‘fundamental’’ substance. In contra- Hendrik Kramers to formulate a fully relativistic version of distinction, Jordan argued that fields were fundamental and (QED) that was finite to the orders of advocated a unitary view of nature in which both matter and that could be readily calculated. Frank Yang radiation were described by wave fields, with particles appearing in his memorial address put it thus: as excitations of the field. Jordan’s view was that the electromag- netic field was to be described by field operators that obeyed ‘‘Renormalization was one of the great peaks of the de- Maxwell equations and satisfied certain commutation relations. velopment of fundamental physics in this century. Scal- In practice this meant exhibiting the electromagnetic field as a ing the peaks was a difficult enterprise. It required superposition of harmonic oscillators, whose dynamical variables technical skill, courage, subtle judgments, and great per- were then required to satisfy the quantum rules [q , p ] ϭ iប␦ . sistence. Many people had contributed to this enterprise. m n mn These commutation rules in turn implied that in any small vol- Many people can climb the peak now. But the person who first conquered the peak was Julian Schwinger’’ (1). ume of space there would be fluctuations of the electric and magnetic field even in the absence of any free photons, and that Schwinger not only climbed the peak but in the process also the rms value of these fluctuations became larger and larger as fashioned the tools that made it possible for others to climb the the volume element one probed became smaller and smaller. peak. In the two articles under review (2, 3) Schwinger indicated Jordan advanced a unitary view of nature in which both matter how the perturbative, diagrammatic approach to the representa- and radiation were described by wave fields. The of tion of quantum field theories (QFTs) that had been given by these wave fields then exhibited the particle nature of their Richard Feynman and , could be generated from quanta and thus elucidated the mystery of the particle-wave du- the functional equations satisfied by what Schwinger called the ality. An immediate consequence of Jordan’s approach was an Green’s functions. The Green’s functions are vacuum expecta- answer to the question: ‘‘Why are all particles of a given species tion values of time-ordered Heisenberg operators, and the field (as characterized by their mass and spin) indistinguishable from theory can be defined nonperturbatively in terms of these func- one another?’’ The answer was that the field quantization rules tions. In a lecture delivered on the occasion of receiving an hon- made them automatically thus, because the ‘‘particles’’ were exci- orary degree from Nottingham University in 1993, Schwinger tations of the same underlying field. (For references to the arti- related his coming to these Green’s functions (4). In the follow- cles of the authors cited in this and the next section, see refs. 5 ing I give a brief account of the background of these two arti- and 6.) cles (2, 3). The creation and annihilation of particles, first encountered in the description of the emission and absorption of photons by QFT 1927–1945 charged particles, was a novel feature of QFT. Dirac’s ‘‘hole 1 The formulation of nonrelativistic quantum mechanics by theory,’’ the theory he constructed to describe relativistic spin ⁄2 , Erwin Schro¨dinger, , Pascual Jor- particles involved the creation and annihilation of matter. Dirac dan, and from 1925 to 1927 was a revolutionary had recognized that the equation he had obtained in 1928 to 1 achievement. Its underlying metaphysics was atomistic. Its suc- describe relativistic spin ⁄2 particles, besides possessing solutions cess derived from the confluence of a theoretical understanding, of positive energy, also admitted solutions with negative energy. the representation of the dynamics of microscopic particles by To avoid transitions to negative energy states, Dirac postulated quantum mechanics, and the apperception of a quasi-stable on- that the state of lowest energy, the vacuum, be the state in which tology, namely, electrons, protons, and nuclei, the building all of the negative energy states were occupied and noted that blocks of the entities (atoms, molecules, and simple solids) that populated the domain that was being carved out. Quasi-stable meant that under normal terrestrial conditions electrons, pro- This Perspective is published as part of a series highlighting landmark papers published in tons, and nuclei could be treated as ahistoric objects, whose PNAS. Read more about this classic PNAS article online at www.pnas.org͞classics.shtml. physical characteristics were seemingly independent of their *E-mail: [email protected]. mode of production and whose lifetimes could be considered as © 2005 by The National Academy of Sciences of the USA

www.pnas.org͞cgi͞doi͞10.1073͞pnas.0405167101 PNAS ͉ May 31, 2005 ͉ vol. 102 ͉ no. 22 ͉ 7783–7788 Downloaded by guest on September 25, 2021 an unoccupied negative energy state (what Dirac called a hole) that only further conceptual revolutions would solve the diver- would correspond to ‘‘a new kind of particle, unknown to experi- gence problem in QFT. In 1938 Heisenberg proposed that the mental physics, having the same mass and opposite charge to an next revolution be associated with the introduction of a funda- ’’ (7). Evidence that such ‘‘’’ existed was given mental unit of length, which would delineate the domain in by Carl Anderson, , and Giuseppe Occhialini which the concept of fields and local interactions would be appli- soon thereafter. cable. The S-matrix theory that he developed in the early 1940s The field theoretic description of ␤-decay phenomena given by was an attempt to make this approach concrete. He observed in 1933–1934 was an important landmark in the that all experiments can be viewed as scattering experiments. In developments of QFT. The process wherein a radioactive nu- the initial configuration the system is prepared in a definite cleus emits an electron (␤-ray) had been studied extensively ex- state. The system then evolves and the final configuration is ob- perimentally. in the early 1930s had suggested served after a time that is long compared with the characteristic 1 the existence of a neutral, very light spin ⁄2 particle that was times pertaining in the interactions. The S-matrix is the operator emitted in the ␤-decay process to maintain energy and angular that relates initial and final states. Its knowledge allows the com- momentum conservation. Fermi then demonstrated that the sim- putation of scattering cross sections and other observable quanti- plest model of a theory of ␤-decay based on Pauli’s suggestion ties. As in 1925, Heisenberg again suggested that only variables implied that electrons do not exist in nuclei before ␤-emission referring to experimentally ascertainable quantities enter theo- occurs, but acquire their existence when emitted in the same way retical descriptions. Because local field operators were not as photons in an atomic transition. measurable, fundamental theories should find new modes of rep- The discovery of the neutron by in 1932 sug- resentations, and with his S-matrix theory he opened a new gested that atomic nuclei are composed of protons and neutrons. chapter in the development of QFTs. The existence of the neutron made possible the application of quantum mechanics to elucidate the structure of the nucleus. QED 1947–1950 Heisenberg, Ettore Majorana, , and others for- Shortly after the end of World War II an important advance in mulated models of nuclear structure based on nucleon–nucleon QFT stemmed from the attempt to explain quantitatively the interactions. Nuclear forces had to be strong and of very short discrepancies between the predictions of the for range. In 1935 proposed a QFT model of nu- the level structure of the hydrogen atom and for the value it as- clear forces. Just as in QED in which the electromagnetic force cribed to the magnetic moment of the electron and the empirical between charged particles was conceptualized as caused by the data. These deviations had been measured in reliable and pre- exchange of ‘‘virtual’’ photons between them (called virtual be- cise molecular beam experiments carried out by and cause these photons do not obey the relation E ϭ h␯ valid for Isidor Rabi and their coworkers at in New free photons) the exchange of a ‘‘meson’’ mediated the force York and were reported at the Shelter Island Conference in between nucleons. Photons being massless made the range of June 1947. Shortly after the conference, it was shown by Hans electromagnetic forces infinite. In Yukawa’s theory, the ex- Bethe that the (the deviation of the 2s and 2p levels changed quanta are massive, resulting in interactions of finite of hydrogen from the values given by the Dirac equation) was of range. The association of interactions with exchanges of quanta quantum electrodynamical origin, and that the effect could be is a feature of all QFTs. computed by making use of what became known as mass renor- QED, Fermi’s theory of ␤-decay, and Yukawa’s theory of malization, an idea that had been put forward by Hendrik nuclear forces established the model on which subsequent devel- Kramers. opments were based. It postulated ‘‘impermanent’’ particles to The parameters for the mass, mo, and that for the charge, eo, account for interactions and assumed that relativistic QFT was that appear in the Lagrangian-defining QED are not the ob- the proper framework for representing processes at ever smaller served charge and mass of an electron. The observed mass, m, distances. But relativistic QFTs are beset by divergence difficul- of the electron is to be introduced in the theory by the require- ties manifested in perturbative calculations beyond the lowest ment that the energy of the physical state corresponding to a order. Higher orders yield infinite results. These difficulties stem free electron moving with momentum p be equal to (p2 ϩ m2)1/2. from the fact that (i) the description is in term of local fields Similarly, the observed charge should be defined by the require- (i.e., fields that are defined at a point in space-time), which are ment that the force between two electrons (at rest) separated by systems with an infinite number of degrees of freedom, and (ii) a large distance r, be described by Coulomb’s law, e2͞r2,withe the interaction between fields is assumed to be local. The local- the observed charge of an electron. It was shown by Schwinger ity of the interaction is expressed by interaction terms in the and Feynman that all of the divergences encountered in the low ␮ ␮ Hamiltonian of the form j (x)A␮(x) in QED, where j (x)isthe orders of perturbation theory could be eliminated by reexpress- current operator of the matter field and A␮(x) is that of the elec- ing the parameters mo and eo in terms of the observed values m tromagnetic field. and e, a procedure that became known as mass and charge In QED the local interaction terms imply that photons will renormalization. [For Schwinger’s contributions to these devel- couple with (virtual) electron- pairs of arbitrarily high opments, see refs. 6 and 8–12.] Feynman’s contribution was a momenta and result in divergences, and similarly electrons and technique by which the perturbative content of a QFT could positrons will couple with (virtual) photons of arbitrary high mo- readily be visualized in terms of diagrams and that for a given menta and give rise to divergences. These problems impeded physical process the contribution of each of the diagrams could progress throughout the 1930s, and most of the workers in the readily be written down by virtue of certain correspondence field doubted the correctness of QFT in view of these divergence rules. Furthermore, these diagrams furnished what Feynman difficulties. Numerous proposals to overcome these problems called the ‘‘machinery’’ of the particular processes: they not only were advanced during the 1930s, but all ended in failure. The indicate the mechanism that explains why certain processes take pessimism of the leaders of the discipline (, Pauli, place in particular systems but also how so by the exchange of Heisenberg, Dirac, and J. Robert Oppenheimer) was partly quanta. Feynman’s approach is defined by a limiting procedure. responsible for the lack of progress. They had witnessed the A cut-off is introduced into the theory so that the physics at mo- overthrow of the classical concepts of space-time and were re- menta higher than some momentum (or equivalently the physics sponsible for the rejection of the classical concept of determin- at distances shorter than some cut-off length) is altered, and all ism in the description of atomic phenomena. They had brought calculated quantities thereby rendered finite but cut-off depen- about the quantum mechanical revolution and were convinced dent. The parameters of the cut-off theory are then expressed in

7784 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0405167101 Schweber Downloaded by guest on September 25, 2021 terms of physically measurable quantities (such as the mass and the field, or fields, being described by variables ␾␣(x) that are ␣ ␣ charge of the particles that are described by the theory). Renor- defined at every point of space-time x, with ␾␮(x) ϭ Ѩ␾ (x)͞ malized QED is the theory that is obtained by letting the cut-off Ѩx␮. The dynamics of the field system is then determined by length go to zero. The success of renormalized QED in account- an action principle that stipulates that the functional ing for the Lamb shift, the anomalous magnetic moment of the electron and the , the radiative corrections to the scattering 1 I͑⍀͒ ϭ ͵ ᏸd4x [2] of photons by electrons, to pair production and bremsstrahlung, c was spectacular. ⍀ In 1948 Dyson was able to show that such charge and mass defined over the space-time volume ⍀ is stationary for the physi- renormalizations were sufficient to absorb all of the divergences ⍀ of the scattering matrix (S-matrix) in QED to all orders of per- cally realizable field configurations. I( ) is said to be stationary if the variation ␾␣ 3 ␾␣ ϩ ␦␾␣ for arbitrary ␦␾␣ that vanish on turbation theory. More generally, Dyson demonstrated that only ⍀ ␦ for certain kinds of QFTs is it possible to absorb all the infinities the boundary of produces no change in I to first order in ␦␾␣. The action principle then yields the fields’ equations of by a redefinition of a finite number of parameters. He called motion: such theories renormalizable. Renormalizability thereafter be- came a criterion for theory selection. Perhaps the most impor- Ѩᏸ Ѩ Ѩᏸ tant legacy of the 1947–1952 period was providing the theoreti- ␣ Ϫ ␣ ϭ 0. [3] Ѩ␾ Ѩx␮ Ѩ␾ cal physics community with a firm foundation for believing that ␮ local QFT was the framework best suited for the unification of ␣ It is usually assumed that ᏸ depends only quadratically on ␾␮(x) quantum theory and special relativity and the description of na- so that the field equations are at most of second order. For ture at a foundational level. physically interesting situations the space-time region ⍀ is The challenge on how to formulate local QFTs based ab initio bounded by two space-like surfaces, ␴1 and ␴2. [A space-like on a quantum action principle was taken up by Schwinger in a surface is one on which every two points, x, xЈ are separated by a series of articles starting in 1950 (refs. 13–15; see also ref. 16). space-like distance, i.e., (x␮ Ϫ x␮Ј)2 Ͼ 0.] The state of the field He wanted to get away from quantization procedures that were system is then completely specified by giving the values of ␾␣(x) expressed as a set of operator prescriptions imposed on a classi- ␣ ␣ on ␴1 and ␴2 or the values of ␾ (x) and n␮␾␮(x)on␴1, n␮ being cal description based on a Hamiltonian. This approach for the the normal to ␴1. field theoretic case produced an asymmetry in the treatment of One can consider more general kinds of variation whereby not time and space and resulted in a formulation that was nonco- only the ␾␣ are varied but also the boundary of ⍀, each point ␣ variant. Additionally, it placed the existence of anticommuting thereof being moved from x␮ to x␮ ϩ ␦x␮; ␦␾ is then defined as Fermi–Dirac fields on a purely empirical basis to be introduced into the formalism on an ad hoc basis. Schwinger’s point of de- ␦␾␣ ϭ ␾␣͑x ϩ ␦x͒ Ϫ ␾␣͑x͒ parture was the articles of Feynman in which he had formulated ϭ ␾␣͑ ͒␦ quantum mechanics by exhibiting the probability amplitude con- ␮ x x␮ . [4] necting the state of a system at one time to that at a later time as a sum of complex unit amplitudes (one for each possible tra- Under this more general variation jectory of the system connecting the initial and final state), the 1 phase of each amplitude being determined by the value of the ␦ ͑⍀͒ ϭ ͩ␲␣␦␾␣ ϩ ͩ ᏸ Ϫ ␾␣␲ ͪ␦ ͪ ␴ I ͵ n␮ ␮ ␣ x␮ d , [5] ͐ c action, L dt, for that trajectory. [See ref. 15.] Furthermore, ⍀ because action is a relativistic invariant the formulation could thus be made covariant. But whereas Feynman gave a global where solution, for Schwinger, 1 Ѩᏸ ␲␣ ϭ ‘‘The idea from the beginning was not, as Feynman n␮ Ѩ␾␣ [6] would do, to write down the answer, but to continue in c ␮ the grand tradition of classical mechanics, but only as a is the momentum conjugate to ␾␣ defined at x and on ␴. historical model, to find a differential, an action princi- Quantization is then imposed by stipulating commutation ple formulation. What is Hamilton’s principle or its gen- rules between ␲␣(x) and ␾␣(xЈ) for x, xЈ on ␴ : eralization in quantum physics? If you want the time 1 ␣ ␤ transformation function, do not ask what it is but how it ͓␾ ͑x͒, ␲ ͑xЈ͔͒ ϭ iប␦␣␤␦͑x Ϫ xЈ͒ infinitesimally changes. The distinction [with Feynman’s ␣ ␤ ␣ ␤ approach] comes [because] this deals with all kinds of ͓␾ ͑x͒, ␾ ͑xЈ͔͒ ϭ ͓␲ ͑x͒, ␲ ͑xЈ͔͒ ϭ 0. [7] quantum variations, on exactly the same footing, which ␾␣ means from a field point of view not only do Bose– In the quantized theory the operators (x) obey the same Einstein fields appear naturally but Fermi–Dirac fields. equations of motion as the classical fields. But because the Whereas with the path integral approach with its clear field operators satisfy commutation rules a state of the field ␾Ј␣ connection to the correspondence principle, the anti- system is specified by only giving the eigenvalues, ,ofthe commuting Fermi system appears out of nowhere, there field operators on one space-like surface, the latter forming a is no logical reason to have it except that one knows one complete set of commuting operators on that surface, because has to. It does not appear as a logical possibility as it fields at different points of a spacelike surface commute with one another as indicated by the commutation rules [␾␣(x), does with the differential.’’ ␾␤ Ј ϭ Ј ␴ Schwinger (ref. 12, p. 316) (x )] 0 for x, x on . Such a state is denoted by the Di- rac ket vector Խ␾Ј␣, ␴͘. The transition probability from the Խ␾Ј␣ ␴ ͘ Խ␾Љ␣ ␴ ͘ Recall that a classical field theory can be fully specified by its state 1 , 1 to the state 2 , 2 is given by the modulus ͗␾Љ␣ ␴ Խ␾Ј␣ ␴ ͘ Lagrangian density squared of the amplitude 2 , 2 1 , 1 . Feynman, based on some previous work of Dirac, derived a ᏸ ϭ ᏸ͑␾␣͑ ͒ ␾␣͑ ͒͒ x , ␮ x [1] readily visualizable expression for the transition amplitude for

Schweber PNAS ͉ May 31, 2005 ͉ vol. 102 ͉ no. 22 ͉ 7785 Downloaded by guest on September 25, 2021 systems with a finite number of degrees of freedom. For a single for t2 Ͼ t Ͼ t1. Similarly one readily derives that particle it reduces to the following prescription for evaluating the ͗ Љ Խ Ј ͘ Љ ប ␦ ប ␦ probability amplitude q , t2 q , t1 for finding the particle at q at ͗ Љ ͉ Ј ͘f ϭ ͗ Љ ͉ ͑ ͑ ͒ ͑ Ј͉͒͒ Ј ͘f q , t2 q , t1 q , t2 T q t q t q , t1 [13] time t2 if it was at qЈ at time t1: i ␦f͑t͒ i ␦f͑tЈ͒

Љ i q t2 in which T(ۙ) is the time-ordered product defined by ͗ Љ ͉ Ј ͘ ϭ ͵Ᏸ ͵ [8] q , t2 q , t1 q expͭ dt Lͮ , ប ͑ ͑ ͒ ͑ Ј͒͒ ϭ ͑ ͒ ͑ Ј͒ Ͼ Ј qЈt1 T q t q t q t q t if t t

where the Ᏸq integration is understood as a summation over ϭ q͑tЈ͒q͑t͒ if tЈ Ͼ t. [14] all curves with fixed end points qЉ at t2 and qЈ at t1. Each curve (path) contributes to the sum of a unit amplitude, the This result generalizes to phase of which being determined by the value of the action ប ␦ ប ␦ ប ␦ ͗ Љ ͉ Ј ͘f ··· q , t2 q , t1 qЈt2 i ␦f͑t͒ i ␦f͑tЈ͒ i ␦f͑tЉ͒ I ϭ ͵ Ldt, ϭ ͗qЉ, t ͉T͑q͑t͒q͑tЈ͒q͑tЉ͒ ···͉͒qЈ, t ͘f qЈt1 2 1

Љ evaluated for that path, with L being the classical Lagrangian i q t2 ϭ ͵Ᏸ ͑ ͒ ͑ Ј͒ ͑ Љ͒ ͵ Ј͓ ϩ ͔ [15] for the system qqt q t q t · · ·expͭ dt L0 fq ͮ ប qЈt1 1 dq 2 L ϭ mͩ ͪ Ϫ V͑q; t͒. [9] 2 dt with T rearranging the operators appearing within the paren- thesis in an ordered form with ascending times to the left: The correspondence between Feynman’s formulation and the ··· usual formulation is established by defining the Heisenberg T͑q͑t͒q͑tЈ͒q͑tЉ͒ ···͒ ϭ q͑tЉЉ͒q͑ٞ͒q͑ЉЉЉ͒ operator q(t)by [with tЉЉ Ͼ tٞ Ͼ tЉЉЉ···. [16 Љ i q t2 ͗ Љ ͉ ͑ ͉͒ Ј ͘ ϭ ͵Ᏸ ͑ ͒ ͵ Ј Feynman’s formulation can readily be formally extended to q , t2 q t q , t1 qqt expͭ dt Lͮ ប Ј the case of a field system. (See in particular ref. 19.) q t1 ␣ ␣ The transition amplitude ͗␾2Љ , ␴2Խ␾1Ј , ␴1͘ is then written as

qЉt2 i i ϭ ͵ dqٞ͑t͒ ͵ Ᏸqexpͭ ͵ dtЈ Lͮ qٞ͑t͒͵ Ᏸq ͗␾Ј␣ ␴ ͉␾Ј␣ ␴ ͘ ϭ ͸ ͫ ͑⍀͒ͬ ប 2 , 2 1 , 1 N exp ប IH , [17] qٞt H

i qЈЉt where the sum on the right extends over all possible histories of expͭ ͵ dtЈ L, [10] ␴ ␴ ␾Ј␣ ប the fields between 1 and 2. A history H is specified by (x) ␣ ␣ qٞt1 taking on the values ␾2Љ on ␴2 and the values ␾1Ј on ␴1 and any value in between. In the limit over all paths, the path-integral is where on the right side of Eq. 10 q(t) is a c–number, whereas on the left side it is a Heisenberg picture operator. a continuously infinite sum to which can be given a rigorous The formalism allows a very useful extension. (See chapter mathematical meaning in some cases, e.g., some field theories in 1 of ref. 18.) If in addition to V(q, t) an external force is one space and one time dimension. N is a normalization factor taken to act on the system and the Lagrangian is assumed to designed to guarantee that be of the form ͸͉͗␾Ј␣ ␴ ͉␾Ј␣ ␴ ͉͘2 ϭ 2 , 2 1 , 1 1. [18] ϭ ϩ ͑ ͒ ͑ ͒ ␾Ј L L0 f t q t 2 [11] 1 dq 2 Again to make explicit the connection between the path-integral L ϭ mͩ ͪ Ϫ V͑q; t͒, 0 2 dt formulation and the usual formulation of QFT, the (Heisenberg picture) operator corresponding to ␾␣ is defined by the formula then under those circumstances the functional derivative of ͗ Љ Խ Ј ͘ i q , t2 q , t1 with respect to the external force coupled to q(t) ͗␾Ј␣ ␴ ͉␾␣ ͑ ͉͒␾Ј␣ ␴ ͘ ϭ ͸ ␾␣ ͑ ͒ ͫ ͑⍀͒ͬ yields the matrix element of the Heisenberg operator q(t) 2 , 2 op x 1 , 1 N H x exp ប IH [19] H qЉt ប ␦ i 2 ប ␦ ␾␣ ␾␣ ͗ Љ ͉ Ј ͘f ϭ ͵Ᏸ ͵ Ј with H being the value (x) takes for the particular path. q , t2 q , t1 q expͭ dt L0ͮ i ␦f͑t͒ ប i ␦f͑t͒ Ј Schwinger’s point of departure was the observation that q t1 ␣ ␣ the states Խ␾Ј2 , ␴2͘ and Խ␾Ј1 , ␴1͘ are related by a unitary Љ i q t2 transformation: ͵ Ј ͑ Ј͒ ͑ Ј͒ expͭ dt f t q t ͮ ␣ ␣ ␣ Ϫ ␣ ប ͗␾Љ ␴ ͉␾Ј ␴ ͘ ϭ ͗␾Љ ␴ ͉ 1͉␾Ј ␴ ͘ 2 , 2 1 , 1 1 , 1 U21 1 , 1 , [20] qЈt1 which for infinitesimal changes becomes Љ i q t2 ϭ ͵ Ᏸ ͑ ͒ ͵ Ј͓ ϩ ͔ ␦͗␾Љ␣ ␴ ͉␾Ј␣ ␴ ͘ ϭ ͗␾Ј␣ ␴ ͉␦ Ϫ1͉␾Ј␣ ␴ ͘ qqt expͭ dt L0 fq ͮ , , , U , . [21] ប 2 2 1 1 1 1 21 1 1 qЈt1 Ϫ1 ͞ប ␦ Ϫ1 Because U21 is unitary (i ) U21 U21 is hermitian, and ϭ ͗ Љ ͉ ͑ ͉͒ Ј ͘f q , t2 q t q , t1 [12] Schwinger thus wrote

7786 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0405167101 Schweber Downloaded by guest on September 25, 2021 i Schwinger’s 1951 PNAS Articles ␦ Ϫ1 ϭ ͩ ͪ Ϫ1␦ U21 ប U21 W21, [22] Schwinger’s article ‘‘The Theory of Quantized Fields,’’ which was submitted to Physical Review in March of 1951 and appeared where ␦W21 is an infinitesimal hermitian operator whose com- in June of 1951 (15), was one of four in which nonperturbative position law is readily established to be given by methods play a central role. His article on ‘‘Gauge Invariance and Vacuum Polarization,’’ which he sent to Physical Review in ␦ ϭ ␦ ϩ ␦ W31 W32 W21. [23] December 1951 (14), deals with the description of the quantized ␦ electron-positron field in an external field. He there introduced a Schwinger’s basic postulate is that W21 can be obtained by vari- description in terms of Green’s functions, what Feynman had ations of the quantities contained in the Hermitian operator called , but one that did not depend on a perturba- tive expansion. The one-particle Green’s function ␴2 W ϭ ͵ ᏸ͑x͒ d4x. [24] 21 G͑x, xЈ͒ ϭ i͗0͉T͑␺͑x͒␺¯͑xЈ͉͒͒0͘ [30] ␴1 is defined in terms of Heisenberg operators that satisfy the The result equation of motion

␣ ␣ i i ␥ ͑ Ϫ Ѩ Ϫ ͑ ͒͒␺͑ ͒ ϩ ␺͑ ͒ ϭ ␦͗␾Љ ␴ ͉␾Ј ␴ ͘ ϭ ͸ ␦ ͫ ͑⍀͒ͬ ␮ i ␮ eA␮ x x m x 0 [31] 2 , 2 1 , 1 ប N IH exp ប IH H and the equal-time anticommutation rules

␴ i 2 ͓␺͑ ͒ ␺¯ ͑ ͔͒ ϭ ␥ ␦ ͑3͒͑ Ϫ Ј͒ ␣ ␣ ␣ 4 ␣ x, x0 , x , x0 ϩ 0 x x . [32] ϭ ͳ␾Ј , ␴ ͯ ͵ ␦ᏸ͑␾ ͑x͒, ␾␮͑x͒͒ d xͯ␾Ј , ␴ ʹ [25] 2 2 ប 1 1 ␴ 1 In Eq. 30 the state Խ0͘ is the vacuum state, the lowest energy state of the interacting system. In Eq. 30 the T product for is, of course, the same as would be obtained formally from anticommuting operators is defined as Feynman’s sum over histories approach, with the right side of the first line being defined by classical variables and the sec- ͑␺͑ ͒␺¯͑ Ј͒͒ ϭ ␺͑ ͒␺¯͑ Ј Ͼ Ј T x x x x ) for x0 x 0 ond line defining the corresponding operators. ϭ Ϫ␺¯ ͑ Ј͒␺͑ ͒ Ј Ͼ If the parameters of the system are not altered, the varia- x x for x 0 x0. [33] tion of the transformation function In the two articles entitled ‘‘On the Green’s Functions of Quan- ␦͗␾Љ␣ ␴ ͉␾Ј␣ ␴ ͘ 2 , 2 1 , 1 tized Fields I and II,’’ which Schwinger submitted in May 1951 to PNAS (2, 3), he gave a preliminary account of a general the- ␴ i 2 ory of Green’s function ‘‘in which the defining property is taken Љ␣ ␣ ␣ 4 ␣ ϭ ͳ␾ , ␴ ͯ ͵ ␦ᏸ͑␾ ͑x͒, ␾␮͑x͒͒ d xͯ␾Ј , ␴ ʹ [26] 2 2 ប 1 1 to be the representation of the fields of prescribed sources.’’ This ␴ 1 is the generalization to field systems of the introduction of exter- nal forces coupled linearly to q(t) in the case of particle motion ␾Ј␣ ␴ ␾Ј␣ ␴ arises only from infinitesimal changes of 2 , 2 and 1 , 1. as described in Eqs. 12–15. Thus, Schwinger took the gauge- These are generated by operators associated with the surfaces invariant Lagrangian for the coupled Dirac and Maxwell field to ␴ ␴ 1 and 2, respectively, include external sources linearly coupled to each field so that ␦W ϭ F͑␴ ͒ Ϫ F͑␴ ͒, [27] 12 1 2 ᏸ ϭ ␺¯͑x͒␥␮(ϪiѨ␮ Ϫ eA␮͑x͒)␺͑x͒ ϩ m␺¯ ͑x͒␺͑x͒ ϩ ␺¯ ͑x͒␩͑x͒

which is the operator principle of stationary action as it states 1 that the action integral operator is not changed by infinitesimal ϩ ␩៮ ͑x͒␺͑x͒ Ϫ F␮␯͑x͒͑Ѩ␮A␯͑x͒ ϪѨ␯ A␮͑x͒͒ variations of the field operators in the interior of the region 2 ␴ ␴ bounded by 1 and 2. The field equations follow 1 ϩ F␮␯͑x͒F␮␯͑x͒ ϩ J␮͑x͒A␮͑x͒, [34] Ѩᏸ Ѩ Ѩᏸ 4 Ϫ ϭ Ѩ␾␣ Ѩ Ѩ␾␣ 0. [28] x␮ ␮ where the source spinor ␩(x) anticommutes with the Dirac field operators. In keeping with his requirement that only first-order Conservation laws are associated with variations that leave variables appear in a quantum Lagrangian, F␮ (x) and A␮(x) are the action integral unchanged, and Eq. 27 then implies that v treated as independent fields. The equations of motion are then the generators that induce the variations are constant. Infini-

tesimal displacements and rotation of the surface are gener- ␥␮(ϪiѨ␮ϪeA␮͑x͒)␺͑x͒ ϩ m␺͑x͒ ϭ ␩͑x͒ ated by the momentum and angular momentum operators, respectively, and as they leave the action invariant they are F␮␯͑x͒ ϭѨ␮A␯͑x͒ ϪѨ␯ A␮͑x͒ [35] constants of the motion. Schwinger was also able to deduce Ѩ ͑ ͒ ϭ ͑ ͒ ϩ ͑ ͒ the commutation rules that hold on a plane surface ␴ ␯ F␮␯ x J␮ x j␮ x

␣ ␤ ͓␾ ͑x͒, ␲ ͑xЈ͔͒Ϯ ϭ iប␦␣␤␦͑x Ϫ xЈ͒ with

␣ ␤ ␣ ␤ ͓␾ ͑x͒, ␾ ͑xЈ͔͒Ϯ ϭ ͓␲ ͑x͒, ␲ ͑xЈ͔͒Ϯ ϭ 0, [29] 1 j␮͑x͒ ϭ e͓␺¯͑x͒, ␥␮␺͑x͔͒. [36] 2 where the commutator applies for Boson fields and the anti- commutator for Fermion fields. (For details see ref. 16; also Schwinger restricted himself to changes in the transformation available at www.nobel.se͞nobel͞nobel-foundation͞publica- function that arise from variations of the external sources. In tions͞lectures.) terms of the notation

Schweber PNAS ͉ May 31, 2005 ͉ vol. 102 ͉ no. 22 ͉ 7787 Downloaded by guest on September 25, 2021 ͗␾Љ␣ ␴ ͉␾Ј␣ ␴ ͘ ϭ ᐃ 2 , 2 1 , 1 exp i In his second PNAS article (3) Schwinger determined the boundary conditions that the Green’s functions satisfied when ͗␾Љ␣ ␴ ͉ ͑ ͉͒␾Ј␣ ␴ ͘ [37] 2 , 2 F x 1 , 1 ␴ ␴ ϭ ͗ ͑ ͒͘ the initial and final states on 1 and 2 were the vacuum ͗␾Љ␣ ␴ ͉␾Ј␣ ␴ ͘ F x 2 , 2 1 , 1 state. When the external fields vanish in the neighborhood of ␴1 and ␴2, the one-particle Green’s function, now denoted by the dynamical principle can be written as Gϩ, contains only positive frequency for x on ␴2 and only negative frequencies for x on ␴ . By letting the surfaces ␴ ␴ 1 1 2 and ␴ recede to the infinite past and infinite future, respec- ␦ᐃ ϭ ͵ 4 ͗␦ᏸ͑ ͒͘ [38] 2 d x x tively, and introducing an integral representation for the ac- ␴ 1 tion of the functional derivatives with ␦ ͩ ϩ ␥ ͪ ͑ Ј͒ ϭ 4 Љ ͑ Љ͒ ͑ Љ Ј͒ m ie ␮ Gϩ x, x ͵d x M x, x Gϩ x , x ͗␦ᏸ͑x͒͘ ϭ ͗␺¯͑x͒͘␦␩͑x͒ ϩ ␦␩៮ ͑x͒͗␺͑x͒͘ ϩ ͗A␮͑x͒͘␦J␮͑x͒, ␦J␮͑x͒ [45] and the one-particle Green’s function ␦ Ϫ1 ⌫␮͑x, xЈ, ␰͒ ϭ Ϫ Gϩ ͑x, xЈ͒, G͑x, xЈ͒ ϭ iT͑␺͑x͒␺¯͑xЈ͒͒ [39] ␦eA␮͑␰͒

can be formally defined as the equations that the one-particle Green’s functions for the electron and the photon, now denoted by Gϩ and Ᏻϩ, can be ␦͗␺͑x͒ͯ͘ written in closed form: ␩ϭ ϭ G(x, xЈ). [40] ␦␩͑xЈ͒ 0 2 4 4 ϭ ϩ ͵ ␰͵ ␰Ј␥͑␰͒ ϩ⌫͑␰Ј͒Ᏻϩ͑␰ ␰Ј͒ Schwinger then proceeded to derive a functional equation for M m ie d d G , the one-particle Dirac Green’s function G(x, xЈ), ␦ and ͫ␥ ͑Ϫ Ѩ Ϫ ͗ ͑ ͒͒͘ ϩ ϩ ͬ ͑ Ј͒ ϭ ␦͑ Ϫ Ј͒ ␮ i ␮ e A␮ x e m G x, x x x , ␦J␮͑x͒ 2 4 ϪѨ␰Ᏻϩ͑␰, ␰Ј͒ ϩ ͵d ␰ЉP͑␰, ␰Љ͒Ᏻϩ͑␰Љ, ␰Ј͒ ϭ ␦͑␰ Ϫ ␰Ј͒ [46] [41]

and for the one-photon Green’s function with

␦͗ ͑ ͒͘ 2 A␮ x P͑␰, ␰Ј͒ ϭ Ϫie Tr͓␥͑␰͒Gϩ⌫͑␰Ј͒Gϩ͔. Ᏻ␮␯͑x, xЈ͒ ϭ ϭ i[͗T͑A␮͑x͒A␯͑xЈ͒͘ Ϫ ͗A␮͑x͒͗͘A␯͑x͔͒͘ ␦J␯͑xЈ͒ (See equations 30–43 of ref. 3 and ref. 16 for a clarification of ␦ the notation.) Eqs. 45 and 46 are ‘‘closed’’ functional differential Ϫ ▫Ᏻ␮␯͑x, xЈ͒ ϭ ␦␮␯␦͑x Ϫ xЈ͒ ϩ ietr␥␮ G͑x, xЈ͒ ␦J␯͑xЈ͒ equations for the electron and photon one-particle propagators and for the vertex and polarization function. They describe the Ѩ␮Ᏻ␮␯͑x, xЈ͒ ϭ 0 [42] theory and lead directly to a perturbation expansions in terms of ‘‘true’’ propagators. Similarly, the graphical, perturbative version as well as for the two-particle Green’s function of QED that Feynman and Dyson had elaborated can readily be obtained from them. ͑ Ј Ј͒ ϭ ͗ ͑␺͑ ͒␺͑ ͒␺¯͑ Ј͒␺¯͑ Ј͒͒͘ G x1, x2; x1, x2 T x1 x2 x1 x2 . [43] Schwinger’s formulation of relativistic QFTs in terms of Green’s functions was a major advance in theoretical physics. The equation satisfied by the two-particle Green’s function It was a representation in terms of elements (the Green’s Ᏺ Ᏺ G͑x , x ; xЈ, xЈ͒ functions) that were intimately related to real physical observ- 1 2 1 2 1 2 ables and their correlation. It gave deep structural insights ϭ ␦͑ Ϫ Ј͒␦͑ Ϫ Ј͒ Ϫ ␦͑ Ϫ Ј͒␦͑ Ϫ Ј͒ x1 x1 x2 x2 x1 x2 x2 x1 [44] into QFTs; in particular, it allowed the investigation of the structure of the Green’s functions when their variables are ␦ Ᏺ ϭ ͫ␥ ͑Ϫ Ѩ Ϫ ͗ ͑ ͒͒͘ ϩ ϩ ͬ analytically continued to complex values, thus establishing ␮ i ␮ e A␮ x e m ␦J␮͑x͒ deep connections with statistical mechanics. The extension of the methods to treat nonrelativistic multiparticle systems from is known as the Bethe–Salpeter equation and was first de- a QFT viewpoint, as presented by Martin and Schwinger (20), rived by . has likewise proven deeply influential.

1. Ng, Y. J., ed. (1996) Julian Schwinger: The , The Teacher, and The Man (World 11. Schwinger, J. S. (2000) in A Quantum Legacy: Seminal Papers of Julian Schwinger, ed. Milton, Scientific, Singapore). K. A. (World Scientific, Singapore). 2. Schwinger, J. (1951) Proc. Natl. Acad. Sci. USA 37, 452–455. 12. Mehra, J. & Milton, K. A. (2000) Climbing the Mountain: The Scientific Biography of Julian 3. Schwinger, J. (1951) Proc. Natl. Acad. Sci. USA 37, 455–460. Schwinger (Oxford Univ. Press, Oxford). 4. Schwinger, J. (1996) in Julian Schwinger: The Physicist, The Teacher, and The Man, ed. Ng, 13. Schwinger, J. (1951) Science 113, 479–482. Y. J. (World Scientific, Singapore), pp. 13–27. 14. Schwinger, J. (1951) Phys. Rev. 82, 664–679. 5. Schwinger, J., ed. (1958) Selected Papers on Quantum Electrodynamics (Dover, New York). 15. Schwinger, J. (1951) Phys. Rev. 82, 914–927. 6. Schweber, S. S. (1994) QED and The Men Who Made It (Princeton Univ. Press, Princeton). 16. Schwinger, J. (1966) Science 153, 949–954. 7. Dirac, P. A. M. (1931) Proc. R. Soc. London Ser. A 133, 60–72. 17. Feynman, R. P. & Hibbs, A. R. (1965) Quantum Mechanics and Path Integrals (McGraw–Hill, 8. Schwinger, J. (1982) J. Phys. 43, Suppl. 12, 409–419. New York). 9. Schwinger, J. (1983) in The Birth of , eds. Brown, L. & Hoddeson, L. 18. Brown, L. S. (1992) (Cambridge Univ. Press, Cambridge, U.K.). (Cambridge Univ. Press, Cambridge, U.K.), pp. 354–369. 19. Dyson, F. J. & Moravcsik, M. J. (1951) Advanced Quantum Mechanics (Cornell Univ., Ithaca, 10. Schwinger, J. (1993) in Most of the Good Stuff: Memories of Richard Feynman, eds. Brown, NY). L. & Rigden, J. S. (Am. Inst. of Physics, New York), pp. 59–63. 20. Martin, P. C. & Schwinger, J. (1959) Phys. Rev. 115, 1342–1373.

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