The Differential Geometry and Physical Basis for the Applications of Feynman Diagrams Samuel L

The Differential Geometry and Physical Basis for the Applications of Feynman Diagrams Samuel L. Marateck

n May 11 of last year, the late Richard General relativity sparked mathematicians’ in- Feynman’s birthday, a stamp was ded- terest in parallel transport, eventually leading to icated to Feynman at the post office in the development of fiber bundles in differential Far Rockaway, New York, Feynman’s geometry. After physicists achieved success using Oboyhood home. (At the same time, the gauge theory, mathematicians applied it to differ- United States Postal Service issued three other ential geometry. The story begins with Maxwell’s stamps honoring the scientists Josiah Willard Gibbs equations. In this story the vector potential A goes and Barbara McClintock, and the mathematician from being a mathematical construct used to fa- John von Neumann.) cilitate problem solution in electromagnetism to The design of the stamp tells a wonderful story. taking center stage by causing the shift in the in- The Feynman diagrams on it show how Feynman’s terference pattern in the Aharonov-Bohm solenoid

work, originally applicable to QED, for which he won effect. As the generalized four-vector Aµ , it be- the Nobel Prize, was then later used to elucidate comes the gauge field that mediates the electro- the electroweak force. The design is meaningful to magnetic interaction and the electroweak and both mathematicians and physicists. For mathe- strong interactions in the standard model of maticians, it demonstrates the application of dif- physics; Aµ is understood as the connection on ferential geometry. For physicists, it depicts the ver- fiber-bundles in differential geometry. The modern ification of QED; the application of the Yang-Mills reader would be unaccustomed to the form in equations; and the establishment and experimen- which Maxwell equations first appeared. They are tal verification of the electroweak force, the first easily recognizable when expressed using vector step in the creation of the standard model. The analysis in the Heaviside-Gibbs formulation. physicists used gauge theory to achieve this and were for the most part unaware of the develop- Maxwell’s Equations ments in differential geometry. Similarly, mathe- The equations used to establish Maxwell’s equations maticians developed fiber bundle theory without in vacuo expressed in Heaviside-Lorentz rational- knowing that it could be applied to physics. We ized units are: should, however, remember that in general relativity, Einstein introduced geometry into physics. (1) ∇·E = ρ (Gauss’s law) And as we will relate below, Weyl did so for electromagnetism. (2) ∇·B = 0 (No magnetic monopoles)

Samuel L. Marateck is a senior lecturer at the Courant In- (3) ∇×B = J (Ampere’s law) stitute of Mathematical Sciences, New York University. His email address is [email protected]. (4) ∇×E =−∂B/∂t (Faraday’s and Lenz’s law)

744 NOTICES OF THE AMS VOLUME 53, NUMBER 7 where E and B are respectively the electric and p-form and M is a p +1 dimension oriented man- magnetic fields; ρ and J are the charge density and ifold with boundary ∂M. For the purposes of this electric current. The continuity equation which dic- article a manifold is simply a space that is locally tates the conservation of charge, Euclidean. The remark is that equation (8), i.e., d2 =0, is related to the principle that the bound- 2 ∅ (5) ∇·J + ∂ρ/∂t = 0, ary of a region has no boundary, i.e., ∂ M = . This last principle once noticed, is geometrically ob- 2 indicates that Maxwell’s equations describe a local vious. The fact that d =0 complements theory, since you cannot destroy a charge locally 2 ∅ ∂ M = can be seen from ∂2M ω = ∂M dω = and recreate it at a distant point instantaneously. 2 = 1 M d ω 0. Indeed Yang notes that equation (8) The concept that the theory should be local is the complements ∂2M = ∅ which, as we show later, is cornerstone of the gauge theory used in quantum in keeping with his championing of geometry’s in- field theory, resulting in the Yang-Mills theory, the fluence on field theory. basis of the standard model. Gauge Invariance Maxwell realized that since In a 1918 article Hermann Weyl2 tried to combine (6) ∇·∇×B = 0, electromagnetism and gravity by requiring the theory to be invariant under a local scale change of equation (3) is inconsistent with (5); he altered (3) the metric, i.e., g → g eα(x), where x is a 4- to read µν µν vector. This attempt was unsuccessful and was (3) ∇×B = J + ∂E/∂t criticized by Einstein for being inconsistent with observed physical results. It predicted that a vec- Thus a local conservation law mandated the addi- tor parallel transported from point p to q would tion of the ∂E/∂t term. Although equations (1), (2), have a length that was path dependent. Similarly, (3) and (4) are collectively known as Maxwell’s the time interval between ticks of a clock would also equations, Maxwell himself was responsible only depend on the path on which the clock was transfor (3). ported. The article did, however, introduce Maxwell calculated the speed of a wave propa- • the term “gauge invariance”; his term was Eich- gated by the final set of equations and found its invarianz. It refers to invariance under his scale velocity very close to the speed of light. He thus change. The first use of “gauge invariance” in hypothesized that light was an electromagnetic English3 was in Weyl’s translation4 of his famous wave. Since the curl of a vector cannot be calculated 1929 paper. in two dimensions, Maxwell’s equations indicate • the geometric interpretation of electromagnet- that light, as we know it, cannot exist in a two- ism. dimensional world. This is the first clue that elec- • the beginnings of nonabelian gauge theory. The tromagnetism is bound up with geometry. In fact, similarity of Weyl’s theory to nonabelian gauge equation (6) is the vector analysis equivalent of the theory is more striking in his 1929 paper. differential geometry result stating that if β is a p-form and dβ is its exterior derivative, then d(dβ) By 1929 Maxwell’s equations had been com- or d2β = 0. bined with quantum mechanics to produce the Unlike the laws of Newtonian mechanics, Maxwell’s start of quantum electrodynamics. In his 1929 ar- equations carry over to relativistic frames. The non- ticle5 Weyl turned from trying to unify electro- homogeneous equations, (1) and (3), become magnetism and gravity to following a suggestion originally thought to have been made by Fritz Lon- (7) ∂ F µν = Jν; µ don in his 1927 article6 and introduced as a phase while the homogeneous equations, (2) and (4), be- factor an exponential in which the phase α is pre- + come ceded by the imaginary unit i, e.g., e iqα(x), in the wave function for the wave equations (for instance, αβγδ (8) ∂βFγδ = 0, µ the Dirac equation is (iγ ∂µ − m)ψ = 0). It is here where αβγδ is the Levi-Civita symbol, F µν = ∂µAν that Weyl correctly formulated gauge theory as a ν µ 0 i −∂ A , A is the scalar potential, and A ’s (i = 1, 1 Yang, C. N., Physics Today 6 (1980), 42. 2, 3) the components of the vector potential A. Note 2 that both equations (7) and (8) are manifestly co- Weyl, Hermann, Sitzwingsber. Preuss. Akad., Berlin, 1919, p. 465. variant. In a later section we will show that equa- 3 tion (8) is due to the principle d2ω =0, where ω See Jackson, J. D., and Okun, L. B., Rev. Mod. Phys. 73 is a p-form. (2001), 663. 4 The following remark can be understood in dif- Weyl, H., Proc. Natl. Acad. Sci. 15 (1929), 32. 5 ferential geometry terms by using the generalized Weyl, Hermann, Z. f Phys. 330 (1929), 56. 6 Stoke’s theorem: M dω = ∂M ω , where ω is a London, Fritz, Z. f Phys. 42 (1927), 375.

AUGUST 2006 NOTICES OF THE AMS 745 symmetry principle from which electromagnetism charge q must be conserved.11 Thus gauge invari- could be derived. It was to become the driving ance dictates charge conservation. By Noether’s force in the development of quantum field theory. theorem, a conserved current is associated with a In their 2001 Rev. Mod. Phys. paper, Jackson and symmetry. Here the symmetry is the nonphysical 7 Okun point out that in a 1926 paper predating Lon- rotation invariance in an internal space called a don’s, Fock showed that for a quantum theory of fiber. In electromagnetism the rotations form the charged particles interacting with the electromag- group U(1), the group of unitary 1-dimensional netic field, invariance under a gauge transforma- matrices. U(1) is an example of a gauge group, and tion of the potentials required multiplication of the the fiber is S1, the circle. wave function by the now well-known phase fac- A fiber bundle is determined by two manifolds tor. Many subsequent authors incorrectly cited the and the structure group G which acts on the fiber: date of Fock’s paper as 1927. Weyl’s 1929 article— the first manifold, called the total space E, consists along with his 1918 one, and Fock’s and London’s, of many copies of the fiber F—one for each point and other key articles—appears in translation in a in the second manifold, the base manifold M which work by O’Raifeartaigh8 with his comments. Yang9 for our discussion is the space-time manifold. The discusses Weyl’s gauge theory results as reported 10 fibers are said to project down to the base mani- by Pauli as a source for Yang-Mills gauge theory 12 (although Yang did not find out until much later fold. A principal fiber bundle is a fiber bundle in that these were Weyl’s results): which the structure group G acts on the total space E in such a way that each fiber is mapped onto it- ... I was very much impressed with the self and the action of an individual fiber looks like idea that charge conservation was re- the action of the structure group on itself by left- lated to the invariance of the theory translation. In particular, the fiber F is diffeomor- under phase changes and even more phic to the structure group G. impressed with the fact the gauge- The gauge principle shows how electromagnet- invariance determined all the electro- ism can be introduced into quantum mechanics. magnetic interactions.... The transformation ∂µ → ∂µ + iqAµ is also called For the wave equations to be gauge invariant, i.e., the minimal principle and the operation ∂µ + iqAµ have the same form after the gauge transformation is the covariant derivative of differential geometry, as before, the local phase transformation D = d + iqA, where A is the connection on a fiber ψ(x) → ψ(x)e+iqα(x) has to be accompanied by the bundle. A connection on a fiber bundle allows one local gauge transformation to identify fibers over points bi M via parallel transport along a path γ from b to b . In general, (9) A → A − q−1∂ α(x). 1 2 µ µ µ the particular identification is path dependent. It (The phase and gauge transformations are local be- turns out that the parallel transport depends only cause α(x) is a function of x.) This dictates that the on the homotopy class of the path if and only if ∂µ in the wave equations be replaced by ∂µ + iqAµ the curvature of the connection vanishes identically. in order for the ∂µα(x) terms to cancel each other Recall that two paths are homotopic if one can be out. Thus gauge invariance determines the type of deformed continuously onto the other keeping the interaction; here, the inclusion of the vector po- end points fixed. tential. This is called the gauge principle, and Aµ In his 1929 paper Weyl also includes an ex- is called the gauge field or gauge potential. Gauge pression for the curvature Ω of the connection A, invariance is also called gauge symmetry. In elec- namely, Cartan’s second structural equation, which tromagnetism A is the space-time vector potential in modern differential geometry notation is representing the photon field, while in electroweak Ω = dA + A ∧ A. It is the same form as the equa- theory A represents the intermediate vector bosons tion used by Yang and Mills, which in modern no- ± W and Z0 fields; in the strong interaction, A rep- = + 1 tation is Ω dA 2 [A, A]. Since the transforma- resents the colored gluon fields. The fact that the tions in (9) form an abelian group U(1), the +iqα(x) q in ψ(x)e must be the same as the q in space-time vector potential A commutes with itself. + ∂µ iqAµ to insure gauge invariance means that the Thus in electromagnetism the curvature of the connection A is just 7 Fock, V., Z. Phys. 39 (1926), 226. (10) Ω = dA 8O’Raifeartaigh, L., The Dawning of Gauge Theory, Prince- ton University, 1997. 11See section 2.6 of Aitchison, I. J. R., and Hey, A. J. G., 9Yang, C. N., “Hermann Weyl’s contribution to physics”, Gauge Theories in Particle Physics, Adam Hilger, 1989. Hermann Weyl (ed. Chandrasekharan, K.), Springer- 12In giving these definitions, we restrict our attention to Verlag, 1980. the smooth manifolds, which are adequate for our dis- 10Pauli, W., Rev. Mod. Phys. 13 (1941), 203. cussion.

746 NOTICES OF THE AMS VOLUME 53, NUMBER 7 which, as we will see in the next section, is the field interference term in the superposition of the so- strength F defined as F = dA. lution for the upper path and of that for the lower path produces a difference in the phase of the Differential Geometry electrons’ wave function called a phase shift. Here the phase shift is (q/c) A(x)dx. By Stoke’s theo- Differential geometry—principally developed by rem, the phase shift is (q/c)φ, where φ is the Levi-Civita, Cartan, Poincaré, de Rham, Whitney, magnetic flux in the solenoid, B · dS. Mathemat- Hodge, Chern, Steenrod and Ehresmann—led to ically, their proposal corresponds to the fact that the development of fiber bundle theory, which is even if the curvature (the electromagnetic field used in explaining the geometric content of strength) of the connection vanishes (as it does out- Maxwell’s equations. It was later used to explain side the solenoid), parallel transport along non- Yang-Mills theory and to develop string theory. homotopic paths can still be path-dependent, pro- The successes of gauge theory in physics sparked ducing a shift in the diffraction pattern. mathematicians’ interest in it. In the 1970s Sir Chambers15 performed an experiment to test the Michael Atiyah initiated the study of the mathe- Aharonov and Bohm (AB) effect. The experiment, matics of the Yang-Mills equations; and in 1983 his however, was criticized because of leakage from the student Simon Donaldson, using Yang-Mills theory, solenoid. Tonomura16 et al. performed beautiful ex- discovered a unique property of smooth mani- periments that indeed verified the AB prediction. 13 R4 folds in . Michael Freedman went on to prove Wu and Yang17 analyzed the prediction of that there exist multiple exotic differential struc- Aharonov and Bohm and comment that different 4 tures only on R . It is known that in other dimen- phase shifts (q/c)φ may describe the same in- n sions the standard differential structure on R is terference pattern, whereas the phase factor e(iq/c)φ unique. provides a unique description as we now show. The 14 In 1959 Aharonov and Bohm established the equation e2πNi = 1, where N is an integer, means primacy of the vector potential by proposing an that electron diffraction experiment to demonstrate a quantum mechanical effect: A long solenoid lies be- e(iq/c)(φ+2πNc/q) = e(iq/c)φe2πNi = e(iq/c)φ. hind a wall with two slits and is positioned between Thus flux of φ, φ + 2πc/q, φ + 4πc/q... all de- the slits and parallel to them. An electron source scribe the same interference pattern. Moreover, in front of the wall emits electrons that follow two they introduced a dictionary relating gauge theory paths: one path through the upper slit and the terminology to bundle terminology. For instance, other path through the lower slit. The first electron the gauge theory phase factor corresponds to the path flows above the solenoid, and the other path bundle parallel transport, and as we shall see, the flows below it. The solenoid is small enough so that Yang-Mills gauge potential corresponds to a con- when no current flows through it, the solenoid nection on a principal fiber bundle. does not interfere with the electrons’ flow. The Let’s see how by using the primacy of the four- two paths converge and form a diffraction pattern vector potential A, we can derive the homogeneous on a screen behind the solenoid. When the current Maxwell’s equations from differential geometry is turned on, there is no magnetic or electric field simply by using the gauge transformation. Then we outside the solenoid, so the electrons cannot be will get the nonhomogeneous Maxwell’s equations affected by these fields. However, there is a vector using the fact that our world is a four-dimensional potential A , and it affects the interference pattern (space-time) world. on the screen. Thus Einstein’s objection to Weyl’s We will also show that Maxwell’s equations are 1918 paper can be understood as saying that there invariant under the transformations Aµ → Aµ+ is no Aharonov-Bohm effect for gravity. Because of ∂µα(x), or expressed in differential geometry terms, the necessary presence of the solenoid, the upper A → A + dα(x) . We want α(x) to vanish when a path cannot be continuously deformed into the function of A is assigned to the E and B fields. lower one. Therefore, the two paths are not ho- Taking the exterior derivative of A will do this, motopically equivalent. since d2α(x) = 0 . Set A to be the 1-form A = 2 The solution of (1/2m)(−i∇−qA/c) ψ+ −A0dt + Axdx + Ay dy + Az dz. Evaluating dA and qVψ = Eψ, the time-independent Schrödinger’s realizing that the wedge product dxi ∧ dxj = −dxj ∧ dxi and therefore dxj ∧ dxj = 0, where dx0 equation for a charged particle, is (iq/c) s(x) A(y)ds (y) 1 2 3 ψ0(x)e where ψ0(x) is the solution is dt, dx is dx, dx is dy and dx is dz, produces of the equation for A equals zero and s(x) repre- a 2-form consisting of terms such as (∂xA0+ sents each of the two paths. Here c is the speed of 15 light, and is Plank’s constant divided by 2π. The Chambers, R. G., Phys. Rev. Lett. 5 (1960), 3. 16Tonomura, Akira, et. al., Phys. Rev. Lett. 48 (1982) , 1443; 13Donaldson, S. K., Bull. Amer. Math. Soc. 8 (1983), 81. and Phys. Rev. Lett. 56 (1986), 792. 14Aharonov, Y., and Bohm, D., Phys. Rev. 115 (1959), 485. 17Wu, T. T., and Yang, C. N., Phys. Rev. D 12( 1975), 3845.

AUGUST 2006 NOTICES OF THE AMS 747 ∂t Ax)dtdx and (∂xAy − ∂y Ax)dxdy. When all the is (-+++). Thus the Hodge star takes a spatial 1-form components are evaluated, these terms become dxi dt into a spatial 2-form and vice versa with a respectively ∇A0 + ∂A/∂t and ∇×A . The analysis sign change. up to now has been purely mathematical. To give The nonhomogeneous Maxwell’s equations are it physical significance we associate these terms then expressed by with the field strengths B and E. In electromagnetic (15) d F = 0 (source-free) theory, two fundamental principles are ∇·B = 0 * (no magnetic monopoles) and for time-indepen- (15 ) d*F = *J (non-source-free) dent fields E =−∇A0 (the electromagnetic field is the gradient of the scalar potential), so consistency where the 2-form *F and the 3-form *J are respec- dictates that in the time-dependent case, we assign tively the Hodge duals of F and J. *F and *J are de- the two terms to B and E respectively: fined as =∇× =−∇ − (11) B A and E A0 ∂t A. ∗F =−Bxdxdt − By dydt − Bz dzdt (16) The gradient, curl, and divergence are spatial op- + Exdydz + Ey dzdx + Ez dxdy erators—they involve the differentials dx, dy, and dz. The exterior derivative of a scalar is the gradi- ∗J = ρdxdydz − Jxdtdydz ent, the exterior derivative of a spatial 1-form is the (17) − Jy dtdzdx − Jz dtdxdy curl, and the exterior derivative of a spatial two- form is the divergence. In the 1-form A, the -A0dt Thus the Hodge star reverses the roles of E and B is a spatial scalar and when the exterior derivative from what they were in F. In *F the coefficient of is applied gives rise to ∇A0. The remaining terms the spatial 1-form is now −B, which will produce in A are the coefficients of dxi constituting a spa- the curl in the nonhomogeneous Maxwell’s equa- tial 1-form and thus produce ∇×A . tions; and the coefficient of the spatial 2-form is We define the field strength F as F = dA, and E, which will produce the divergence. In Rn the from equation (10) we see that the field strength Hodge star operation on a p-form produces an is the curvature of the connection A. Using the (n-p)-form. Thus the form of Maxwell’s equations equations in (11) and the 2-form dA, we get is dictated by the fact that we live in a four- dimensional world. When the 1-form A undergoes F = Exdxdt + Ey dydt + Ez dzdt (12) the local gauge transformation A → A + dα(x), dA + + + Bxdydz By dzdx Bdxdy remains the same, since d2α = 0. Since B and E are where, for example, dxdt is the wedge product unchanged, Maxwell’s theory is gauge invariant. dx ∧ dt. Since d2A = 0 The Dirac and Electromagnetism (13) dF = 0. Lagrangians Evaluating dF gives the homogeneous Maxwell’s To prepare for the discussion of the Yang-Mills equations. In equation (12), since the E part is a spa- equations, let’s investigate the Dirac and Electro- tial 1-form, when the exterior derivative is applied magnetism Lagrangians. The Dirac equation is it produces the ∇×E part of Maxwell’s homoge- µ − = neous equations. Since the B part of equation (12) (18) (iγ ∂µ m)ψ 0, is a spatial 2-form, it results in the ∇·B part. Since where the speed of light, c, and Plank’s constant dF = 0, F is said to be a closed 2-form. are set to one. Its Lagrangian density is To get the expression for the nonhomogeneous L= µ − Maxwell’s equations, i.e., the equivalent of equation (19) ψ¯(iγ ∂µ m)ψ. (7), we use The Euler-Lagrange equations minimize the action S where S = Ldx. Using the Euler-Lagrange equa- (14) J = ρdt + Jxdx + Jy dy + Jz dz tion where the differentiation is with respect to ψ¯ , and calculate the Hodge dual using the Hodge star i.e., operator. The Hodge duals are defined18 by ∂L ∂L = γδ = δ (20) ∂µ( ) − = 0, *Fαβ 1/2 αβγδF and *Jαβγ αβγδJ . The ∂(∂µ ψ¯) ∂ψ¯ Hodge star19 operates on the differentials in equa- yields equation (18). tions (12) and (14) using *(dxi dt) = dxj dxk and The same gauge invariant argument used in the *(dxj dxk) =−dxi dt, where i, j, and k refer to x, y, “Gauge Invariance” section applies here. In order and z and are taken in cyclic order. The metric used for the Lagrangian to be invariant under the phase + 18Misner, C. W., Thorne, K. S., and Wheeler, J. A., Gravi- transformation ψ(x) → ψ(x)e iqα(x), this transfor- tation, Freeman, San Francisco 1973. mation has to be accompanied by the local gauge 19 −1 Flanders, H., Differential Forms, Academic Press, 1963. transformation Aµ → Aµ − q ∂µα(x) and ∂µ has

748 NOTICES OF THE AMS VOLUME 53, NUMBER 7 to be replaced by ∂µ + ieAµ. The Lagrangian den- 1/137, for a good approximation we can neglect the sity becomes fact that the electromagnetic forces break this symmetry. By Noether’s theorem, if there is a rota- (21) L=ψ¯(iγµ∂ − m)ψ − eψγ¯ µψA . µ µ tional symmetry in isospin space, the total iso- The last term is the equivalent of the interaction en- topic spin is conserved. This hypothesis enables us µ ergy with the electromagnetic field, j Aµ. In order to estimate relative rates of the strong interactions for the Euler-Lagrangian equation differentiated in which the final state has a given isospin. The spin with respect to Aµ to yield the inhomogeneous matrices turn out to be the Pauli matrices σi. The − 1 2 Maxwell equation (7), we must add ( 4 )(Fµν) , get- theory just described is a global one; i.e., the iso- ting topic spin rotation is independent of the space-time coordinates, and thus no connection is used. We (22) L=ψ¯(iγµ∂ − m)ψ − eψγ¯ µψA − ( 1 )(F )2 . µ µ 4 µν will see that Yang and Mills21 elevated this global The Euler-Lagrange equation yields theory to a local one. In 1954 they proposed ap- plying the isospin matrices to electromagnetic the- (23) ∂ F µν = eψγ¯ ν ψ, µ ory in order to describe the strong interactions. Ul- ν which equals J . Note that the gauge field Aµ does timately their theory was used to describe the not carry a charge and that there is no gauge field interaction of quarks in the electroweak theory22 self-coupling, which would be indicated by an and the gluons fields of the strong force. In the next [Aµ,Aν ] term in (23). The Lagrangian density does section we will give an example using the up quark 2 not yield the homogeneous Maxwell equations. u, which has a charge of 3 e, and the down 1 They are satisfied trivially, because the definition quark d, which has a charge of -3 e. of F µν satisfies the homogeneous equations auto- We have seen that the field strength (which is matically.20 also the curvature of the connection on the fiber) From this it is apparent that the Lagrangian den- is given by F = dA + A ∧ A. In electromagnetism A sity for the electromagnetic field alone, is a 1-form with scalar coefficients for dxi, so A ∧ A vanishes. If, however, the coefficients are non- (24) L=−JµA − ( 1 )(F )2 , µ 4 µν commuting matrices, A ∧ A does not vanish and yields all of Maxwell’s equations. provides for gauge field self-coupling. Yang and In differential geometry, if j = 0, this Lagrangian Mills formulated the field strength using the letter density becomes B instead of A, so we will follow suit. The field is L=−1 ∧∗ (26) F = (∂ B − ∂ B ) + i (B B − B B ) (25) 2 F F. µν µ ν ν µ µ ν ν µ or equivalently F = (∂ B − ∂ B ) + i [B ,B ] , The Yang-Mills Theory µν µ ν ν µ µ ν where B is the connection on a principal fiber bun- The Yang-Mills theory incorporates isotopic spin dle, i.e., the gauge potential and where is the symmetry introduced in 1932 by Heisenberg, who coupling constant analagous to q in (9). Therefore, observed that the proton and neutron masses are as opposed to the electromagnetic field strength almost the same (938.272 MeV versus 939.566 MeV which is linear, their field strength is nonlinear. respectively). He hypothesized that if the electro- They proposed using a local phase, magnetic field were turned off, the masses would − j be equal and the proton and neutron would react (27) ψ(x) → ψ(x)e i αj (x)σ , identically to the strong force, the force that binds where σ j are the Pauli matrices and j goes from 1 the nucleus together and is responsible for the to 3. Thus the isotopic spin rotation is space-time formation of new particles and the rapid (typical dependent, i.e., local. The Pauli matrices do not com- lifetimes are about 10−20 seconds) decay of others. mute: In a nonphysical space (also known as an internal σ i σ j σ k [ , ] = i ijk. space) called isospin space, the proton would have 2 2 2 = i = · i isospin up, for instance, and the neutron, isospin Since Bµ bµσi or Bµ b σ (where bµ is called the down but other than that they would be identical. isotopic spin vector gauge field), the four- The wave function for each particle could be trans- vectors Bµ and Bν in (26) do not commute leading formed to that for the other by a rotation using the to nonabelian theory. The purpose of the Pauli spin matrices of the nonabelian group SU(2). Be- spin matrices in the connection B is to rotate the cause of charge independence, the strong interac- particles in isospin space so that they retain their tions are invariant under rotations in isospin space. identities at different points in R4. Equation (26) Since the ratio of the electromagnetic to strong can be rewritten so that the curvature is defined force is approximately α, where α = e2/4πc = 21Yang, C. N., and Mills, R. L., Phys. Rev. 96 (1954), 91. 20Jackson, J. D., Classical Electrodynamics, 3rd ed., John 22The weak and electromagnetic forces are the two man- Wiley and Sons, 1998. ifestations of the electroweak force.

AUGUST 2006 NOTICES OF THE AMS 749 as F = dB + 1 i [B, B]. As opposed to the Maxwell’s intermediate vector bosons, where GeV represents 2 ± equations case, the exterior derivative of the cur- a billion electron volts. The W was discovered23 vature dF does not equal zero because of the com- in 1983 (its mass is now reported at 80.425 Gev ± mutator in the expression for the curvature. Thus 0.033 GeV), and later that year the Z0 was discov- the exterior derivative for the 2-form F has to be ered24 (its mass is now reported at a mass of 91.187 altered to include the connection B in order to gen- ± 0.002 GeV). eralize the homogeneous Maxwell’s equations. The Euler-Lagrange equations for equation (28) The Yang-Mills Lagrangian give the Dirac equation µ − + = L= ¯ µ − − 1 µν (31) γ (∂µ ieBµ)ψ mψ 0, (28) ψ(iγ (Dµ m)ψ ( 4 )Tr(FµνF ) and also the vector equation for the vector field F, is invariant under the gauge transformation for namely the covariant derivative given as µ µ (32) ∂ Fµν − i [B , Fµν] =−i ψγ¯ ν ψ =−Jν , (29) Dµ = ∂µ − i Bµ. which, if it weren’t for the commutator, is the same The connection Bµ transforms as form as the nonhomogeneous four-vector Maxwell −1 equation. The commutator causes the gauge par- (30) Bµ → Bµ + ∂µα − i[α, Bµ], ticles to interact with themselves. 2 the fiber is the sphere S , and the gauge group is The effect of these equations is explained by SU(2). ’t Hooft,25 who with Veltman proved the renor- Since there are three components of the vector malizability of Yang-Mills theories: i gauge field bµ, there are three vector gauge fields representing three gauge particles having spin one. ...The B quanta would be expected to They were later identified as the intermediate vec- be exchanged between any pair of par- tor bosons W ± and Z0 which mediate the elec- ticles carrying isospin, generating not troweak interactions. The fact that there are three only a force much like the electro- gauge particles is dictated by the fact that the magnetic force, but also a force that ro- gauge field is coupled with the three Pauli spin ma- tates these particles in isospin space, trices. Also, since the charges of the up quark and which means that elementary reactions down quark differ by one, the gauge field particles envolving the transmutation of particles that are absorbed and emitted by them in quark- into their isospin partners will result. A quark interactions can have charges of ±1 or zero. novelty in the Yang-Mills theory was It is astonishing that Yang and Mills in their 1954 that the B quanta are predicted to in- paper predicted the existence of the three inter- teract directly with one another. These mediate vector bosons. interactions originate from the com- Independently of Yang and Mills and slightly mutator term in the Fµν field [equation after them, Shaw and Utiyama separately developed (32)], but one can understand physi- nonabelian gauge theories similar to the Yang-Mills cally why such interactions have to one. Utiyama’s theory also included a gauge the- occur: in contrast with ordinary pho- ory for gravity and electromagnetism. Their pa- tons, the Yang-Mills quanta also carry pers are included in O’Raifeartaigh’s The Dawning isospin, so they will undergo isospin of Gauge Theory. transitions themselves, and further- The gauge particles predicted by the Lagrangian more, some of them are charged, so the (28) have zero mass, since any mass term added neutral components of the Yang-Mills to (28) would make the Lagrangian noninvariant under a gauge transformation. So the force asso- fields cause Coloumb-like interactions ciated with the particles would have infinite range, between these charged particles. as the photons of the electromagnetic interaction So the Yang-Mills equations indicate that, for in- do. Of course the weak force (the force responsi- stance, for the up quark-down quark doublet, the ble for particles decaying slowly; typically their W − generates a force that rotates the d into the u −10 lifetimes are about 10 seconds or much less) and in isospin space exhibited by the transition strong nuclear force are short range. This dis- d → u + W −. The commutator in equation (32) is re- crepency was corrected some years later by the in- sponsible for interactions like W → W + Z troduction of spontaneous symmetry breaking in the electroweak SU(2) × U(1) theory of Weinberg, 23 Salam and Glashow (WSG) using the Higgs mecha- Arnison, G., et al., Phys. Lett. 122B (1983), 103. nism. The WSG theory, which explains the electro- 24Arnison, G., et. al., Phys. Lett. 126B (1983), 398. magnetic and weak forces, predicts the mass of the 25’t Hooft, Gerardus, ed., 50 Years of Yang-Mills Theory, W ± (80.37 ± 0.03 GeV) and Z0 (92 ± 2 GeV) World Scientific, 2005.

750 NOTICES OF THE AMS VOLUME 53, NUMBER 7 occurring,26 and the W can radiate, producing a photon in W → W + γ. The Yang-Mills field strength contribution (also called the kinetic term) to the differential geometry Lagrangian density, where k is a constant is: (33) L=−kT r(F ∧∗F).

The Euler-Lagrange equations produce dB F = 0 (it is called the Bianchi Identity, which is purely geometric), and in the absence of matter fields the field equation dB ∗ F = 0, where dB is the exterior covariant derivative. These are the Yang-Mills equations in compact form.

The Feynman Stamp In QED after Schwinger, Tomonaga and Feynman addressed the singularities produced by the self- would not conserve energy and momentum. To energy of the electron by renormalizing the theory. see this you must first remember that since the pho- They were then exceedingly successful in predict- ton has zero mass due to the gauge invariance of ing phenomena such as the Lamb shift and anom- electromagnetic theory, its energy and momentum alous magnetic moment of the electron. p are equal. Thus β, which equals E , has the value 1; Feynman introduced27 schematic diagrams, v but β = c , so that the photon’s velocity is always today called Feynman diagrams, to facilitate cal- c, the speed of light. In the electron-positron cen- culations of particle interaction parameters. Ex- ter of mass frame (more aptly called the ternal particles, represented by lines (edges) con- center of momentum frame, since the nected to only one vertex are real, i.e., observable. net momentum of all the particles is They are said to be on the mass shell, meaning their zero there), the electron and positron four-momentum squared equals their actual mass, momenta are equal and are in opposite i.e., m2 = E2 − p2. Internal particles are represented directions. The photon travels at the by lines that connect vertices and are therefore in- speed of light, and therefore its mo- termediate states—that is why they are said to me- mentum cannot be zero; but there is no diate the interaction. They are virtual and are con- particle to cancel its momentum, so the sidered to be off the mass shell. This means their interaction cannot occur. (For it to occur four-momentum squared differs from the value requires a Coloumb field from a nearby Figure 1. A pair of their actual mass. This is done so that four- nucleus to provide a virtual photon that production vertex. momentum is conserved at each vertex. The ra- transfers momentum, producing a nu- tionale for this difference is the application of the clear recoil.) Therefore the γ in the di- uncertainty principle ∆E · ∆t = . Since ∆t, the agram is internal. Its mass is off the time spent between external states is very small for mass shell and cannot equal its nor- that short time period, ∆E, and thus the difference mal value, i.e., zero. between the actual and calculated mass can be The diagram on the lower-left of the large. In the following Feynman diagrams, the time stamp (Figure 2) is also a vertex diagram axis is vertical upwards. and represents an electron-positron The diagram on the upper-left of the stamp (Fig- pair annihilation producing a γ. Again, ure 1) is a vertex diagram and as such represents if all the particles are external, conser- Figure 2. A pair a component of a Feynman diagram. It illustrates vation energy and momentum prohibit annihilation vertex. the creation of an electron-positron pair from a pho- the reaction from occurring; therefore ton, γ; it is called pair production. The γ is repre- the γ must be virtual. sented by a wavy line. The Feynman-Stuckelberg in- The diagram on the bottom to the terpretation of negative-energy solutions indicates right of Feynman (Figure 3) was meant that here the positron, the electron’s antiparticle, to represent an electron-electron scat- which is propagating forward in time, is in all ways tering with a single photon exchange. equivalent to an electron going backwards in time. This is called Møller scattering. (It can, If all the particles here were external, the process however, represent any number of interactions exchanging a photon.) The 26This is indicated in Figure 1 of the Yang-Mills paper. See diagram represents the t-channel of also F. Halzen and A. D. Martin, Quarks and Leptons, Møller scattering; there is another dia- Figure 3. Electron- John Wiley & Sons, 1984, p. 343. gram not shown here representing the electron (Møller) 27Feynman, R. P., Phys. Rev. 76 (1949), 769. u-channel contribution, where u, t and scattering.

AUGUST 2006 NOTICES OF THE AMS 751 another variable s are called the Mandelstam vari- d → W − + u shown in Figure 6, and flavor-con- ables. They are used in general to describe 2-body serving transitions, e.g., d → Z0 + d of the elec- → 2-body interactions. If you rotate the diagram troweak force—the u and d quarks have different in Figure 3 by 90o, you have the s-channel diagram values of flavor. The process in Figure 6 occurs for for electron-positron scattering, called Bhabha scat- instance in β decay, where a neutron (udd) decays tering, shown in Figure 4 but not on the stamp. Here into a proton (udu) and electron and an antineu- an electron and positron annihilate, trino. What happens is that the transition producing a virtual photon which in d → u + W − corresponds to a rotation in isospin turn produces an electron-positron pair. space. This rotation is caused by the virtual W − There is also a t-channel contribution which mediates the decay. It in turn decays into an to Bhabha scattering. The cross-section electron and an antineutrino. The calculations for for Bhabha scattering can be easily ob- these transitions all use the Yang-Mills theory. Al- tained from the one for Møller scatter- though quarks are confined in hadrons (particles ing by interchanging the s and u in the that undergo strong interactions like the proton and cross-section expression in a process neutron), they are free to interact with the inter- called crossing. Small angle Bhabha scat- mediate vector bosons. Figure 4. Electron- tering is used to test the luminosity in positron (Bhabha) + − e -e colliding beam accelerators. Who Designed the Stamp? scattering. To the right of the Møller scattering Feynman’s daughter, Michelle, was sent a provi- diagram is a vertex correction to elec- sional version of the stamp by the United States tron scattering, shown in Figure 5, Postal Service and advised on the design of it by, where the extra photon forms a loop. among others, Ralph Leighton, coauthor with It is used to calculate both the anom- Richard Feynman of two popular books, and Cal- alous magnetic moment of the electron Tech’s Steven Frautschi and Kip Thorne. Frautschi and muon, also the anomalous mag- and Leighton edited the Feynman diagrams, and netic moment contribution to the Lamb Frautschi rearranged them and composed the final shift.28 The other two contributions to design. The person who chose the original Feynman the Lamb shift are the vacuum polar- diagrams that form the basis for the stamp re- Figure 5. Radiative ization and the electron mass renor- mains a mystery. correction. malization. The Lamb shift explains the splitting in the spectrum of the 2S 1 and Acknowledgements 2 2P 1 levels of hydrogen, whereas Dirac The author thanks Jeff Cheeger, Bob Ehrlich, J. D. 2 theory alone incorrectly predicted that Jackson, and C. N. Yang for their suggestions on these two levels should be degenerate. the article. Their contributions improved the paper. The low-order solution of the Dirac The author especially thanks J. D. Jackson for gen- equation predicts a value of 2 for the erously giving of his time, and periodically pro- g-factor used in the expression for the viding advice since the author first corresponded magnetic moment of the electron. The with him many years ago. vertex correction shown in Figure 5, Figure 6. A flavor non- however, alters the g-factor, producing conserving transition an anomalous magnetic moment con- g−2 vertex. tribution, written as 2 . When this and higher-order contributions are included, g−2 the calculated value of 2 for the electron is 1159 652 460(127)(75) × 10−12 and the ex- perimental value is 1159 652 193(10) × 10−12 , where the numbers in parenthesis are the errors. This seven-significant figure agreement is a spec- tacular triumph for QED. We need not emphasize that the calculations for all these diagrams use the gauge principal for quantum electrodynamics. The other diagrams on the stamp are all vertex diagrams and show how Feynman’s work, originally applicable to QED, was then later used to elucidate the electroweak force. This is exemplified on the stamp by flavor-changing transitions, e.g.,

28See, for instance, Griffith, David, Introduction to Ele- mentary Particles, John Wiley and Sons, 1987, p. 156.

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