Relativistic Particle Orbits

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H. Kleinert, PATH INTEGRALS August 12, 2014 (/home/kleinert/kleinert/books/pathis/pthic19.tex) Agri non omnes frugiferi sunt Not all ﬁelds are fruitful Cicero (106 BC–43 BC), Tusc. Quaest., 2, 5, 13

19 Relativistic Particle Orbits

Particles moving at large velocities near the speed of light are called relativistic particles. If such particles interact with each other or with an external potential, they exhibit quantum effects which cannot be described by fluctuations of a single particle orbit. Within short time intervals, additional particles or pairs of particles and antiparticles are created or annihilated, and the total number of particle orbits is no longer invariant. Ordinary quantum mechanics which always assumes a fixed number of particles cannot describe such processes. The associated path integral has the same problem since it is a sum over a given set of particle orbits. Thus, even if relativistic kinematics is properly incorporated, a path integral cannot yield an accurate description of relativistic particles. An extension becomes necessary which includes an arbitrary number of mutually linked and branching fluctuating orbits. Fortunately, there exists a more efficient way of dealing with relativistic particles. It is provided by quantum field theory. We have demonstrated in Section 7.14 that a grand-canonical ensemble of particle orbits can be described by a functional integral of a single fluctuating field. Branch points of newly created particle lines are accounted for by anharmonic terms in the field action. The calculation of their effects proceeds by perturbation theory which is systematically performed in terms of Feynman diagrams with calculation rules very similar to those in Section 3.18. There are again lines and interaction vertices, and the main difference lies in the lines which are correlation functions of fields rather than position variables x(t). The lines and vertices represent direct pictures of the topology of the worldlines of the particles and their possible collisions and creations. Quantum field theory has been so successful that it is generally advantageous to describe the statistical mechanics of many completely different types of line-like objects in terms of fluctuating fields. One important example is the polymer field theory in Section 15.12. Another important domain where field theory has been extremely successful is in the theory of line-like defects in crystals, superfluids, and superconductors. In the latter two systems, the defects occur in the form of quantized vortex lines or quantized magnetic flux lines, respectively. The entropy of their classical shape fluctuations determines the temperature where the phase transitions take place. Instead of the usual way of describing these systems as ensembles of

1380 1381 particles with their interactions, a field theory has been developed whose Feynman diagrams are the direct pictures of the line-like defects, called disorder field theory [1]. The most important advantage of field theory is that it can describe most easily phase transitions, in which particles form a condensate. The disorder theory is therefore particularly suited to understand phase transitions in which defect-, vortex-, or flux-lines proliferate, which happens in the processes of crystal melting, superfluid to normal, or superconductor to normal transitions, respectively. In fact, the disorder theory is so far the only theory in which the critical behavior of the superconductor near the transition is properly understood [2]. A particular quantum field theory, called quantum electrodynamics describes with great success the electromagnetic interactions of electrons, muons, quarks, and photons. It has been extended successfully to include the weak interactions among these particles and, in addition, neutrinos, using only a few quantized Dirac fields and a quantized electromagnetic vector potential. The inclusion of a nonabelian gauge field, the gluon field, is a good candidate for explaining all known features of strong interactions. It is certainly unnecessary to reproduce in an orbital formulation the great amount of results obtained in the past from the existing field theory of weak, electromagnetic, and strong interactions. The orbital formulation was, in fact, proposed by Feynman back in 1950 [3], but never pursued very far due to the success of quantum field theory. Recently, however, this program was revived in a number of publications [4, 5]. The main motivation for this lies in another field of fundamental research: the string theory of fundamental particles. In this theory, all elementary particles are supposed to be excitations of a single line-like object with tension, and various difficulties in obtaining a consistent theory in the physical spacetime have led to an extension by fermionic degrees of freedom, the result being the so-called superstring. Strings moving in spacetime form worldsurfaces rather than worldlines. They do not possess a second-quantized field theoretic formulation. Elaborate rules have been developed for the functional integrals describing the splitting and merging of strings. If one cancels one degree of freedom in such a superstring, one has a theory of splitting and merging particle worldlines. As an application of the calculation rules for strings, processes which have been known from calculations within the quantum field theory have been recalculated using these reduced superstring rules. In this textbook, we shall give a small taste of such calculations by evaluating the change in the vacuum energy of electromagnetic fields caused by fluctuating relativistic spinless and spin-1/2 particles. It should be noted that since up to now, no physical result has emerged from superstring theory,1 there is at present no urgency to dwell deeper into the subject. By giving a short introduction into this subject we shall be able to pay tribute to some historic developments in quantum mechanics, where the relativistic generaliza-

1This theory really deserves a price for having the highest popularity-per-physicality ratio in the history of science, enjoying a great amount of financial support. The situation is very similar to the geocentric medieval picture of the world. 1382 19 Relativistic Particle Orbits tion of the Schrödinger equation was an important step towards the development of quantum field theory [6]. For this reason, many textbooks on quantum field theory begin with a discussion of relativistic quantum mechanics. By analogy, we shall incorporate relativistic kinematics into path integrals. It should be noted that an esthetic possibility to give a path Fermi statistics is based on the Chern-Simons theory of entanglement of Chapter 16. However, this approach is still restricted to 2 + 1 spacetime dimensions [7], and an extension to the physical 3 + 1 spacetime dimensions is not yet in sight.

19.1 Special Features of Relativistic Path Integrals

Consider a free point particle of mass M moving through the 3+1 spacetime dimensions of Minkowski space at relativistic velocity. Its path integral description is conveniently formulated in four-dimensional Euclidean spacetime where the ﬂuctuating worldlines look very similar to the ﬂuctuating polymers discussed in Chapter 15. Thus, time is taken to be imaginary, i.e.,

t = iτ = ix4/c, (19.1) − − and the length of a four-vector x =(x, x4) is given by

x2 = x2 +(x4)2 = x2 + c2τ 2. (19.2)

If xµ(λ) is an arbitrarily parametrized orbit, the well-known classical Euclidean action is proportional to the invariant length of the orbit in spacetime:

λb S = dλ x′2(λ), (19.3) λ Z a q and reads

= McS, (19.4) Acl,e or, explicitly,

λb cl,e = Mc ds(λ), (19.5) A Zλa with

ds(λ) dλ x′2(λ)= dλ x′2(λ)+ c2τ ′2(λ). (19.6) ≡ q q The prime denotes the derivative with respect to the parameter λ. The action is independent of the choice of the parametrization. If λ is replaced by a new parameter

λ λ¯ = f(λ), (19.7) →

H. Kleinert, PATH INTEGRALS 19.1 Special Features of Relativistic Path Integrals 1383 then 1 x′2 x′2, (19.8) → f ′2 dλ dλ f ′, (19.9) → so that ds and the action remain invariant. We now calculate the Euclidean amplitude for the worldline of the particle to run from the spacetime point xa =(xa, cτa) to xb =(xb, cτb). For the sake of generality, we treat the case of D Euclidean spacetime dimensions. Before starting we observe that the action (19.5) does not lend itself easily to a calculation of the path integral over e−Acl,e/¯h. There exists an alternative form for the classical action that is more suitable for this purpose. It involves an auxiliary ﬁeld h(λ) and reads:

λb Mc ′2 Mc ¯e = dλ x (λ)+ h(λ) . (19.10) A Zλa "2h(λ) 2 # This has the advantage of containing the particle orbit quadratically as in a free nonrelativistic action. The auxiliary ﬁeld h(λ) has been inserted to make sure that the classical orbits of the action (19.10) coincide with those of the original action (19.5). Indeed, extremizing ¯ in h(λ) gives the relation Ae h(λ)= x′2(λ). (19.11) q Inserting this back into ¯ renders the classical action Ae

λb ′2 cl,e = Mc dλ x (λ), (19.12) A λa Z q which is the same as (19.5). At this point the reader will worry that although the new action (19.10) describes the same classical physics as the original action (19.12), it may lead to a completely different quantum physics of a relativistic partile. With a little effort, however, it can be shown that this is not so. Since the proof is quite technical, it will be given in Appendix 19A. The new action (19.10) shares with the old action (19.12) the reparametrization invariance (19.7) for arbitrary fluctuating path configurations. We only have to assign an appropriate transformation behavior to the extra field h(λ). If λ is replaced by a new parameter λ¯ = f(λ), then x′2 and dλ transform as in (19.8) and (19.9), and the action remains invariant, if h(λ) is simultaneously changed as

h h/f ′. (19.13) → We now set up the path integral of a relativistic particle associated with the action (19.10). First we sum over the orbital fluctuations at a fixed h(λ). To find 1384 19 Relativistic Particle Orbits the correct measure of integration, we use the canonical formulation in which the Euclidean action reads

λb ′ h(λ) 2 2 2 ¯e[p, x]= dλ ipx + p + M c . (19.14) A λa "− 2Mc # Z This must be sliced in the length parameter λ. We form N + 1 slices as usual, choosing arbitrary small parameter diﬀerences ǫn = λn λn−1 depending on n, and write the sliced action as −

N+1 2 ¯N pn Mc e [p, x]= ipn(xn xn−1)+ hnǫn + ǫnhn . (19.15) A "− − 2Mc 2 # nX=1 This path integral has a universal phase space measure [recall (2.28)]

D N N+1 D p ¯ d p ¯N D −Ae[p,x]/¯h D n −Ae [p,x]/¯h x D e d xn e . (19.16) D (2πh¯)D ≈ " (2πh¯)D # Z Z nY=1 Z nY=1 Z

The momentum variables pn are integrated out to give the conﬁguration space integrals (setting λ λ , h h ) [compare (2.79)] N+1 ≡ b N+1 ≡ b 1 N dDx 1 n exp ¯N [x] , (19.17) D  D  −h¯ Ae 2πhǫ¯ bhb/Mc n=1 Z 2πhǫ¯ nhn/Mc Y   q  q  with the time-sliced action in conﬁguration space

N+1 ¯N Mc 2 Mc e [x]= (∆xn) + ǫnhn . (19.18) A 2hnǫn 2 nX=1

The Gaussian integrals over xn in (19.17) can now be done successively using For- mula (2.75), and we ﬁnd [as in (2.80)]

2 1 Mc (xb xa) Mc D exp − S , (19.19) 2πhS/Mc¯ "− 2¯h S − 2¯h # q where S is the total sliced length of the orbit

N+1 S ǫ h , (19.20) ≡ n n nX=1 whose continuum limit is

λb S = dλ h(λ). (19.21) Zλa The result (19.19) does not depend on the function h(λ) but only on S, this being a consequence of the reparametrization invariance of the path integral. While

H. Kleinert, PATH INTEGRALS 19.1 Special Features of Relativistic Path Integrals 1385 the total λ-interval changes under the transformation, the total length S of (19.21) is invariant under the joint transformations (19.7) and (19.13). This invariance permits only the invariant length S to appear in the integrated expression (19.19), and the path integral over h(λ) can be reduced to a simple integral over S. The appropriate path integral for the time evolution amplitude reads ∞ D −A¯e/¯h (xb xa)= dS h Φ[h] x e , (19.22) | N Z0 Z D Z D where is some normalization factor and Φ[h] an appropriate gauge-ﬁxing func- N tional.

19.1.1 Simplest Gauge Fixing The simplest choice for the gauge-ﬁxing functional is a δ-functional,

Φ[h]= δ[h 1], (19.23) − which ﬁxes h(λ) to be equal to the light velocity everywhere, and relates S = λ λ . (19.24) b − a This Lorentz-invariant length parameter is the so-called proper length of special relativity, equal to c times the proper time. By analogy with the discussion of thermodynamics in Chapter 2 we shall then denote λ λ by chβ¯ and write (19.24) b − a as S = λ λ c hβ.¯ (19.25) b − a ≡

If we further use translational invariance to set λa = 0, we arrive at the gauge-ﬁxed path integral

∞ 2 −βMc /2 D −A0,e/¯h (xb xa)= c h¯ dβ e x e , (19.26) | N Z0 Z D with ¯hβ M 2 0,e = dλ x˙ . (19.27) A Z0 2 Here we have gone to a timelike paramter τ = λ/c und therefore used a dot to denote the derivative:x ˙(τ) dx(λ)/dτ. Remarkably, the gauge-ﬁxed action coincides with the action of a free≡ nonrelativistic particle in D Euclidean spacetime dimensions. ¯hβ 2 Having taken the trivial term 0 dτ Mc /2¯h out of the action (19.14), the expression −βMc2/2 (19.26) contains a BoltzmannR weight e for each particle orbit of mass M. The solution of the path integral is then given by

∞ 2 2 1 M (xb xa) Mc (xb xa)= c h¯ dβ D exp − β . (19.28) 0 2 2¯h hβ¯ 2 | N Z 2πh¯ β/M "− − # q 1386 19 Relativistic Particle Orbits

By Fourier-transforming the x-dependence, this amplitude can also be written as

∞ D 2 2 −βMc2/2 d k h¯ k (xb xa)= c h¯ dβ e D exp ik(xb xa) β , (19.29) | N Z0 Z (2π) " − − 2M # and evaluated to 2Mc dDk 1 (x x ) = eik(xb−xa). (19.30) b a D 2 2 2 2 | N h¯ Z (2π) k + M c /h¯ Upon setting = λC /2, where N M λC h/Mc,¯ (19.31) M ≡ is the well-known Compton wavelength of a particle of mass M [recall Eq. (4.377)], this becomes the Green function of the Klein-Gordon ﬁeld equation in Euclidean time:

( ∂2 + M 2c2/h¯2)(x x )= δ(D)(x x ). (19.32) − b b| a b − a In the Fourier representation (19.30), the integral over k [or the integral over β in (19.28)] can be performed with the explicit result for the Green function

D/2−1 1 Mc 2 (xb xa)= KD/2−1 Mc√x /h¯ , (19.33) | (2π)D/2 h¯√x2 ! where K (z) denotes the modiﬁed Bessel function and x x x . In the nonrela- ν ≡ b − a tivistic limit c , the asymptotic behavior K (z) π/2ze−z [see Eq. (1.357)] →∞ ν → leads to q

c→∞ h¯ 2 (x x )=(x τ x τ ) e−Mc (τb−τa)/¯h (x τ x τ ) , (19.34) b| a b a| a a −−−→ 2Mc b b| a a Schr with the usual Euclidean time evolution amplitude of the free Schr¨odinger equation

2 1 M (xb xa) (xbτb xaτa)Schr = D−1 exp − . (19.35) | 2πh¯(τ τ )/M (−2¯h τb τa ) b − a − q The exponential prefactor in (19.34) contains the eﬀect of the rest energy Mc2 which is ignored in the nonrelativistic Schr¨odinger theory. Note that the same limit may be calculated conveniently in the saddle point approximation to the β-integral (19.28). For c , the exponent has a sharp → ∞ extremum at

2 2 2 2 2 (xb xa) (xb xa) +c (τb τa) τ τ (x x ) β = − = − − b − a + b − a + ...,(19.36) q ch¯ q ch¯ c→∞→ h¯ 2c2h¯(τ τ ) b − a and the β-integral can be evaluated in a quadratic approximation around this value. This yields once again (19.34).

H. Kleinert, PATH INTEGRALS 19.1 Special Features of Relativistic Path Integrals 1387

19.1.2 Partition Function of Ensemble of Closed Particle Loops

The diagonal amplitude (19.26) with xb = xa contains the sum over all lengths and shapes of a closed particle loop in spacetime. This sum can be made a partition function of a closed loop if we remove a degeneracy factor proportional to 1/L from the integral over L. Then all cyclic permutations of the points of the loop are counted only once. Apart from an arbitrary normalization factor to be ﬁxed later, the partition function of a single closed loop reads

∞ 2 dβ −βMc /2 D −A0,e/¯h Z1 = e x e . (19.37) Z0 β Z D

Inserting the right-hand integral in (19.29) for the path integral (with xb = xa), this becomes

∞ D 2 2 dβ −βMc2/2 d k h¯ k Z1 = VD e D exp β , (19.38) Z0 β Z (2π) − 2M ! where VD is the total volume of spacetime. This can be evaluated immediately. The Gaussian integral gives for each of the D dimensions a factor 1/ 2πh¯2β/M, after which formula (2.498) leads to q

∞ dβ −βMc2/2 1 VD Γ(1 D/2) Z1 = VD e D = D − D/2 , (19.39) 0 β 2 C (4π) Z 2πh¯ β/M λM q C where λM is the Compton wavelength (19.31). With the help of the sloppy formula (2.506) of analytic regularization which implies the minimal subtraction explained in Subsection 2.15.1, the right-hand side of (19.38) can also be written as dDk Z = V log k2 + M 2c2/h¯2 . (19.40) 1 − D (2π)D Z The right-hand side can be expressed in functional form as

Z = Tr log ∂2 + M 2c2/h¯2 = Tr log h¯2∂2 + M 2c2 , (19.41) 1 − − − − the two expressions being equal in the analytic regularization of Section 2.15, since a constant inside the logarithm gives no contribution by Veltman’s rule (2.508). The partition function of a grand-canonical ensemble is obtained by exponentiating this:

2 2 2 2 Z = eZ1 = e−Tr log(−¯h ∂ +M c ). (19.42)

In order to interprete this expression physically, we separate the integral dDk/(2π)D into an integral over the temporal component kD and a spatial re- Rmainder, and write

2 2 2 2 D 2 2 2 k + M c /h¯ = k + ωk/c , (19.43) 1388 19 Relativistic Particle Orbits with the frequencies

2 ωk c k2 + M 2c2/h¯ . (19.44) ≡ q Recalling the result (2.505) of the integral (2.491) we obtain

D−1 d k hω¯ k Z1 = 2VD D−1 . (19.45) − Z (2π) 2c The exponent is the sum of two ground state energies of oscillators of energyhω ¯ k/2, which are the vacuum energies associated with two relativistic particles. In quantum field theory one learns that these are particles and antiparticles. Many neutral particles are identical to their antiparticles, for example photons, gravitons, and the pion with zero charge. For these, the factor 2 is absent. Then the integral (19.38) contains a factor 1/2 accounting for the fact that paths running along the same curve in spacetime but in the opposite sense are identified. Comparing (19.42) with (3.559) and (3.622) for j = 0 we identify Z1h¯ with W [0] and the Euclidean effective action Γ of the ensemble of loops: − − Z = W [0]/h¯ =Γ /h.¯ (19.46) − 1 − e 19.1.3 Fixed-Energy Amplitude The fixed-energy amplitude is related to (19.22) by a Laplace transformation: ∞ E(τb−τa)/¯h (xb xa)E i dτb e (xb xa) , (19.47) | ≡ − Zτa | where τb, τa are once more the time components in xb, xa. As explained in Chapter 9, the poles and the cut along the energy axis in this amplitude contain all information on the bound and continuous eigenstates of the system. The fixed-energy amplitude has the reparametrization-invariant path integral representation which reads with the conventions of Eq. (19.10):

∞ h¯ D −A¯e,E /¯h (xb xa)E = dL h Φ[h] x e , (19.48) | 2Mc Z0 Z D Z D with the Euclidean action 2 λb Mc E Mc ¯ x′2 e,E = dλ (λ) h(λ) 3 + h(λ) . (19.49) A Zλa "2h(λ) − 2Mc 2 # To prove this, we write the temporal xD-part of the sliced D-dimensional action (19.18) in the canonical form (19.15). In the associated path integral (19.16), we D integrate out all xn -variables, producing N δ-functions. These remove the integrals D D over N momentum variables pn , leaving only a single integral over a common p . The Laplace transform (19.47), ﬁnally, eliminates also this integral making pD equal to iE/c. In the continuum limit, we thus obtain the action (19.49). −The path integral (19.48) forms the basis for studying relativistic potential prob- lems. Only the physically most relevant example will be treated here.

H. Kleinert, PATH INTEGRALS 19.2 Tunneling in Relativistic Physics 1389

19.2 Tunneling in Relativistic Physics

Relativistic harbors several new tunneling phenomena, of which we want to discuss two especially interesting ones.

19.2.1 Decay Rate of Vacuum in Electric Field In relativistic physics, an empty Minkowski space with a constant electric ﬁeld E is unstable. There is a ﬁnite probability that a particle-antiparticle pair can be created. For particles of mass M, this requires the energy

2 Epair =2Mc . (19.50)

This energy can be supplied by the external electric ﬁeld. If the pair of charge e C ± is separated by a distance which is roughly twice the Compton wavelength λM = C h/Mc¯ of Eq. (19.31), it gains an energy 2 E λM e. The decay will therefore become signiﬁcant when | |

M 2c3 E > E = . (19.51) | | c eh¯

Euclidean Action In Chapter 17 we have shown that in the semiclassical limit where the decay-rate −Acl,e/¯h is small, it is proportional to a Boltzmann-like factor e , where cl,e is the action of a Euclidean classical solution mediating the decay. Such a solutionA is easily found. We use the classical action in the form (19.12) and choose the parameter λ to measure the imaginary time λ = τ = it = x4/c. Then the action takes the form

τb 2 2 2 cl,e = dτ Mc 1+ x˙ (τ)/c e E x(τ) . (19.52) A τa − · Z q This is extremized by the classical equation of motion d x˙ (τ) M = eE, (19.53) dτ 1+ x˙ 2(τ)/c2 − q whose solutions are circles in spacetime of a ﬁxed E-dependent radius lE:

2 2 2 2 2 2 Mc (x x0) + c (τ τ0) = lE , E E . (19.54) − − ≡ eE ! ≡| |

To calculate their action we parametrize the circles in the Eˆ τ -plane, where Eˆ is the unit vector in the direction of E, by an angle θ as −

l x(θ)= l Eˆ cos θ + x , τ(θ)= E sin θ + τ . (19.55) E 0 c 0 1390 19 Relativistic Particle Orbits

A closed circle has an action 2π 2 lE 1 Ec cl,e = Mc dθ cos θ cos θ = MclEπ =¯h π. (19.56) A c Z0 cos θ − E The decay rate of the vacuum is therefore proportional to

Γ e−πEc/E. (19.57) ∝ The circles (19.54) are, of course, the space-time pictures of the creation and the annihilation of particle-antiparticle pairs at times τ l /c and τ + l /c and 0 − E 0 E positions x0, respectively (see Fig. 19.1). A particle can also run around the circle

(x , cτ + l ) 0 0 r E

❄ ✻

(x0cτ0)

r (x , cτ l ) 0 0 − E Figure 19.1 Spacetime picture of pair creation at the point x and time τ l /c and 0 0 − E annihilation at the later time τ0 + lE/c. repeatedly. This leads to the formula

∞ Γ F e−nπEc/E, (19.58) ∝ n nX=1 with ﬂuctuation factors Fn which we are now going to determine.

Fluctuations As explained above, ﬂuctuations must be calculated with the relativistic Euclidean action (19.10), in which we have to include the electric ﬁeld by minimal coupling:

λb M ′2 Mc e ′ ¯e = dλ x (λ)+ h(λ) i A(x(λ)) x (λ) . (19.59) A Zλa "2h(λ) 2 − c # The vacuum will decay by the creation of an ensemble of pairs which in Euclidean spacetime corresponds to an ensemble of particle loops. For free particles, the partition function Z was given in (19.42) as an exponential of the one-loop partition function Z1 in (19.37). The corresponding Z1 in the presence of an electromagnetic ﬁeld has the one-loop partition function

∞ 2 dβ −βMc /2 4 −A¯e/¯h Z1 = e x e , (19.60) Z0 β Z D

H. Kleinert, PATH INTEGRALS 19.2 Tunneling in Relativistic Physics 1391 with the Euclidean action

¯hβ M ′2 e ′ ¯e = dτ x (τ) i A(x(τ)) x (τ) . (19.61) A Z0 2 − c The equations of motion are now e e x′′ = (x′ E i x′ B) , x′′ = x′ E, (19.62) Mc 4 − × 4 −Mc · If both a constant electric and a constant magnetic ﬁeld are present, the vector potential is A = F xν /2 and the action (19.61) takes the simple quadratic form µ − µν ¯hβ M ′2 e µ ′ν ¯e = dτ x i Fµν x x , (19.63) A Z0 2 − 2c and the equations of motion (19.62) are e x′′µ = i F µν x′ , with F = ǫ Bk, F i4 = iF i0 = iEi. (19.64) − c ν ij − ijk In a purely electric ﬁeld, the solutions are circular orbits:

x(τ)= Eˆ A cos ωE(τ τ )+ c , x (τ)= A sin ωE(τ τ )+ c , (19.65) L − 0 2 4 L − 0 4 with the electric version of the Landau or cyclotron frequency (2.648) eE ωE . (19.66) L ≡ Mc The circular orbits are the same as those in the previous formulation (19.55). If E points in the z-direction, the action (19.63) decomposes into two decoupled quadratic ¯(12) ¯(34) 1 2 3 4 actions e + e for the motion in the x x and x x -planes, respectively, and theA one-loopA partition function (19.60) factorizes− as follows:−

∞ dβ 2 (12) (34) −βMc /2 2 (12) −A¯e /¯h 2 (34) −A¯e /¯h Z1 = e x e x e Z0 β Z D Z D ∞ dβ 2 e−βMc /2Z(12)(0)Z(34)(E). (19.67) ≡ Z0 β The path integral for Z(12)(0) collecting the ﬂuctuations in the x1 x2 -plane have − the trivial action

¯hβ ¯(12) M ′ 2 ′ 2 e = dτ (x1 + x2 ), (19.68) A Z0 2 with the trivial ﬂuctuation determinant Det ( ∂2) = 1, so that we obtain for Z(12)(0) − τ the free-particle partition function in two dimensions

2 (12) M Z (0) = ∆x1∆x2 . (19.69) s2πh¯2β 1392 19 Relativistic Particle Orbits

1 2 The factor ∆x1∆x2 is the total area of the system in the x x -plane. Note that upper and lower indices are the same in the present Euclidean− metric. For the motion in the x3 x4 -plane, the quadratic ﬂuctuations with periodic boundary conditions have a functional− determinant

2 E E ∂τ ωL ∂τ 2 2 E2 sinhω ¯ L β/2 Det −E − 2 = Det ∂τ Det ∂τ ωL =1 E , (19.70) ω ∂τ ∂ ! − × − − × hω¯ β/2 L − τ L leading to the partition function

2 E (34) M sinhω ¯ L β/2 Z (E) = ∆x3∆x4 2 E ,. (19.71) s2πh¯ β hω¯ L β/2 This result can, of course, also be obtained without calculation from the observation that the Euclidean electric path integral is completely analogous to the real-time magnetic path integral solved in Section 2.18. Indeed, with the E-ﬁeld pointing in the z-direction, the action (19.63) becomes for the motion in the x3 x4 -plane − ¯hβ ¯(34) M ′ 2 ′ 2 e ′ ′ e = dτ x3 + x4 + E(x3x4 x4x3) . (19.72) A Z0 2 c − This coincides with the magnetic action (2.635) in real time, if we insert there the magnetic vector potential (2.636) and replace B by E. The equations of motion (19.62) reduces to

x′′ = ωEx′ , x′′ = ωE x′ , (19.73) 3 L 4 4 − L 3 in agreement with the magnetic equations of motion (2.672) in real time. Thus the motion in the x3 x4 -plane as a function of the pseudotime τ is the same as the real-time motion− in the x y -plane, if the magnetic field B in the z-direction is − exchanged by an E-field of the same size pointing in the x3-axis. The amplitude can therefore be taken directly from (2.668), — we must merely replace the real time difference t t byhβ ¯ . Anyhow, it is zero for any closed orbit. b − a Yet another way of obtaining the result (19.71) is by summing over the imaginary- iβEm 1 E time Boltzmann factors e of the energies m =(m + 2 )¯hωL of the Hamiltonian associated with the Euclidean action (19.72):E

2 ∞ M 1 E (34) E −i(m+ 2 )¯hωL Z (E) = ∆x3∆x4 hω¯ L β e . (19.74) s2πh¯2β mX=0 Inserting (19.69) and (19.71) into (19.67), we obtain the partition function of a single closed particle orbit in four Euclidean spacetime dimensions:

4 ∞ E dβ M ωL hβ/¯ 2 −βMc2/2 Z1 = ∆x4 V 2 E e , (19.75) Z0 β s2πh¯ β sin ωL hβ/¯ 2

H. Kleinert, PATH INTEGRALS 19.2 Tunneling in Relativistic Physics 1393 where V ∆x ∆x ∆x is the total spatial volume. We now go over to real times ≡ 1 2 3 by setting ∆x4 = ic∆t. By exponentiating the subtracted expression (19.75) as in Eq. (19.42), we go to a grand-canonical ensemble, and may identify Z1 with i times the effective electromagnetic action caused by fluctuating ensemble of particle loops Z = i∆ eff /h¯ = i∆tV ∆ eff /h.¯ (19.76) 1 A L The integral over β in (19.75) is divergent. In order to make it convergent, we perform two subtractions. In the subtracted expression we change the integration variable to the dimensionless ζ = βMc2/2, and obtain the effective Lagrangian density

4 ∞ 2 eff Mc 1 dζ Eζ/Ec 1 E −ζ ∆ =¯hc 2 3 1 e . (19.77) L h¯ 4(2π) Z0 ζ "sin Eζ/Ec − − 6 Ec # The first subtraction has removed the divergence coming from the 1/ζ3-singularity in the integrand. This produces a real infinite field-independent contribution to the effective action which can be omitted since it is unobservable by electromagnetic experiments.2 After subtracting this divergence, the integral still contains a loga- rithmic divergence. It can be interpreted as a contribution proportional to E2 to the Lagrangian density

4 2 ∞ ∞ eﬀ Mc 1 E dζ −ζ α dζ −ζ ∆ div =¯hc 2 e = e , (19.78) L h¯ 24(2π) Ec Z0 ζ 24π Z0 ζ 2 2 which changes the original Maxwell term E /2 to ZAE /2 with

∞ α dζ −ζ ZA =1+ e . (19.79) 12π Z0 ζ According to the rules of renormalization theory, the prefactor is removed by renor- 1/2 malizing the field strength, replacing E E/ZA , and identifying the replaced field →1/2 with the physical, renormalized field E/ZA ER. Due to the presence of the affective action,≡ the vacuum is no longer time independent, put depends on time like e−i(H−iΓ¯h/2)∆t/¯h. Thus the decay rate of the vacuum per unit volume is given by the imaginary part of the effective Lagrangian density Γ 2 = Im ∆ eff . (19.80) V h¯ L

In order to calculate this we replace Eζ/Ec by z in (19.77), and expand in the integrand

∞ 2 ∞ 2 z n z n ζ Ec =1+2 ( 1) 2 2 2 =2 ( 1) 2 2 , ζn nπ . (19.81) sin z − z n π − ζ ζn ≡ E nX=1 − nX=1 − 2This energy would, however, be observable in the cosmological evolution to be discussed in Subsection 19.2.3. 1394 19 Relativistic Particle Orbits

Adding to the poles the usual inﬁnitesimal iη-shifts in the complex plane (see p. 115), we replace with the help of the decomposition (1.329) ζ ζ π π = i δ(ζ + ζ ) i δ(ζ ζ )+ ζ P . (19.82) ζ2 ζ2 → ζ2 ζ2 + iη 2 n − 2 − n ζ2 ζ2 − n − n − n The δ-functions yield imaginary parts and thus directly the decay rate

4 2 ∞ n−1 Γ 2 eﬀ Mc E 1 1 n−1 ( 1) −nπEc/E = Im = c 3 ( 1) − 2 e V h¯ L h¯ Ec 4π 2 − n nX=1 1 e2 ∞ ( 1)n−1 = E2 − e−nπEc/E. (19.83) 8π3 hc¯ n2 nX=1 The principal values produce a real eﬀective Lagrangian density

4 ∞ 2 2 eﬀ Mc 1 dζ Eζ/Ec E ζ −ζ ∆ P =hc ¯ 2 3 1 2 e . (19.84) L h¯ 4(2π) P Z0 ζ sin Eζ/Ec − − 6Ec ! If we expand

z z2 7 31 =1+ + z4 + z6 + ... (19.85) sin z 6 360 15120 and perform the integrals over ζ, we ﬁnd

4 4 6 eﬀ Mc 1 7 E 31 E ∆ =¯hc 2 + + ... , (19.86) L h¯ 4(2π) " 360 Ec 2520 Ec # or

1 7α2 (¯hc)3 31πα3 (¯hc)3 2 eff = E2 + E4 + E6 + ... , (19.87) L 2  180 (Mc2)4 315 "(Mc2)4 #    The subscript of has been omitted since the imaginary part of the effective P L Lagrangian density (19.83) has a vanishing Taylor expansion. Each coefficient in (19.87) is exact to leading order in α. The extra expansion terms in (19.87) imply that the physical vacuum has a nontrivial E-dependent dielectric constant ǫ(E). This is caused by the virtual creation and annihilation of particle-antiparticle pairs. Since the dielectric displacement D(E) is obtained from the first derivative of eff , the dielectric constant is given by ǫ(E)= D(E)/E = ∂ eff /E∂E. From (19.87)L we find the lowest expansion terms L 7α2 (¯hc)3 31πα3 (¯hc)3 2 ǫ(E)=1+ E2 + E4 + .... (19.88) 90 (Mc2)4 105 " (Mc2)4 #

Such a term gives rise to a small amplitude for photon-photon scattering in the vacuum, a process which has been observed in the laboratory.

H. Kleinert, PATH INTEGRALS 19.2 Tunneling in Relativistic Physics 1395

Another way of evaluating (19.83) is based on the representation (19.74) of the partition function in terms of the energy eigenvalues = (m + 1 )¯hωE of the Em 2 L Hamiltonian associated with (19.72). Then Z1 is given by

4 ∞ ∞ M 1 E E −i(n+ 2 )¯hωL β Z1 = ∆x4 V dβ 2 ωL h¯ e , (19.89) 0 s2πh¯ β Z nX=0 Then the integral representation (19.77) becomes (before the subtraction)

4 ∞ Mc 1 ∞ dζ E 1 eﬀ i(m+ 2 )2Eζ/Ec −ζ ∆ =¯hc 2 2 e e . (19.90) L h¯ 2(2π) 0 ζ Ec " # Z mX=0 This can be rewritten as

4 ∞ k/2−1 k eﬀ Mc i ( 1) 1−k k−1 E ∆ reg =¯hc 2 − (2 1)ζ(1 k)2 , L h¯ 2(2π) " (k 1)(k 2) − − Ec # k=4X,6,... − − α=1 (19.91)

Expanding the terms in the sum in powers of E/Ec yields two divergent terms plus a regular series

4 ∞ eﬀ Mc i ∆ reg =hc ¯ dα dα L h¯ 2(2π)2 " Z Z k=4X,6,... ∞ k/2−1 k ( 1) k/2−1 1 k−1 k−1 E − k ( 1) [(m + 2 )] 2 . (19.92) × α − Ec # mX=0 α=1 Performing the sum over m using Riemann’s zetafuntions (2.521), this becomes

4 ∞ k/2−1 k eﬀ Mc i ( 1) 1−k k−1 E ∆ reg =¯hc 2 − (2 1)ζ(1 k)2 (19.93) L h¯ 2(2π) " (k 1)(k 2) − − Ec # k=4X,6,... − − α=1 and coincides with the previous result (19.87).

Including Constant Magnetic Field parallel to Electric Field Let us see how the decay rate (19.83) and the effective Lagrangian (19.77) are influenced by the presence of an additional constant B-field. This will at first be assumed to be parallel to the E-field, with both fields pointing in the z-direction. Then the action (19.68) for the motion in the x1 x2 -plane becomes − ¯hβ ¯(12) M ′ 2 ′ 2 e ′ ′ e = dτ x1 + x2 + iB(x1x2 x2x1) . (19.94) A Z0 2 c − The partition function in the x1 x2 -plane will therefore have the same form as E − B (19.71), except with ωL replaced by iωL : Z(12)(B)= Z(34)(iB). (19.95) 1396 19 Relativistic Particle Orbits

Thus the B-ﬁeld changes the partition function (19.89) of a single closed orbit to

4 ∞ E B dβ M ωL hβ/¯ 2 ωL hβ/¯ 2 −βMc2/2 Z1 = ∆x4 V 2 E B e , (19.96) Z0 β s2πh¯ β sin ωL hβ/¯ 2 sinh ωL hβ/¯ 2 and the eﬀective Lagrangian (19.77) becomes

4 ∞ 2 2 2 eﬀ Mc 1 dζ Eζ/Ec Bζ/Ec (E B )ζ −ζ ∆ =¯hc 2 3 1 − 3 e . L h¯ 4(2π) Z0 ζ " sin Eζ/Ec sinh Bζ/Ec − − 6Ec # (19.97)

In the subtracted free-field term (19.78), the term E2 is changed to the Lorentz- invariant combination E2 B2. The decay rate (19.83) is modified to − 4 2 ∞ Γ 2 eff Mc E 1 1 n−1 1 nπB/E −nπEc/E = Im ∆ = c 3 ( 1) 2 e . (19.98) V h¯ L h¯ Ec 4π 2 − n sinh nπB/E nX=1 In Eqs. (7.521)–(7.525) we have shown that all results derived for parallel constant electric and magnetic fields remain valid if we replace E , B , where and are given in Eq. (7.524). After this we may expand the→E integrand→ in B (19.97) E B in powers of ε and β using Eq. (19.85), 1 eβτ eετ 1 e2 τ = ε2 β2 + e4 7ε4 10ε2β2 +7β4 τ 3 sin eβτ sinh eετ τ 3 − 6τ − 360 − τ 3 e6 31ε6 49ε4β2 + 49ε2β4 31β6 + .... (19.99) − 1520 − − and obtain the effective Lagrangian the fields generalizing (19.87):

1 7α2 (¯hc)3 eﬀ = (E2 B2)+ (E2 B2)2 L 2 ( − 180 (Mc2)4 − 31πα3 (¯hc)3 2 + (E2 B2)[2(E2 B2)2 4(EB)2]+ ... ,(19.100) 315 "(Mc2)4 # − − −   For strong ﬁelds, a saddle point approximation to the generalized version of the integral (19.97) yields the asymptotic form

e2 (¯hc)3 eﬀ (E2 B2) log 4e2 (E2 B2) + .... (19.101) L ≡ −192π2 − "− (Mc2)4 − #

Spin-1/2 Particles We anticipate the small modiﬁcation which is necessary to obtain the analogous result for spin-1/2 fermions: The tools for this will be developed in Subsections 19.5.1– 19.5.3. The relevant formula has actually been derived before in Eq. (7.520). Ex- tending that equation to a constant ﬁeld tensor that contains both magnetic and

H. Kleinert, PATH INTEGRALS 19.2 Tunneling in Relativistic Physics 1397 electric ﬁelds, and going to imaginary time τ = λ/c and antiperiodic boundary conditions in the interval t t = ihβ¯ , we ﬁnd the functional determinant b − a

1/2 e 1/2 e hβ¯ 4Det gµν ∂λ + i 2 Fµν = 4 det cosh Fµν , (19.102) − Mc Mc 2 ! with an ordinary determinant of the 4 4-dimensional matrix of the cosin on the right-hand side. According to Eqs. (7.520)× and (7.525), the result is

1/2 e 4Det gµν ∂λ + i 2 Fµν = 4cosh(µB β/2) cos(µB β/2). (19.103) − Mc B E where µB = eh/Mc¯ is the Bohr magneton (2.649), and and are defined in Eq. (7.525) B E For a purely electric field, the right-hand side becomes 4 cos(µBEβ/2) = E 4 cos(ωL hβ/¯ 2) [recall (19.66)]. Multiplying the cos-factor into the expansion (19.81), this is modified to

∞ 2 ∞ 2 cos z z ζ Ec z =1+2 2 2 2 =2 2 2 , ζn nπ . (19.104) sin z z n π ζ ζn ≡ E nX=1 − nX=1 − Performing now the singular integral over ζ in (19.77), we obtain for the decay rate the same formula as in (19.83), except that the alternating signs are absent. The factor 4 in (19.103) reduces to a factor 2 in the eﬀective action. The resulting eﬀective Lagrangian density for spin-1/2 fermions is therefore

4 ∞ 2 2 eﬀ Mc 1 dζ Eζ/Ec E ζ −ζ ∆ 1 = hc¯ 1+ e , (19.105) spin 2 2 3 3 L − h¯ 2(2π) Z0 ζ tan Eζ/Ec − 3Ec ! a result ﬁrst derived by Heisenberg and Euler in 1935 [8]. Its imaginary part yields the decay rate of the vacuum due to pair creation

Γ 1 4 2 ∞ spin 2 2 eﬀ Mc E 1 1 −nπEc/E = Im ∆ spin 1 = c 3 2 e V h¯ L 2 h¯ Ec 4π n nX=1 1 e2 ∞ 1 = E2 e−nπEc/E. (19.106) 4π3 hc¯ n2 nX=1 The reason for the reduction from 4 to 2 is that the sum over bosonic paths has to be ﬁrst divided by a factor 2 to remove their orientation, before the fermionic factor 4 is applied. This procedure is not so obvious at this point but will be understood later in Subsection 19.5.2. The remaining factor 2 accounts for the two spin orientations of the charged particles. The Taylor series

z z2 1 2 =1 z4 z6 ... (19.107) tan z − 3 − 45 − 945 − 1398 19 Relativistic Particle Orbits in the integrand of (19.105) leads to the expansion

4 4 6 eﬀ Mc 2 1 E 4 E ∆ =¯hc 2 + + ... , (19.108) L h¯ 16π "45 Ec 315 Ec # so that

1 4α2 (¯hc)3 64πα3 (¯hc)3 2 eﬀ = E2 + E4 + E6 + ... . (19.109) L 2  45 (Mc2)4 315 "(Mc2)4 #    The term proportional to α2-term implies a small amplitude for photon-photon scattering which can be observed in the laboratory [9]. As in the boson result (19.100), each coeﬃcient is exact to leading order in α, and the virtual creation and annihilation of fermion-antifermion pairs gives the physical vacuum a nontrivial E-dependent dielectric constant

1 ∂ eﬀ 8α2 (¯hc)3 64πα3 (¯hc)3 2 ǫ(E)= L =1+ E2 + E4 + .... (19.110) E ∂E 45 (Mc2)4 105 "(Mc2)4 #

If we admit also a constant B-ﬁeld parallel to E, the formulas (19.106) and 1 (19.105) for the spin- 2 -particles are modiﬁed in the same way as the bosonic formulas (19.83) and (19.84), except that the determinant (19.102) in spinor space introduces a further factor cosh(eB/Mc). Thus we obtain

4 ∞ 2 2 2 eﬀ Mc 1 dζ Eζ/Ec Bζ/Ec (E B )ζ −ζ ∆ 1 = hc¯ 1+ − e . spin 2 2 3 3 L − h¯ 2(2π) Z0 ζ "tan Eζ/Ec tanh Bζ/Ec − 3Ec # (19.111)

For a general combination of constant electric and magnetic ﬁelds, we simply ex- change E and B by the Lorentz invariants ε and β. From the imaginary part we obtain the decay rate

Γ 1 4 2 ∞ spin 2 2 eﬀ Mc ε 1 1 nπβ/ε −nπEc/ε = Im ∆ spin 1 = c 3 2 e .(19.112) V h¯ L 2 h¯ Ec 4π n tanh nπβ/ε nX=1 For strong ﬁelds, a saddle point approximation to the generalized integral (19.105) yields the asymptotic form

e2 (¯hc)3 eﬀ (E2 B2) log 4e2 (E2 B2) + .... (19.113) L ≡ −48π2 − "− (Mc2)4 − #

19.2.2 Birth of Universe A similar tunneling phenomenon could explain the birth of the expanding universe [10].

H. Kleinert, PATH INTEGRALS 19.2 Tunneling in Relativistic Physics 1399

As an idealization of the observed density of matter, the universe is usually assumed to be isotropic and homogeneous. Then it is convenient to describe it by a coordinate frame in which the metric is rotationally invariant. To account for the expansion, we have to allow for an explicit time dependence of the spatial part of the metric. In the spatial part, we choose coordinates which participate in the expansion. They can be imagined as being attached to the gas particles in a homogenized universe. Then the time passing at each coordinate point is the proper time. In this context it is the so-called cosmic standard time to be denoted by t. We imagine being an observer at a coordinate point with dxi/dt = 0, and measure t by counting the number of orbits of an electron around a proton in a hydrogen atom, starting from the moment of the big bang (forgetting the fact that in the early time of the universe, the atom does not yet exist).

Geometry

With this time calibration, the component g00 of the metric tensor is identically equal to unity

g (x) 1, (19.114) 00 ≡ such that at a ﬁxed coordinate point, the proper time coincides with the coordinate time, dτ = dt. Moreover, since all clocks in space follow the same prescription, there is no mixing between time and space coordinates, a property called time orthogo- nality, so that

g (x) 0. (19.115) 0i ≡ µ As a consequence, the Christoﬀel symbol Γ¯00 [recall (10.7)] vanishes identically: 1 Γ¯ µ gµν(∂ g + ∂ g g ) 0. (19.116) 00 ≡ 2 0 0ν 0 0ν − 00 ≡ This is the mathematical way of expressing the fact that a particle sitting at a coordinate point, which has dxi/dt = 0, and thus dxµ/dt = uµ =(c, 0, 0, 0), experiences no acceleration duµ = Γ¯ µc2 =0. (19.117) dτ − 00 The coordinates themselves are trivially comoving. Under the above condition, the invariant distance has the general form

ds2 = c2dt2 (3)g (x)dxidxj. (19.118) − ij

We now impose the spatial isotropy upon the spatial metric gij. Denote the spatial length element by dl, so that

2 (3) i j dl = gij(x)dx dx . (19.119) 1400 19 Relativistic Particle Orbits

The isotropy and homogeneity of space is most easily expressed by considering the (3) l (3) spatial curvature Rijk calculated from the spatial metric gij(x). The space corresponds to a spherical surface. If its radius is a, the curvature tensor is, according to Eq. (10.161), 1 (3)R (x)= (3)g (x)(3)g (x) (3)g (x)(3)g (x) . (19.120) ijkl a2 il jk − ik jl h i The derivation of this expression in Section 10.4 was based on the assumption of a spherical space whose curvature K 1/a2 is positive. If we allow also for hyperbolic ≡ and parabolic spaces with negative and vanishing curvature, and characterize these by a constant

1 spherical k =  0 parabolic  universe, (19.121)  1 hyperbolic  −   then the prefactor 1/a2 in (19.120) is replaced by K k/a2. For k = 1 and 0, the ≡ − space has an open topology and an inﬁnite total volume. The Ricci tensor and curvature scalar are in these three cases [compare (10.163) and (10.156)] 2 6 (3)R = k g (x), (3)R = k . (19.122) il a2 il a2 By construction it is obvious that for k = 1, the three-dimensional space has a closed topology and a ﬁnite spatial volume which is equal to the surface of a sphere of radius a in four dimensions

4Sa =2π2a3. (19.123)

A circle in this space has maximal radius a and a maximal circumference 2πa. A sphere with radius r0 < a has a volume

2π π r 2 (3) a r Vr0 = dϕ dθ sin θ dr (19.124) Z0 Z0 Z0 1 r2/a2 − q a3 r a2r r 2 = 4π arcsin 0 0 1 0 .  2 a − 2 s − a2    For small r0, the curvature is irrelevant and the volume depends on r0 like the usual volume of a sphere in three dimensions: 4π (3)V a V = r 2. (19.125) r0 ≈ r0 3 0 For r a, however, (3)V a approaches a saturation volume 2πa3. 0 → r0 The analogous expressions for negative and zero curvature are obvious.

H. Kleinert, PATH INTEGRALS 19.2 Tunneling in Relativistic Physics 1401

Robertson-Walker Metric In spherical coordinates, the four-dimensional invariant distance (19.118) deﬁnes the Robertson-Walker metric.

ds2 = c2dt2 dl2 (19.126) − dr2 dl2 = + r2(dθ2 + sin2 θdϕ2). (19.127) 1 kr2/a2 − It will be convenient to introduce, instead of r, the angle α on the surface of the four-sphere, such that

r = a sin α. (19.128)

Then the metric has the four-dimensional angular form

ds2 = c2dt2 a2(t)[dα2 + f 2(α)(dθ2 + sin2 θdϕ2)], (19.129) − where for spherical, parabolic, and hyperbolic spaces, f(α) is equal to

sin α k =1, f(α)=  α k =0, (19.130)  sinh α k = 1. −  In order to have maximal symmetry, it is useful to absorb a(t) into the time and deﬁne a new timelike variable η by

c dt = a(η) dη, (19.131) so that the invariant distance is measured by

ds2 = a2(η)[dη2 dα2 f 2(α)(dθ2 + sin2 θdϕ2)]. (19.132) − − Then the metric is simply

1 2 1 gµν = a (η)  −  (19.133) f 2(α)    − 2 2   f (α) sin θ,   −  and the Christoﬀel symbols become a a a Γ 0 = η , Γ i =0, Γ 0 =0, Γ j = η δ j, Γ 0 = η g , Γ k =0, (19.134) 00 a 00 0i 0i a i ij −a3 ij ij where the subscripts denote derivatives with respect to the corresponding variables: da a da a a = a . (19.135) η ≡ dη c dt ≡ c t 1402 19 Relativistic Particle Orbits

We now calculate the 00-component of the Ricci tensor:

R = ∂ Γ µ ∂ Γ µ Γ νΓ µ +Γ µΓ ν. (19.136) 00 µ 00 − 0 µ0 − µ0 0ν 00 νµ Inserting the Christoﬀel symbols (19.134) we ﬁnd d a 1 ∂ Γ µ ∂ Γ µ = ∂ Γ i = 3 η = 3 a a a2 , (19.137) µ 00 − 0 µ0 − 0 i0 − dη a − a2 ηη − η 2 2 ν µ 0 0 i 0 0 i j i aη aη Γµ0 Γ0ν =Γ00 Γ00 +Γ00 Γ0i +Γi0 Γ00 +Γi0 Γ0j = +3 , (19.138) a a 2 2 µ ν 0 0 0 i i 0 i k aη aη Γ00 Γνµ =Γ00 Γ00 +Γ00 Γi0 +Γ00 Γ0i +Γ00 Γki = +3 , (19.139) a a so that 3 3 R = (aa a2), R 0 = g00R = (aa a2). (19.140) 00 −a2 ηη − η 0 00 −a4 ηη − η The other components can be determined by relating them to the three-dimensional (3) curvature tensor Rij which has the simple form (19.120). So we calculate

µ k 0 Rij = Rµij = Rkij + R0ij (19.141) = (3)R Γ 0Γ k +Γ 0Γ k + R 0. ij − kj i0 ij k0 0ij Inserting

R 0 = ∂ Γ 0 ∂ Γ 0 Γ lΓ 0 Γ 0Γ 0 +Γ lΓ 0 +Γ 0Γ 0, (19.142) 0ij 0 ij − i 0j − 0j il − 0j i0 ij 0l ij 00

2 (3)R = k g (19.143) ij a2 ij and the above Christoﬀel symbols (19.134) gives 1 R = (2ka2 + a2 + aa )g (19.144) ij −a4 η ηη ij and thus a curvature scalar

00 ij 1 3 2 3 2 2 R = g R00 + g Rij = 2 2 (aaηη aη) 4 (2ka + aη + aaηη) −a a − − a 6 = (a + ka). (19.145) −a3 ηη

Action and Field Equation In the absence of matter, the Einstein-Hilbert action of the gravitational ﬁeld is

f f 1 = d4x√ g = d4x√ g(R +2λ), (19.146) A Z − L −2κ Z −

H. Kleinert, PATH INTEGRALS 19.2 Tunneling in Relativistic Physics 1403 were κ is related to Newton’s gravitational constant

G 6.673 10−8 cm3g−1s−2 (19.147) N ≈ · by 1 c3 = . (19.148) κ 8πGN A natural length scale of gravitational physics is the Planck length, which can be formed from a combination of Newton’s gravitational constant (19.147), the light velocity c 3 1010 cm/s, and Planck’s constanth ¯ 1.05459 10−27: ≈ × ≈ × 3 −1/2 c −33 lP = 1.615 10 cm. (19.149) GN h¯ ! ≈ × It is the Compton wavelength l h/m¯ c associated with the Planck mass P ≡ P 1/2 ch¯ −5 22 2 mP = 2.177 10 g=1.22 10 MeV/c . (19.150) GN ! ≈ × × The constant 1/κ in the action (19.146) can be expressed in terms of the Planck length as 1 h¯ = 2 . (19.151) κ 8πlP If we add to (19.146) a matter action and vary to combined action with respect to the metric gµν, we obtain the Einstein equation 1 1 Rµν gµν R λgµν = Tµν , (19.152) κ − 2 − where Tµν is the energy-momentum tensor of matter. The constant λ is called the cosmological constant. It is believed to arise from the zero-point oscillations of all quantum fields in the universe. f A single field contributes to the Lagrangian density in (19.146) a term Λ λ/κ which is typically of the order ofh/l ¯ 4 . For bosons,L the sign is positive,− for≡ − P fermions negative, reflecting the filling of all negative-energy in the vacuum. A constant of this size is much larger than the present experimental estimate. In the literature one usually finds estimates for the dimensionless quantity λc2 Ωλ0 2 . (19.153) ≡ 3H0 where H0 is the Hubble constant, whose inverse is roughly the lifetime of the universe

H−1 14 109 years. (19.154) 0 ≈ × 1404 19 Relativistic Particle Orbits

Present ﬁts to distant supernovae and other cosmological data yield the estimate [11]

Ω 0.68 0.10. (19.155) λ0 ≈ ± The associated cosmological constant λ has the value

3H2 Ω Ω Ω λ = Ω 0 λ0 λ0 λ0 . (19.156) λ0 c2 ≈ (6.55 1027 cm)2 ≈ (6.93 109 ly)2 ≈ (2.14 R )2 × × universe Note that in the presence of λ, the Schwarzschild solution around a mass M has the metric

ds2 = B(r)c2dt2 B−1dr2 r2dθ2 r2 sin2 θdφ2, (19.157) − − − with

2 2MGN 2 2 M lP 2 r B(r)=1 2 λr =1 Ωλ0 2 . (19.158) − c r − 3 − mP r − 3 (2.14 Runiverse) The λ-term adds a small repulsion to Newton’s force between mass points. if the distances are the order of the radius of the universe. The value of the constant Λ associated with (19.155) is

2 2 λ 3H0 lP −122 h¯ Λ= = Ωλ0 2 10 4 . (19.159) κ c 8π ≈ lP Such a small prefactor can only arise from an almost perfect cancellation of the contributions of boson and fermion ﬁelds. This cancellation is the main reason for postulating a broken supersymmetry in the universe, in which every boson has a fermionic counterpart. So far, the known particle spectra show no trace of such a symmetry. Thus there is need to explain it by some other not yet understood mechanism. The simplest model of the universe governed by the action (19.146) is called Friedmann model or Friedmann universe.

19.2.3 Friedmann Model Inserting (19.140) and (19.145) into the 00-component of the Einstein equation (19.152), we obtain the equation for the energy 3 a2 + ka2 λ = κT 0. (19.160) a4 η − 0 In terms of the cosmic standard time t, the general equation reads

2 2 at c 2 2 0 3 + k 2 λc = c κT0 . (19.161) " a a # −

H. Kleinert, PATH INTEGRALS 19.2 Tunneling in Relativistic Physics 1405

0 The simplest Friedmann model is based on an energy-momentum tensor T0 of an ideal pressure-less gas of mass density ρ:

ν ν Tµ = cρuµu , (19.162) where uµ are the velocity four-vectors uµ =(γ,γv/c) of the particles whose components are uµ = (γ,γv/c) with γ 1/ 1 v2/c2 if the four components transform ≡ − like (dx0 = cdt,dx). The gas is assumedq to be at rest in our comoving coordinates. 0 so that only the T0 -component is nonzero:

0 T0 = cρ. (19.163) This component is invariant under the time transformation (19.131). As a fortunate accident, this component of the Einstein equation has no aηηa term. Thus we may simply study the first-order differential equation 3 a2 + ka2 λ = cκρ. (19.164) a4 η − Since the total volume of the universe is 2πa3, we can express ρ in terms of the total mass M as follows M ρ = . (19.165) 2π2a3 In this way we arrive at the differential equation 3 κMc 4G M (a2 + ka2) λ = = N . (19.166) a4 η − 2π2a3 πc2a3 This equation of motion can also be obtained in another way. We express the action (19.146) in terms of a(η) using the equation (19.180) for R. We use the volume (19.123) and the relation (19.131) to rewrite the integration measure as

d4x √ g = dt (4)Sa =2π2 dη a4(η), (19.167) Z − Z Z so that

f 2 2 2π 4 2π 2 2 4 = dη 6a(aηη + ka) 2λa ˆ= dη 3a +3ka λa .(19.168) A 2κ − κ − η − Z h i Z h i The second expression arises from the ﬁrst by a partial integration and ignoring the boundary terms which do not inﬂuence the equation of motion. The above matter is described by the action

m Mc 4 √ 2 = d x gcρ = 2π dη 2 a(η). (19.169) A − Z − − Z 2π Variation with respect to a yields the Euler-Lagrange equation κMc 6(a + ka) 4λa3 =0. (19.170) ηη − − 2π2 1406 19 Relativistic Particle Orbits

Note that in terms of the Robertson-Walker time t [recall (19.131)], the equation of motion reads λ 1 κMc a¨ = a . (19.171) 3 − 6 2π2a2 As one should expect, the cosmological expansion is slowed down by matter, due to the gravitational attraction. A positive cosmological constant, on the other hand, accelerates the expansion. At the special value κMc 4G M 4πG ρ λ = λ = N = N , (19.172) Einstein ≡ 2π2a3 πc2a3 c the two eﬀects cancel each other and there exist a time-independent solution at a radius a and a density ρ. This is the cosmological constant which Einstein chose before Hubble’s discovery of the expanding universe to agree with Hoyle’s steady- state model of the universe (a choice which he later called the biggest blunder of his life). Multiplying (19.170) by aη and integration over η yields the pseudo-energy conservation law κMc 3(a2 + ka2) λa4 a = const, (19.173) η − − 2π2 in agreement with (19.166) for a vanishing pseudo-energy. This equation may also be written as λ a2 + ka2 a a a4 =0, (19.174) η − max − 3 where κMc 4G M a = N . (19.175) max ≡ 6π2 3πc2 This looks like the energy conservation law for a point particle of mass 2 in an eﬀective potential of the universe λ V univ(a)= ka2 a a a4, (19.176) − max − 3 at zero total energy. The potential is plotted for the spherical case k = 1 in Fig. 19.2.

Friedmann neglects the cosmological constant and considers the equation a2 + ka2 a a =0. (19.177) η − max The solution of the diﬀerential equation for this trajectory is found by direct integration. Assuming k = 1 we obtain da da 2a η = = = arccos . (19.178) univ 2 2 a Z V (a) Z (a amax/2) + a /4 − max − − − max q q

H. Kleinert, PATH INTEGRALS 19.2 Tunneling in Relativistic Physics 1407

univ 2 4 V (a)/amax

2 a 1 2 3 4 5 6 ✻ amax -2

-4 a0 amax -6

Figure 19.2 Potential of closed Friedman universe as a function of the reduced radius 2 a/amax for λamax = 0.1. Note the metastable minimum which leads to a possible solution a a . A tunneling process to the right leads to an expanding universe. ≡ 0

With the initial condition a(0) = 0, this implies a a(η)= max (1 cos η). (19.179) 2 − Integrating Eq. (19.131), we ﬁnd the relation between η and the physical (=proper) time 1 a a t = dη a(η)= max dη (1 cos η)= max (η sin η). (19.180) c Z 2c Z − 2c − The solution a(t) is the cycloid pictured in Fig. 19.3. The radius of the universe bounces periodically with period t0 = πamax/c from zero to amax. Thus it emerges from a big bang, expands with a decreasing expansion velocity due to the gravitational attraction, and recontracts to a point.

2.5 a 2 amax 1.5

0.5

0.2 0.4 0.6 0.8 1 1.2 t/t0

Figure 19.3 Radius of universe as a function of time in Friedman universe, measured in terms of the period t πa /c. (solid curve=closed, dashed curve=hyperbolic, dotted 0 ≡ max curve =parabolic). The curve for the closed universe is a cycloid.

Certainly, for high densities the solution is inapplicable since the pressure-less ideal gas approximation (19.163) breaks down. 1408 19 Relativistic Particle Orbits

Consider now the case of negative curvature with k = 1. Then the diﬀerential equation (19.177) reads −

λ a2 a2 a a a4 =0. (19.181) η − − max − 3 In order to compare the curves we shall again introduce a mass parameter M and rewrite the density as in (19.165), although M has no longer the meaning of the total mass of the universe (which is now inﬁnite). The solution for λ = 0 is now a a(η) = max (cosh η 1), (19.182) 2 − a t = max (sinh η η). (19.183) 2c − The solution is again depicted in Fig. 19.3. After a big bang, the universe expands forever, although with decreasing speed, due to the gravitational pull. The quantity amax is no longer the maximal radius nor is t0 the period. Consider ﬁnally the parabolic case k = 0, where the equation of motion (19.177) reads

λ a2 a a a4 =0, (19.184) η − max − 3 where M is a mass parameter deﬁned as before in the negative curvature case. Now the solution for λ = 0 is simply

a η =2 , (19.185) samax which is inverted to

η2 a(η)= a . (19.186) max 4

Now we solve (19.131) with

a t = max η3, (19.187) 12c so that

1/3 9 2/3 a(t)= amax (ct) . (19.188) 4 This solution is simply the continuation of leading term in the previous two solutions to large t.

H. Kleinert, PATH INTEGRALS 19.3 Relativistic Coulomb System 1409

19.2.4 Tunneling of Expanding Universe It is now interesting to observe that the potential for the spherical universe in Fig. 19.2 allows for a time-independent solution in which the radius lies at the metastable minimum which we may call a . The solution a a is a time- 0 ≡ 0 independent universe. We may now imagine that the expanding universe arises from this by a tunneling process towards the abyss on the right of the potential [10]. Its rate can be calculated from the Euclidean action of the associated classical solution in imaginary time corresponding to the motion from a0 towards the right in the inverted potential V univ(a). − Observe that this birth can only lead to universe of positive curvature. For a negative curvature, where the a2-term in (19.176) has the opposite sign, the metastable minimum is absent.

19.3 Relativistic Coulomb System

An external time-independent potential V (x) is introduced into the path integral (19.48) by substituting the energy E by E V (x). In the case of an attractive Coulomb potential, the second term in the action− (19.49) becomes

λb 2 2 int (E + e /r) = dλ h(λ) 3 , (19.189) A − Zλa 2Mc where r = x . The associated path integral is calculated via a Duru-Kleinert | | transformation as follows [12]. Consider the three-dimensional Coulomb system where the spacetime dimension is D = 4. Then we increase the three-dimensional space in a trivial way by a dummy fourth component x4, just as in the nonrelativistic treatment in Section 13.4. The 4 4 additional variable x is eliminated at the end by an integral dxa/ra = dγa, as in (13.120) and (13.127). Then we perform a Kustaanheimo-StiefelR transformationR ref(13.120) µ µ ν ′µ2 2 ′ 2 (13.106) dx =2A(u) νdu . This changes x into 4~u ~u , with the vector symbol lab(13.re) indicating the four-vector nature. The transformed action reads: est(13.114) ref(13.127)

λb 2 4 lab(13.3d) ˜ 4Mc ~u ′ 2 h(λ) 2 4 2 2 2 e e,E = dλ ~u (λ)+ 3 2 (M c E )~u 2Ee 2 . (19.190) est(13.121) A Zλa 2h(λ) 2Mc ~u " − − − ~u # ref(13.106) We now choose the gauge h(λ) = 1, and go from λ to a new dimensionless parameter lab(13.diﬀe) est(13.101) s via the path-dependent time transformation dλ = fds with f = ~u2. Result is the DK-transformed action

sb 2 4 ¯DK ds 4Mc ′ 2 1 2 4 2 2 2 e e,E = ~u (s)+ 2 M c E ~u 2Ee 2 . (19.191) A sa c 2 2Mc " − − − ~u # Z It describes a particle of mass µ =4M moving as a function of the “pseudotime” s in a harmonic oscillator potential of dimensionless “frequency” 1 ω = √M 2c4 E2. (19.192) 2Mc2 − 1410 19 Relativistic Particle Orbits

The oscillator possesses an additional attractive potential e4/2Mc2~u2, which is conveniently parametrized in the form of a centrifugal barrier− l2 V =¯h2 extra , (19.193) extra 2µ~u2 whose squared angular momentum has the negative value

l2 4α2. (19.194) extra ≡ − Here α denotes the ﬁne-structure constant α e2/hc¯ 1/137. In addition, there is also a trivial constant potential ≡ ≈ E V = e2. (19.195) const −Mc2

If we ignore, for the moment, the centrifugal barrier Vextra, the solution of the path ref(13.127)integral can immediately be written down [see (13.127)]: lab(13.3d) ∞ 4π est(13.121) h¯ 1 e2ES/Mc2¯h (xb xa)E = i dS e dγa (~ubS ~ua0) , (19.196) | − 2Mc 16 Z0 Z0 | where (~u S ~u 0) is the pseudotime evolution amplitude of the four-dimensional har- b | a monic oscillator. There are no time slicing corrections for the same reason as in the three- dimensional case. This is ensured by the aﬃne connection of the Kustaanheimo- Stiefel transformation satisfying

µλ µν λ i Γµ = g ei ∂µe ν = 0 (19.197)

(see the discussion in Section 13.6). Performing the integral over γa in (19.196), we obtain

h¯ Mκ 1 ̺−ν 2√̺ (xb xa)E = i d̺ I0 2κ (rbra + xbxa)/2 | − 2Mc πh¯ 0 (1 ̺)2 1 ̺ ! Z − − q 1+ ̺ exp κ (rb + ra) , (19.198) × "− 1 ̺ # − with the variable

h e−2ωS , (19.199) ≡ and the parameters e2 E α ν = = , (19.200) 2ωh¯ Mc2 M 2c4/E2 1 − µω 1 q E α κ = = √M 2c4 E2 = . (19.201) 2¯h hc¯ − hc¯ ν

H. Kleinert, PATH INTEGRALS 19.3 Relativistic Coulomb System 1411 ref(13.203)As in the further treatment of (13.203), the use of formula (13.207) lab(13.n1) est(13.192) 2 ∞ I (z cos(θ/2)) = (2l + 1)P (cos θ)I (z) (19.202) ref(13.207) 0 z l 2l+1 lab(13.n5) Xl=0 est(13.197)provides us with a partial wave decomposition 1 ∞ 2l +1 (xb xa)E = (rb ra)E,l Pl(cos θ) | rbra | 4π Xl=0 ∞ l 1 ∗ = (rb ra)E,l Ylm(xˆb)Ylm(xˆa). (19.203) rbra | Xl=0 mX=−l The radial amplitude is normalized slightly diﬀerently from (13.210): ref(13.210) lab(13.n9) ∞ h¯ 2M 1 2νy est(13.201) (rb ra)E,l = i √rbra dy e (19.204) | − 2Mc h¯ Z0 sinh y 1 exp [ κ coth y(rb + ra)] I2l+1 2κ√rbra . × − sinh y ! At this place, we incorporate the additional centrifugal barrier via the replace- ment

2l +1 2˜l +1 (2l + 1)2 + l2 , (19.205) → ≡ extra q as in Eqs. (8.146) and (14.223). The integration over y according to (9.29) yields ref(8.146) lab(x8.146) h¯ M Γ( ν + l˜+ 1) est([email protected]) (r r ) = i − W ˜ (2κr ) M ˜ (2κr ) . (19.206) b| a E,l − 2Mc hκ¯ ˜ ν,l+1/2 b ν,l+1/2 a ref(14.223) (2l + 1)! lab(14.239) This expression possesses poles in the Gamma function whose positions satisfy est(14.239) ref(9.29) the equations ν ˜l 1=0, 1, 2,... . These determine the bound states of the − − lab(intf3) Coulomb system. To simplify subsequent expressions, we introduce the small posi- est(9.52) tive l-dependent parameter α2 δ l ˜l = l +1/2 (l +1/2)2 α2 + (α4). (19.207) l ≡ − − − ≈ 2l +1 O q Then the pole positions satisfy ν =n ˜l n δl, with n = l +1,l +2,l +3,... . Using the relation (19.200), we obtain the≡ bound-state− energies:

2 −1/2 2 α Enl = Mc 1+ 2 ± " (n δl) # 2− 4 2 α α 1 3 6 Mc 1 2 3 + (α ) . (19.208) ≈ ± " − 2n − n 2l +1 − 8n O # Note the appearance of the plus-minus sign as a characteristic property of energies in relativistic quantum mechanics. A correct interpretation of the negative energies as 1412 19 Relativistic Particle Orbits

positive energies of antiparticles is straightforward only within quantum field theory, and will not be discussed here. Even if we ignore the negative energies, there is poor agreement with the experimental spectrum of the hydrogen atom. The spin of the electron must be included to get more satisfactory results. To find the wave functions, we approximate near the poles ν n˜ : ≈ l ( )nr 1 Γ( ν + l˜+ 1) − , − ≈ − n ! ν n˜ r − l 1 2 h¯2κ2 E 2 2Mc2 , ν n˜ ≈ n˜ 2M Mc2 E2 E2 − l l − nl E 1 1 κ 2 , (19.209) ≈ Mc aH n˜l with the radial quantum number n = n l 1. By analogy with the nonrelativistic r − − ref(13.213)equation (13.213), the last equation can be rewritten as lab(13.n11) 1 1 est(13.203) κ = , (19.210) a˜H ν where Mc2 a˜ a (19.211) H ≡ H E denotes a modified energy-dependent Bohr radius [compare (4.376)]. It sets the length scale of relativistic bound states in terms of the energy E. Instead of being 1/α 137 times the Compton wavelength of the electronh/Mc ¯ , the modified Bohr radius≈ is equal to 1/α timeshc/E ¯ . Near the positive-energy poles, we now approximate M ( )nr 1 E 2 2Mc2ih¯ iΓ( ν + l˜+ 1) − . (19.212) − − hκ¯ ≈ n˜2n ! a˜ Mc2 E2 E2 l r H − nl ref(9.48) Using this behavior and formula (9.48) for the Whittaker functions [together with lab(9.for) (9.50)] we write the contribution of the bound states to the spectral representation est(9.73) of the fixed-energy amplitude as ref(9.50) lab(x9.75) h¯ ∞ E 2 2Mc2ih¯ est(9.75) (rb ra)E,l = 2 2 2 Rnl(rb)Rnl(ra)+ .... (19.213) | Mc Mc E Enl n=Xl+1 − A comparison between the pole terms in (19.206) and (19.213) renders the radial wave functions

1 1 (ñl + l˜)! Rnl(r) = 1/2 v a˜ n˜ (2˜l + 1)!u(n l 1)! H l u − − ˜ t (2r/n˜ a˜ )l+1e−r/n˜la˜H M( n + l +1, 2˜l +2, 2r/n˜ a˜ ) (19.214) × l H − l H 1 (n l 1)! ˜ ˜ = − − e−r/nã˜H (2r/n˜ a˜ )l+1L2l+1 (2r/n˜ a˜ ). 1/2 v ˜ l H n˜l−l−1 l H a˜ n˜l u (ñ + l)! H u t

H. Kleinert, PATH INTEGRALS 19.4 Relativistic Particle in Electromagnetic Field 1413

The properly normalized total wave functions are 1 ψ (x)= R (r)Y (xˆ). (19.215) nlm r nl lm The continuous wave functions are obtained in the same way as from the nonrelativistic amplitude in formulas (13.221)–(13.229). ref(13.221) lab(x13.210) est(13.210) 19.4 Relativistic Particle in Electromagnetic Field ref(13.229) lab(x13.218) Consider now the relativistic particle in a general spacetime-dependent electromag- est(13.218) netic vector ﬁeld Aµ(x).

19.4.1 Action and Partition Function An electromagnetic ﬁeld Aµ(x) is included into the canonical action (19.14) in the usual way by the minimal substitution (2.644):

λb 2 h(λ) e 2 2 ¯e[p, x]= dλ ipx˙ + p A + M c , (19.216) A Zλa (− 2Mc " − c #) and the amplitude (19.22):

∞ h¯ D −A¯e/¯h (xb xa)= dS h Φ[h] x e , (19.217) | 2Mc Z0 Z D Z D with the minimally coupled action [compare (2.706)]

λb Mc 2 e Mc ¯e = dλ x˙ (λ)+ i x˙(λ)A(x(λ)) + h(λ) , (19.218) A Zλa "2h(λ) c 2 # which reduces in the simplest gauge (19.23) to the obvious extension of (19.26):

2 ∞ 2 h¯ −βMc /2 4 −Ae (xb xa)= dβ e x e , (19.219) | 2M Z0 Z D with the action

¯hβ M 2 e e = e,0 + e,int dτ x˙ (τ)+ i x˙(τ)A(x(τ)) . (19.220) A A A ≡ Z0 2 c The partition function of a single closed particle loop of all shapes and lengths in an external electromagnetic ﬁeld is from (19.37)

∞ 2 dβ −βMc /2 D −Ae/¯h Z1 = e x e . (19.221) Z0 β Z D As in (19.45) and (19.46) this yields, up to a factor 1/h¯ the eﬀective action of an ensemble of closed particle loops in an external electromagnetic ﬁeld. 1414 19 Relativistic Particle Orbits

19.4.2 Perturbation Expansion Since the electromagnetic coupling is rather small, we can split the exponent e−A/¯h into e−A0/¯he−Aint/¯h and expand the second factor in powers of : Aint ∞ ( ie/hc¯ )n n ¯hβ −Aint/¯h e = − dτi x˙(τi)A(x(τi)) . (19.222) n! 0 nX=0 iY=1 Z If the noninteracting eﬀective action implied by Eqs. (19.39), (19.37), and (19.46) is denoted by

∞ Γe,0 dβ −βMc2/2 VD VD 1 Z1 = e D = D D/2 Γ(1 D/2), (19.223) h¯ ≡− − 0 β 2 − C (4π) − Z 2πh¯ β/M λM q C with the Compton wavelength λM of Eq. (19.31), we obtain the perturbation expansion

∞ ∞ n n ¯hβ Γe Γe,0 dβ −βMc2/2 VD ( ie/c) = e D − dτi x˙(τi)A(x(τi)) , h¯ h¯ − 0 β 2 n! * 0 + Z 2πh¯ β/M nX=1 iY=1 Z 0 q (19.224) where ... 0 denotes the free-particle expectation values [compare (3.483)–(3.486)] taken inh thei free path integral with periodic paths with a ﬁxed β [compare (19.38) and (19.41)]:

Dx [x] e−Ae,0/¯h [x] 0 D O . (19.225) hO i ≡ Dx e−Ae,0/¯h R D R D 2 The denominator is equal to VD/ 2πh¯ β/M . The free eﬀective action in theq expansion (19.224) can be omitted by letting the sum start with n = 0. The evaluation of the cumulants proceeds by Fourier decomposing the vector ﬁelds as D d k ikx A(x)= D e A(k), (19.226) Z (2π) and rewriting (19.224) as

∞ Γe Γe,0 dβ −βMc2/2 VD = e D h¯ h¯ 0 β 2 − Z 2πh¯ β/M ∞ n n q D n ¯hβ ( ie/hc¯ ) d ki µi µi ikix(τi) − A (ki) dτi x˙ (τi)e .(19.227) × n! " (2π)D #* 0 + nX=1 iY=1 Z iY=1 Z 0 First we evaluate the expectation values

ik1x(τ1) iknx(τn) x˙(τ1)e x˙(τn)e . (19.228) ··· 0 D E

H. Kleinert, PATH INTEGRALS 19.4 Relativistic Particle in Electromagnetic Field 1415

Due to the periodic boundary conditions, we separate, as in Section 3.25, the path average x0 =x ¯(τ) [recall (3.807)], writing

x(τ)= x0 + δx(τ), (19.229) and factorize (19.228) as

i(k1+...+kn)x0 ik1δx(τ1) iknδx(τn) e δx˙(τ1)e δx˙(τn)e . (19.230) 0 ··· 0 D E D E The ﬁrst average can be found as an average with respect to the x0-part of the path integral whose measure was given in Eq. (3.811). It yields a δ function ensuring the conservation of the total energy and momenta of the n photons involved:

i(k1+...+kn)x0 1 D (D) e = (2π) δ (k1 + ... + kn). (19.231) 0 V D E D The denominator comes from the normalization of the expectation value which has D an integral d x0 in the denominator. The secondR average is obtained using Wick’s theorem. The correlation function δxµ(τ )δxν (τ ) , is obtained from Eq. (3.842) in the limit Ω 0 for τ , τ (0, β). h 1 2 i0 → 1 2 ∈ It is the periodic propagator with subtracted zero mode: h¯ δxµ(τ )δxν (τ ) = δµνG(τ , τ )= δµν ∆(¯ τ , τ ), (19.232) h 1 2 i0 1 2 M 1 2 ¯ a ′ where ∆(τ1 τ2) is the subtracted periodic Green function Gω,e(τ τ ) of the − 2 − diﬀerential operator ∂τ calculated in Eq. (3.254) in the short notation of Subsection 10.12.1 [see Eq. (10.565)].− In the presently used physical units it reads: (τ τ ′)2 τ τ ′ hβ¯ ∆(¯ τ, τ ′) ∆(¯ τ τ ′)= − − + , τ [0, hβ¯ ]. (19.233) ≡ − 2¯hβ − 2 12 ∈ The time derivatives of (19.233) are from (10.566): τ τ ′ ǫ(τ τ ′) ˙∆(¯ τ, τ ′)= ∆˙(¯ τ, τ ′) − − , τ,τ ′ [0, hβ¯ ]. (19.234) − ≡ hβ¯ − 2 ∈ With these functions it is straightforward to calculate the expectation value using the Wick rule (3.310) for j(τ)= n k δ(τ τ ): i i − i 1 n ik δx(τ ) ik Pδx(τ ) − kikj G(τi,τj ) e 1 1 e n n = e 2 i,j=1 . (19.235) ··· 0 D E P By rewriting the right-hand side as

1 n 1 n 1 n 2 − kikj G(τi,τj ) − kikj [G(τi,τj )−G(τi,τi)]− ( ki) G(τi,τi) e 2 i,j=1 = e 2 i,j=1 2 i=1 , (19.236)

P P n P we see that if the momenta ki add up to zero, i=1 ki = 0, we can replace (19.235) by P n ik1δx(τ1) iknδx(τn) e e = exp kikj[G(τi, τj) G(τi, τi)] . (19.237) ··· 0 − −  D E  Xi

It is therefore useful to introduce the subtracted Green function

G′(τ , τ ) G(τ , τ ) G(τ , τ ). (19.238) i j ≡ i j − i i Recall the similar situation in the evaluation (5.377). An obvious extension of (19.235) is

ei[k1δx(τ1)+q1x˙(τ1)] ei[knδx(τn)+qnx˙(τn)] (19.239) 0 n ··· n n n D − 1 k k G(τ ,τ )− 1 q Ek ˙G (τ ,τ )− 1 k q G˙(τ ,τ )− 1 q q ˙G˙(τ ,τ ) = e 2 i,j=1 i j i j 2 i,j=1 i j i j 2 i,j=1 i j i j 2 i,j=1 i j i j , P P P P where the dots have the same meaning as in (10.395).

19.4.3 Lowest-Order Vacuum Polarization Consider the lowest nontrivial case n = 2. By diﬀerentiating (19.237) with respect to iq and iq , and setting q = 0, we obtain for k = k = k: 1 2 i 2 − 1 − 2 ikδx(τ1) −ikδx(τ2) 2 k [G(τ1,τ2)−G(τ1,τ1)] x˙(τ1)e x˙(τ2)e = ˙G˙(τ1, τ2)+k ˙G (τ1, τ2)G˙(τ1, τ2) e . 0 D E h i (19.240)

Inserting this into (19.227) after factorization according to (19.230), we obtain the lowest correction to the eﬀective action

2 ∞ 2 D e dβ −βMc2/2 1 d ki D (D) µ ν ∆Γe= 2 e D D (2π) δ (k1+k2)A (k1)A (k2) −2¯hc 0 β 2 " (2π) # Z 2πh¯ β/M iY=1 Z

¯hβ ¯hβ q 2 ′ µν µ ν k1G (τ1,τ2) dτ1 dτ2 [˙G˙(τ1, τ2)δ + k1 k1 ˙G (τ1, τ2) G˙(τ1, τ2)] e , (19.241) × Z0 Z0 where we have displayed the proper vector indices. A partial integration over τ1 brings the second line to the form

¯hβ ¯hβ 2 ′ 2 µν µ ν k1G (τ1,τ2) dτ1 dτ2 k1δ k1 k1 ˙G (τ1, τ2) G˙(τ1, τ2) e . (19.242) − 0 0 − Z Z Expressing G′(τ , τ ) as (¯h/M) ∆(¯ τ τ ) ∆(0)¯ and using the periodicity in τ , 1 2 1 − 2 − 2 we calculate the integral h i

2 ¯hβ h¯ µ 2 ′ ′ 2 µν ν ˙ ′ 2 ¯hk1M[∆p(τ)−∆p(0)] k1δ k1 k1 hβ¯ dτ ∆p (τ)e . (19.243) M 2 − 0 Z We now introduce reduced times u τ/hβ¯ and rewrite the Green functions for τ (0, hβ¯ ), u (0, 1) as ≡ ∈ ∈ hβ¯ 1 ∆(¯ τ1 τ2) = u(1 u) , (19.244) − − 2 − − 6 1 ∆(¯˙ τ τ ) = u , (19.245) 1 − 2 − 2

H. Kleinert, PATH INTEGRALS 19.4 Relativistic Particle in Electromagnetic Field 1417 such that the integral in (19.243) becomes

1 1 2 −β¯h2k2u(1−u)/2M du (2u 1) e 1 . (19.246) 4 Z0 −

Inserting this into (19.241) and dropping the irrelevant subscript of k1 we arrive at

2 ∞ D e dβ −βMc2/2 1 d k µ ν ∆Γe = 2 e D D A (k)A ( k) 2¯hc 0 β 2 (2π) Z 2πh¯ β/M Z − 4q2 1 h¯ β 2 2 k2δµν kµkν du (2u 1)2e−β¯h k 2Mu(1−u)/2M . (19.247) × − 4M 2 0 − Z After replacingh ¯2β/2M β, the integral over β can easily be performed using the → formula (2.498) with the result e2h¯ 1 1 dDk ∆Γ = Aµ(k)Aν ( k) k2δµν kµkν e c2 (4π)D/2 2c (2π)D − − Z 1 D/2−2 Γ (2 D/2) du (2u 1)2 u(1 u)k2 + M 2c2/h¯2 . (19.248) × − 0 − − Z h i In the prefactor we recognize the ﬁne structure constant α = e2/hc¯ [recall (1.505)]. The momentum integral can be rewritten as 1 dDk 1 dDk Aµ(k)Aν( k) k2δµν kµkν = F ( k)F (k), (19.249) 2 (2π)D − − 4 (2π)D µν − µν Z Z where

F (x)= ∂ A (x) ∂ A (x) (19.250) µν µ ν − ν µ is the tensor of electromagnetic ﬁeld strengths. We now abbreviate the integral over u as follows: 4π 1 D/2−2 Π(k2) α Γ(2 D/2) du (2u 1)2 u(1 u)k2 + M 2c2/h¯2 .(19.251) ≡ (4π)D/2 − 0 − − Z h i This allows us to re-express (19.248) in conﬁguration space:

1 4 2 ∆Γe = d xFµν (x)Π( ∂ )Fµν(x), (19.252) 16πc Z − where Fµν(x) is the Euclidean version of the gauge-invariant 4-dimensional curl of the vector potential:

F (x)= ∂ A (x) ∂ A (x). (19.253) µν µ ν − ν µ

In Minkowski space, the components of Fµν are the electric and magnetic ﬁelds: F = F 0i = ∂0Ai + ∂iA0 = ∂ Ai ∂ A0 = Ei, (19.254) 0i − − − 0 − i − F = F ij = ∂iAj + ∂jAi = ∂ Aj + ∂ Ai = ǫ Bk. (19.255) ij − i j − ijk 1418 19 Relativistic Particle Orbits

This is in accordance with the electrodynamic deﬁnitions 1 E A˙ ∇φ, B ∇ A, (19.256) ≡ − c − ≡ × where A0(x) is identiﬁed with the electric potential φ(x). In terms of Fµν (x), the Maxwell action in the presence of a charge density ρ(x) and a electric current density j(x) 1 1 em = dtd3x E2(x) B2(x) ρ(x)φ(x) j(x) A(x) , (19.257) A 4π − − − c · Z h i can be written covariantly as

em 4 1 2 1 µ = d x Fµν(x)+ 2 j (x)Aµ(x) , (19.258) A − Z 8πc c where

jµ(x)=(cρ(x), j(x)) (19.259) is the four-vector formed by charge density and electric current. By extremizing the action (19.258) in the vector field Aµ(x) we find the Maxwell equations in the covariant form 1 ∂ F νµ(x)= jµ(x), (19.260) ν c whose zeroth and spatial components reduce to the time-honored laws of Gauss and Ampère: ∇ E = 4πρ (Gauss′s law), (19.261) · 4π ∇ B = j (Ampère’s law). (19.262) × c Expressing E(x) in terms of the potential using Eq. (19.256) and inserting this into Gauss’s law, we obtain for a static point charge e at the origin the Poisson equation

∇2φ(x)=4πeδ(3)(x). (19.263) − An electron of charge e experiences an attractive mechanical potential V (x) = − eφ(x). In momentum space this satisﬁes the equation reads − k2V (k)= 4πe2. (19.264) − From this we ﬁnd directly the Coulomb potential of a hydrogen atom

d3k 4πe2 e2 x ∇2 −1 (3) x x V ( )=( ) 4πeδ ( )= 3 2 = , r , (19.265) − Z (2π) k − r ≡| |

H. Kleinert, PATH INTEGRALS 19.4 Relativistic Particle in Electromagnetic Field 1419 where e2 can be expressed in terms of the fine-structure constant α as e2 =hc ¯ α. (1.505). The Euclidean result (19.252) implies that a fluctuating closed particle orbit changes the first term in the Maxwell action (19.258) to 1 eff = dDx F (x) 1+Π( ∂2) F (x). (19.266) Aem − 16πc µν − µν Z h i The quantity Π( ∂2) is the self-energy of the electromagnetic field caused by the − fluctuating closed particle orbit. The self-energy changes the Maxwell equations (19.261) and (19.262) into

1+Π( ∂2) ∇ E = 4πρ, − · h i 4π 1+Π( ∂2) ∇ B = j. (19.267) − × c h i The static equation (19.264) for the atomic potential changes therefore into

1+Π(k2) k2V (k)= 4πe2. (19.268) − h i Since Π(k2) is of order α 1/137, this can be solved approximately by ≈ 1 V (k) 4πe2 1 Π(k2) . (19.269) ≡ − − k2 h i In real space, the attractive atomic potential is changed to lowest order in α as α α 1 Π(∇2) . (19.270) − r → − − r h i Set us calculate this change explicitly. For small ǫ and k2, we expand the self- energy (19.251) in D =4 ǫ dimensions as − 2 2 γ 2 2 2 2 α 2 M c e αh¯ k k Π(k )= + log 2 + ǫ, 2 . (19.271) 24π "− ǫ 4πh¯ # − 160πM 2c2 O M 2c2/h¯ !

Inserting this into (19.269), or into (19.269) and using the Poisson equation ∇2 − × 1/r = 4πδ(3)(x), we see that the self energy changes the Coulomb potential as follows:

2 2 γ 2 2 α ∇2 α α 2 M c e α α h¯ (3) 1 Π( ) 1 +log 2 δ (x). − r → − − r ≈ − ( − 24π2 "− ǫ 4πh¯ #) r − 40M 2c2 h i (19.272) The ﬁrst term amounts to a small renormalization of the electromagnetic coupling by the factor in curly brackets, which is close to unity for ﬁnite ǫ since α is small. We are, however, interested in the result in D = 4 spacetime dimensions where ǫ 0 and → (19.272) diverges. The physical resolution of this divergence problem is to assume 1420 19 Relativistic Particle Orbits

the initial point charge e0 in the electromagnetic interaction to be diﬀerent from the experimentally observed e to precisely compensate the renormalization factor, i.e.,

2 2 γ 2 2 α 2 M c e e0 = e 1+ +log 2 . (19.273) ( 24π2 "− ǫ 4πh¯ #)

Thus, the result Eq. (19.272) is really obtained in terms of e0, i.e., with α replaced 2 by α0. Then, using (19.273), we ﬁnd that up to order α , the atomic potential is α α2h¯2 V eﬀ (x)= δ(3)(x). (19.274) − r − 40M 2c2 The second is an additional attractive contact interaction. It shifts the energies of the s-wave bound states in Eq. (19.208) slightly downwards.

19.5 Path Integral for Spin-1/2 Particle

For particles of spin 1/2 the path integral formulation becomes algebraically more involved. Let us ﬁrst recall a few facts from Dirac’s theory of the electron.

19.5.1 Dirac Theory

In the Dirac theory, electrons are described by a four-component field ψα(x) in spacetime parametrized by xµ = (ct, x) with µ = 0, 1, 2, 3. The field satisfies the wave equation (ih/∂¯ Mc) ψ(x)=0, (19.275) − where /∂ is a short notation for γµ∂ and γµ are 4 4 Dirac matrices satisfying the µ × anticommutation rules γµ,γν =2gµν, (19.276) { } where gµν is now the Minkowski metric 1 0 0 0 0 1 0 0 gµν =  −  . (19.277) 0 0 1 0    −   0 0 0 1   −  An explicit representation of these rules is most easily written in terms of the Pauli matrices (1.448): σ0 0 0 σi γ0 = , γi = , (19.278) 0 σ0 σi 0 − ! − ! where σ0 is a 2 2 unit matrix. The anticommutation rules (19.276) follow directly from the multiplication× rules for the Pauli matrices: σiσj = δij + iǫijkσk. (19.279)

H. Kleinert, PATH INTEGRALS 19.5 Path Integral for Spin-1/2 Particle 1421

The action of the Dirac ﬁeld is

= d4x ψ¯(x)(ih/∂¯ Mc) ψ(x), (19.280) A Z − where the conjugate ﬁeld ψ¯(x) is deﬁned as

ψ¯(x) ψ†(x)γ0. (19.281) ≡ It can be shown that this makes ψ¯(x)ψ(x) a scalar ﬁeld under Lorentz transformations, ψ¯(x)γµψ(x) a vector ﬁeld, and an invariant. If we decompose ψ(x) into its A Fourier components

1 ikx ψ(x)= e ψk(t), (19.282) k √V X where V is the spatial volume, the action reads

t b † = dt ψk(t)[ih∂¯ t H(¯hk)] ψk(t), (19.283) A ta k − Z X with the 4 4 Hamiltonian matrix × 0 0 2

H(p) γ p c + γ Mc . (19.284) ≡ This can be rewritten in terms of 2 2-submatrices as ×

Mc p H(p)= c. (19.285)

p Mc − − ! Since the matrix is Hermitian, it can be diagonalized by a unitary transformation to

εk 0 Hd(p)= , (19.286) 0 εk − ! where

εk c p2 + M 2c2 (19.287) ≡ q are energies of the relativistic particles of mass M and momentum p. Each entry in (19.286) is a 2 2-submatrix. This is achieved× by the Foldy-Wouthuysen transformation

Hd = eiSHe−iS, (19.288)

where

S = i /2, arctan (v/c), v p/M = velocity. (19.289) − · ≡ ≡ 1422 19 Relativistic Particle Orbits

The vector points in the direction of the velocity v and has the length ζ = arctan (v/c), such that Mc p cos ζ = , sin ζ = | | . (19.290) √p2 + M 2c2 √p2 + M 2c2 A function of a vector v is deﬁned by its Taylor series where even powers of v are scalars v2n = v2n and odd powers are vectors v2n+1 = v2nv. If vˆ denotes as usual

the direction vector vˆ v/ v , the matrix vˆ has the property that all even powers ≡ | | · 2n n of it are equal to a 4 4 unity matrix up to an alternating sign: ( vˆ) =( 1) . × iS · − Thus S = i vˆζ and the Taylor series of e reads explictly − · n n n−1 n

iS ( 1) ζ ( 1) ζ ζ ζ e = − +( vˆ) − = cos + vˆ sin . (19.291) n! 2! · n! 2! 2 · 2 n=0X,2,4,... n=1X,3,5,...

ˆ 0

Now, S commutes trivially with p = p , while anticommuting with γ due to the anticommutation rules (19.276).· · Hence| | we can move the right-hand transformation in (19.288) simply to the left-hand side with a sign change of S, and obtain

Hd = e2iSH. (19.292)

It is easy to calculate e2iS: we merely have to double the rapidity in (19.291) and obtain

2iS Mc e = cos ζ + vˆ sin ζ = (1 + p/Mc) . (19.293) · √p2 + M 2c2 · Hence we obtain

d 2iS Mc 2 0 H = e H = (1 + p/Mc) Mc γ (1 + p/Mc) . (19.294) √p2 + M 2c2 · ·

0 Taking the right-hand parentheses to the left of γ changes the sign of . The

2 2 2 product (1 + p/Mc) (1 p/Mc) is simply 1 + p /M c , such that · − · d 0 0 0 H = c p2 + M 2c2 γ = εk γ =¯hωk γ . (19.295) q Remembering γ0 from Eq. (19.278) shows that Hd has indeed the diagonal form (19.286). d iS Going to the diagonal ﬁelds ψk(t)= e ψk(t), the action becomes

tb d† d d = dt ψk (t) ih∂¯ t H (¯hk) ψk(t). (19.296) A ta k − Z X h i Thus a Dirac ﬁeld is equivalent to a sum of inﬁnitely many momentum states, each being associated with four harmonic oscillators of the Fermi type. The path integral is a product of independent harmonic path integrals of frequencies ωk. ±

H. Kleinert, PATH INTEGRALS 19.5 Path Integral for Spin-1/2 Particle 1423

It is then easy to calculate the quantum-mechanical partition function using the result (7.419) for each oscillator, continued to real times:

4 ZQM = 2 cosh [ωk(tb ta)/2] . (19.297) k − Y n o This can also be written as

ZQM = exp 4 log 2 cosh[ωk(tb ta)/2] , (19.298) k { − }! X or as

d ZQM = exp 4 Tr log (ih∂¯ t hω¯ k) = exp Tr log ih∂¯ t H (¯hk) . (19.299) " k − # ( k − ) X X h i Since the trace is invariant under unitary transformations, we can rewrite this as

ZQM = exp Tr log [ih∂¯ t H(¯hk)] , (19.300) ( k − ) X or, since the determinant of γ0 is unity, as

0 0 d 0 2 ZQM =exp Tr log ihγ¯ ∂t γ H (¯hk) =exp Tr log ihγ¯ ∂t hc¯ k Mc . ( k − ) ( k − − ) X h i X h i If we include the spatial coordinates into the functional trace, this can also be written as

0 2

Z = exp Tr log ihγ¯ ∂ ihc¯ ∇ Mc = exp Tr log [c (ih/∂¯ Mc)] . QM t − − { − } h i In analytic regularization of Section 2.15, the factor c in the tracelog can be dropped. Moreover, there exists a simple algebraic identity (ih/∂¯ + Mc)(ih/∂¯ Mc)= h¯2∂2 M 2c2. (19.301) − − − The factors on the left-hand side have the same functional determinant since [compare (7.338) and (7.420)]

d4p d4p V4 Tr log(¯h /p −Mc) V4 Tr log(−¯h/p −Mc) Det(ih/∂¯ Mc) = eTr log(i¯h/∂ −Mc) = e (2π)4 = e (2π)4 − = Det( ih/∂¯ Mc). R R (19.302) − − This allows us, as a generalization of (7.420), to write

2 2 2 2 Det(ih/∂¯ Mc) = Det(ih/∂¯ + Mc)= Det( h¯ ∂ M c )14×4, (19.303) − q − − where 14×4 is a 4 4 unit matrix. In this way we arrive at the quantum-mechanical partition function×

1 2 2 2 2 iΓf /¯h ZQM = exp 4 Tr log( h¯ ∂ M c ) e 0 . (19.304) × 2 − − ≡ 1424 19 Relativistic Particle Orbits

The factor 4 comes from the trace in the 4 4 matrix space, whose indices have disappeared in the formula. The exponent determines× the effective action Γ of the quantum system by analogy with the Euclidean relation Eq. (19.46). The Green function of the Dirac equation (19.275) is a 4 4 -matrix defined by × (ih/∂¯ Mc) (x x ) = ihδ¯ (D)(x x )δ . (19.305) − αβ | a βγ − a αγ Suppressing the Dirac indices, it has the spectral representation d4p ih¯ (x x )= e−ip(xb−xa)/¯h. (19.306) b| a (2πh¯)4 /p Mc + iη Z − This can be written more formally as a functional matrix ih¯ (x x )= x x , (19.307) b| a h b|ih/∂¯ Mc| ai − which obviously satisfies the differential equation (19.305).

19.5.2 Path Integral It is straightforward to write down a path integral representation for the amplitude (19.307):

∞ xb=x(λb) 4 4 p iA/¯h (xb xa)= dS x D 4 e , (19.308) | Z0 Zxa=x(λa) D Z (2πh¯) with the action

S [x, p]= dλ [ px˙ +(/p Mc)] . (19.309) A Z0 − − As in Section 19.1, the parameter λ is a lengthalong the orbits, but in contrast to that section we shall work here with real time t, and λ is related t by λ = ct. and S is the total reparametrization-invariant length. The dot denotes the derivative x˙ dx(λ)/dλ. The path integral over x(λ) ensures that the momentum is λ- independent,≡ so that the path integral over p(λ) reduces to an ordinary Fourier integral [compare Eq. (2.41)]:

xb=x(λb) 4 4 4 p d p ip(xb−xa)+iS(/p −Mc)/¯h x D 4 = 4 e (19.310) Zxa=x(λa) D Z (2πh¯) Z (2πh¯) Performing the integral over S in (19.308) leads to

d4p ih¯ (x x )= eip(xb−xa) , (19.311) b| a (2πh¯)4 /p Mc Z − in agrrement with the amplitude (19.307). The minus sign in front of px˙ is necessary to have the positive sign for the spatial part px˙ in the Minkowski metric (19.277).

H. Kleinert, PATH INTEGRALS 19.5 Path Integral for Spin-1/2 Particle 1425

The action (19.309) can immediately be generalized to

S ¯[x, p] = dλ [ px˙ + h(λ)(/p Mc)] , (19.312) A Z0 − − with any function h(λ) > 0. This makes it invariant under the reparametrization λ f(λ), h(λ) h(λ)/f(λ). (19.313) → → The path integral (19.308) contains then an extra functional integration over h(λ) with some gauge-fixing functional Φ[h], as in (19.22), which has been chosen in (19.309) as Φ[h]= δ[h 1]. − The path integral alone yields an amplitude x eiS(i¯h/∂ −Mc)/¯h x , (19.314) h | | ai and the integral over S in (19.308) produces indeed the propagator (19.307). In evaluating this we must assume, as usual, that the mass carries an infinitesimal negative imaginary part iη. This is also necessary to guarantee the convergence of the path integral (19.308). Electromagnetism is introduced as usual by the minimal substitution (2.644). In the operator version, we have to substitute e ∂ ∂ + i A . (19.315) µ −−−→ µ hc¯ µ Thus we obtain the gauge-invariant action S eh¯ ¯[x, p]= dλ px˙ + h(λ) /p /A Mc . (19.316) A Z0 "− − c − !# Another path integral representation which is closer to the spinless case is obtained by rewriting (19.307) as ih¯ (x x )=(ih/∂¯ + Mc) x x , (19.317) | a h | h¯2∂2 M 2c2 | ai − − where we have omitted the negative infinitesimal imaginary part ih¯ of the mass, for brevity, and used the fact that − (ih/∂¯ + Mc)(ih/∂¯ Mc)= h¯2∂2 M 2c2, (19.318) − − − on account of the anticommutation relation (19.276). By rewriting (19.317) as a proper-time integral

∞ 1 iS(−¯h2∂2−M 2c2)/2Mc¯h (x xa)= (ih/∂¯ + Mc) dS x e xa , (19.319) | 2Mc Z0 h | | i we ﬁnd immediately the canonical path integral

∞ x=x(λb) 4 1 4 p iA/¯h (x xa)= (ih/∂¯ + Mc) dS x D 4 e , (19.320) | 2Mc Z0 Zxa=x(λa) D Z (2πh¯) 1426 19 Relativistic Particle Orbits with the action S 1 [x, p]= dλ px˙ + p2 M 2c2 . (19.321) A 0 − 2Mc − Z The suppressed Dirac indices of the 4 4 -amplitude on the left-hand side, (x x ) , × | a αβ are entirely due to the prefactor (ih/∂¯ + Mc)αβ on the right-hand side. As in the generalization of (19.309) to (19.309), this action can be generalized to S h(λ) ¯[x, p] = dλ px˙ + p2 M 2c2 , (19.322) A 0 "− 2Mc − # Z with any function h(λ) > 0, thus becoming invariant under the reparametrization (19.313), and the path integral (19.308) contains then an extra functional integration h(λ) Φ[h]. The action (19.322), is precisely the Minkowski version of the path D integralR of a spinless particle of the previous section [see Eq. (19.14)]. Introducing here electromagnetism by the minimal substitution (19.315) in the prefactor of (19.317) and on the left-hand side of (19.318), the latter becomes then

2 e e 2 e e µν 2 2 ih/∂¯ /A + Mc ih/∂¯ /A Mc =¯h i∂ A Σ Fµν M c , − c − c − " − hc¯ − hc¯ # − (19.323) where i Σµν [γµ,γν]= Σνµ (19.324) ≡ 4 − are the generators of Lorentz transformations in the space of Dirac spinors. For any ﬁxed index µ, they satisfy the commutation rules: [Σµν , Σµκ]= igµµΣνκ. (19.325) Due to the antisymmetry in the two indices, this determines all nonzero commutators of the Lorentz group. Using Eqs. (19.254), we can write the last interaction term in (19.323) as Σµν F = 2ΣiBi + 2Σ0iEi, (19.326) µν − where Σi are the generators of rotation

i i 1 jk 1 σ 0 Σ ǫijkΣ = i , (19.327) ≡ 2 2 0 σ ! and i 0i i 0 i σ 0 Σ iα iγ γ = i − i (19.328) ≡ ≡ 0 σ ! are the generators of rotation-free Lorentz transformations. Thus

µν (B + iE) 0 Σ Fµν = . (19.329) − 0 (B iE) − !

H. Kleinert, PATH INTEGRALS 19.5 Path Integral for Spin-1/2 Particle 1427

19.5.3 Amplitude with Electromagnetic Interaction The obvious generalization of the path integral (19.320) which includes minimal electromagnetic interactions is then

∞ x=x(λb) 4 1 e 4 p ˆ iA/¯h (x xa)= ih/∂¯ /A +Mc dS h(λ) Φ[h] x D 4 T e , | 2M − c Z0 Z D Zxa=x(λa) D Z (2πh¯) (19.330) with the action

S 2 h(λ) e he¯ µν 2 2 ¯[x, p]= dλ px˙ + p A Σ Fµν M c . (19.331) A Z0 (− 2Mc " − c − c − #) The symbol Tˆ is the time-ordering operator defined in (1.241), now with respect to the proper time λ, which has to be present to account for the possible noncommu- µν tativity of FµνΣ /2 at different λ. Integrating out the momentum variables yields the configuration-space path integral

∞ x=x(λb) 1 e 4 iA/¯h (x xa)= ih/∂¯ /A +Mc dS h(λ) Φ[h] x Teˆ ,(19.332) | 2M − c 0 D xa=x(λa) D with the action Z Z Z

S ¯ Mc 2 e he¯ µν Mc [x]= dλ x˙ xA˙ h(λ) 2 Σ Fµν h(λ) . (19.333) A Z0 "−2h(λ) − c − 2Mc − 2 # The coupling to the magnetic ﬁeld adds to the rest energy Mc2 an interaction energy he¯

H = B. (19.334) int −Mc · From this we extract the magnetic moment of the electron. We compare (19.334) with the general interaction energy (8.316), and identify the magnetic moment as

he¯ = . (19.335) Mc Recall that in 1926, Uhlenbeck and Goudsmit explained the observed Zeeman splitting of atomic levels by attributing to an electron a half-integer spin. However, the magnetic moment of the electron turned out to be roughly twice as large as what one would expect from a charged rotating sphere of angular momentum L, whose magnetic moment is L

= µ , (19.336) B h¯ where µB he/Mc¯ is the Bohr magneton (2.649). On account of this relation, it is customary≡ to parametrize the magnetic moment of an elementary particle of spin S as follows: S

= gµ . (19.337) B h¯ 1428 19 Relativistic Particle Orbits

The dimensionless ratio g with respect to (19.336) is called the gyromagnetic ratio

or Land´efactor. For a spin-1/2 particle, S is equal to /2, and comparison with (19.335) yields the gyromagnetic ratio

g =2, (19.338) the famous result found first by Dirac, predicting the intrinsic magnetic moment µ of an electron to be equal to the Bohr magneton µB, thus being twice as large as expected from the relation (19.336), if we insert there the spin 1/2 for the orbital angular momentum. In quantum electrodynamics one can calculate further corrections to this Dirac result as a perturbation expansion in powers of the fine-structure constant α [recall (1.505)]. The first correction to g due to one-loop Feynman diagrams was found by Schwinger: α g =2 1+ 2 1.001161, (19.339) × 2π ≈ × where α is the fine-structure constant (1.505). Experimentally, the gyromagnetic ratio has been measured to an incredible accuracy:

g =2 1.001 159 652 193(10), (19.340) × in excellent agreement with (19.339). If the perturbation expansion is carried to higher orders, one is able to reach agreement up to the last experimentally known digits [14]. In the literature, there exist other representations of path integrals for Dirac particles involving Grassmann variables. For this we recall the discussion in Sub- section 7.11.3 that a path integral over four real Grassmann ﬁelds θµ, µ =0, 1, 2, 3

4 i ih¯ ˙µ θ exp dt θµθ , (19.341) Z D "h¯ Z − 4 !# generates a matrix space corresponding to operators θˆµ with the anticommutation rules

θˆµ, θˆν =2gµν, (19.342) n o and the matrix elements

β θˆµ α =(γ γµ) , β,α =1, 2, 3, 4. (19.343) h | | i 5 βα It is then possible to replace path integral (19.332) by 1 ∞ (x xa) = dS h Φ[h] | 2M Z0 Z D p ¯ µ 4 i(A[x]+AG[θ ,A])/¯h χΦ[χ] θ x D 4 e , (19.344) × Z D Z D Z D Z (2πh¯)

H. Kleinert, PATH INTEGRALS 19.5 Path Integral for Spin-1/2 Particle 1429 with the action of a relativistic spinless particle [the action (19.333) without the spin coupling]

S h(λ) e 2 ¯[x, p]= dλ px˙ + p A M 2c2 , (19.345) A Z0 (− 2Mc " − c − #) and an action involving the Grassmann ﬁelds θµ:

S µ ih¯ ˙µ ihe¯ µ ν G[θ , A]= dλ θµ(λ)θ (λ)+ h(λ) 2 Fµν(x(λ))θ (λ)θ (λ) . (19.346) A Z0 (− 4 4Mc ) This follows directly from Eq. (7.513). The function h(λ) is the same as in the bosonic actions (19.14) and the path integral (19.22) guaranteeing the reparametrization invariance (19.13). After integrating out the momentum variables in the path integral (19.344), the canonical action is of course replaced by the conﬁguration space action (19.218). In the simplest gauge (19.23), the total action reads

S µ Mc 2 e h¯ µ ν Mc ih¯ ˙µ ¯[x, θ ]= dλ x˙ xA˙ i Fµν θ θ + θµ(τ)θ (τ) . A Z0 "− 2 − c − 4Mc ! − 2 4 # (19.347) The Grassmann variables can be integrated out using formula (19.103), which reads here [compare (7.501)]:

i µ µ ν 4 dt −iθµ(t)θ˙ (t)+(ie/4Mc)Fµν θ θ 1/2 ie θ e 4 [ ] = 4Det iδ ∂ + F (x(λ)) . (19.348) D − µν t Mc µν Z R

For a constant ﬁeld tensor Fµν and with the usual antiperiodic boundary conditions in the interval ct = λ (0,S), the right-hand side has been given before in Eq. (19.103). In the present∈ case it reads:

1/2 e e S e S 4Det igµν ∂λ + i 2 Fµν = 4 cos 2 cosh 2 . (19.349) − Mc Mc B 2 Mc E 2 19.5.4 Effective Action in Electromagnetic Field In the absence of electromagnetism, the effective action of the fermion orbits is given by (19.304). Its Euclidean version differs from the Klein-Gordon expression in (19.38) only by a factor 2: − Γf e,0 = 2 Tr log h¯2∂2 + M 2c2 . (19.350) h¯ − − h i Explicitly we have from (19.39), (19.41), and (19.46):

f ∞ Γe,0 dβ −βMc2/2 1 VD 1 =2VD e D =2 D D/2 Γ(1 D/2). (19.351) h¯ 0 β 2 C (4π) − Z 2πh¯ β/M λM q 1430 19 Relativistic Particle Orbits

The factor 2 may be thought of as 4 1/2 where the factor 4 comes from the free path integral over the Grassmann ﬁeld,×

Dθ e−Ae,0[θ]/¯h =4. (19.352) Z D counts the four components of the Dirac field. Recall that by (19.286), the Dirac field carries four modes, one of energyhω ¯ k, with two spin degrees of freedom, the other of energy hω¯ k with two spin degrees. The latter are shown in quantum field theory to correspond− to an antiparticle with spin 1/2. The path integral over x (λ) which counts paths in opposite directions with the ground state energy (19.39) describes particles and antiparticles [recall the remarks after Eq. 19.45]. This explains why only the spin factor 2 remains in (19.351). By including the vector potential via the minimal substitutionp ˆ pˆ (e/c)A, we → − obtain the Euclidean effective action from Eq. (19.221), and thus obtain immediately the path integral representation

f ∞ Γ˜ dβ 2 e =2 e−βMc /2 Dx e−Ae/¯h, (19.353) h¯ Z0 β Z D with the Euclidean action (19.220). This is not yet the true partition function Γe of the spin-1/2 particle, since the proper path integral contains the additional Grassmann terms of the action (19.347). In the Euclidean version, the full interaction is

¯hβ e i µ ν e,int[x, θ]= dλ i x˙ µ(λ)Aµ(x(λ)) Fµν(x(λ))θ (λ)θ (λ) . (19.354) A Z0 c − 4M Thus we obtain the path integral representation

f ∞ Γ dβ 2 e =2 e−βMc /2 Dx Dθ e−{Ae,0[x,θ]+Ae,int[x,θ]}/¯h, (19.355) h¯ Z0 β Z D Z D where the free part of the Euclidean action is

¯hβ ¯hβ M 2 h¯ µ µ e,0[x, θ]= e,0[x]+ e,0[θ] dτ x˙ (τ)+ dτ θ (τ)θ˙ (τ). (19.356) A A A ≡ Z0 2 Z0 4 19.5.5 Perturbation Expansion The perturbation expansion is a straightforward generalization of the expansion (19.224):

f f ∞ ∞ n Γe Γe,0 dβ −βMc2/2 2VD ( ie/c) = + e D − (19.357) h¯ h¯ 0 β 2 n! Z 2πh¯ β/M nX=1 n ¯hβ q h¯ µ ν dτi x˙ µ(τi)Aµ(x(τi)) Fµν (x(τi))θ (τi)θ (τi) . × * ( 0 " − 4Mc #) + iY=1 Z 0

H. Kleinert, PATH INTEGRALS 19.5 Path Integral for Spin-1/2 Particle 1431

The leading free effective action coincides, of course, with the n = 0 -term of the sum [compare (19.351)]. The expectation values are now defined by the Grassmann extension of the Gaus- sian path integral (19.225): Dx Dθ [x, θ] e−Ae,0[x,θ]/¯h [x, θ] 0 D D O . (19.358) hO i ≡ Dx e−Ae,0[x]/¯h Dθ e−Ae,0[θ]/¯h RD R D R R D where the denominator is equal to (1/2)V / 2πh¯2β/M 4. D × There exists also an expansion analogousq to (19.227), where the vector potentials have been Fourier decomposed according to (19.226). Then we obtain an expansion just like (19.227), except for a factor 2 and with the expectation values replaced as follows: − n ¯hβ µi ikix(τi) dτi x˙ (τi)e * 0 + iY=1 Z 0 n ¯hβ µi ih¯ νi νi µi ikix(τi) dτi x˙ (τi)+ ki θ (τi)θ (τi) e . (19.359) → * 0 " 2Mc # + iY=1 Z 0 The evaluation of these expectation values proceeds as in Eqs. (19.228)–(19.239), except that we also have to form Wick contractions of Grassmann variables which have the free correlation functions θµ(τ)θν (τ ′) =2δµν Ga (τ τ ′), (19.360) h i ω,e − where 1 Ga (τ τ ′)= ǫ(τ), τ [ hβ,¯ hβ¯ ) (19.361) ω,e − 2 ∈ − is the Euclidean version of the antiperiodic Green function (3.109) solving the inho- mogeneous equation a ∂τ Gω,e(τ)= δ(τ). (19.362) Outside the basic interval [ hβ,¯ hβ¯ ) the function is to be continued antiperiodically. in accordance with the fermionic− nature of the Grassmann variables. In operator language, the correlation function (19.360) is the time-ordered ex- ′ ′ pectation value Tˆθˆ(τ)θˆ(τ ) 0 [recall (3.299)]. By letting τ τ once from above and once from below,h the correlationi function shows agreement→ with the anticommutation rule (19.342). In verifying this we must use the fact that the time ordered product of fermion operators is defined by the following modification of the bosonic definition in Eq. (1.241): Tˆ(Oˆ (t ) Oˆ (t )) ǫ Oˆ (t ) Oˆ (t ), (19.363) n n ··· 1 1 ≡ P in in ··· i1 i1 where tin ,...,ti1 are the times tn,...,t1 relabeled in the causal order, so that

tin > tin−1 >...>ti1 . (19.364) The diﬀerence lies in the sign factor ǫ which is equal to 1 for an even and 1 for P − an odd number of permutations of fermion variables. 1432 19 Relativistic Particle Orbits

19.5.6 Vacuum Polarization Let us see how the ﬂuctuations of an electron loop change the electromagnetic ﬁeld action. To lowest order, we must form the expectation value (19.359) for n = 0 and k = k k: 1 − 2 ≡

µ1 ih¯ ν1 ν1 µ1 ikδx(τ1) µ2 ih¯ ν2 ν2 µ2 −ikδx(τ2) x˙ (τ1)+ k θ (τ1)θ (τ1) e x˙ (τ2) k θ (τ2)θ (τ2) e . *" 2Mc # " − 2Mc # +0 (19.365)

µ1 µ2 From the contraction of the velocitiesx ˙ (τ1) andx ˙ (τ2) we obtain again the spinless result (19.240) leading in (19.242) to the integrand

h¯2 k2δµ1ν2 kµ1 kµ2 ˙G 2(τ , τ )= k2δµ1ν2 kµ1 kµ2 (u 1/2)2. (19.366) 1 − 1 1 1 2 1 − 1 1 M 2c2 − In addition, there are the Wick contractions of the Grassmann variables:

h¯ ν1 ν1 µ1 ikδx(τ1) ih¯ ν2 ν2 µ2 −ikδx(τ2) k θ (τ1)θ (τ1) e k θ (τ2)θ (τ2) e *" 2Mc # "2Mc # +0 h¯2 1 = k2δµ1ν2 kµ1 kµ2 ǫ2(τ τ ). (19.367) − 1 − 1 1 M 2c2 4 1 − 2 Since ǫ2(τ τ ) = 1, this changes the spinless result (19.366) to 1 − 2 h¯2 k2δµ1ν2 kµ1 kµ2 ˙G 2(τ , τ )= k2δµ1ν2 kµ1 kµ2 [(u 1/2)2 1/4]. (19.368) 1 − 1 1 1 2 1 − 1 1 M 2c2 − − Remembering the factor 2 in the expansion (19.358) with respect to the spinless − one, we ﬁnd that the vacuum polarization due to ﬂuctuating spin-1/2 orbits is obtained from the spinless result (19.251) by changing the factor 4(u 1/2)2 = (2u 1)2 in the integrand to 2 4u(u 1) = 8u(1 u). The resulting− function − − × − − Π(k2) has the expansion

2 2 γ 2 2 2 2 1 2 M c e h¯ k k Π(k )= log 2 + ǫ, 2 . (19.369) 3π " ǫ − 4πh¯ # − 15πM 2c2 O M 2c2/h¯ !

The ﬁrst term produces a renormalization of the charge which is treated as in the bosonic case [recall (19.271)–(19.274)], which causes an additional contact interaction α α 4α2h¯2 δ(3)(x). (19.370) − r → − r − 15M 2c2

There, the vacuum polarization has the eﬀect of lowering the state 2S1/2, which is the s-state of principal quantum number n = 2, against the p-state 2P1/2 by 27.3 MHz. The experimental frequency shift is positive 1057 MHz [recall Eq. (18.600)], ≈

H. Kleinert, PATH INTEGRALS 19.6 Supersymmetry 1433 and is mainly due to the effect of the electron moving through a bath of photons as calculated in Eq. (18.599). The effect of vacuum polarization was first calculated by Uehling [13], who assumed it to be the main cause for the Lamb shift. He was disappointed to find only 3% of the experimental result, and a wrong sign. The situation in muonic atoms is different. There the vacuum polarization does produce the dominant contribution to the Lamb shift for a simple reason: The other 2 2 effects contain in a factor M/Mµ, where Mµ is the mass of the muon, whereas the vacuum polarization still involves an electron loop containing only the electron mass M, thus being enhanced by a factor (M /M)2 2102 over the others. µ ≈ The calculations for the electron in an atom have been performed to quite high orders [14] within quantum electrodynamics. We have gone through the above calculation only to show that it is possible to re-obtain quantum field-theoretic result within the path integral formalism. More details are given in the review article [5], As mentioned in the beginning, the above calculations are greatly simplified version of analogous calculations within superstring theory, which so far have not produced any physical results. If this ever happens, one should expect that also in this field a second-quantized field theory would be extremely useful to extract efficiently observable consequences. Such a theory still need development [15].

19.6 Supersymmetry

It is noteworthy that the various actions for a spin-1/2 particle is invariant under certain supersymmetry transformations.

19.6.1 Global Invariance Consider ﬁrst the ﬁxed-gauge action (19.347). Its appearance can be made somewhat more symmetric by absorbing a factor h/¯ 2M into the Grassmann variables θµ(τ), so that it reads q

2 µ M 2 e i µ ν Mc M ˙µ ¯[x, θ ]= dτ x˙ xA˙ + Fµν θ θ + iθµ(τ)θ (τ) . (19.371) A Z "− 2 − c 2 − 2 2 # The correlation functions (19.360) of the θ-variables are now h¯ θµ(τ)θν (τ ′) = δµνGf (τ, τ ′) δµν ∆f (τ τ ′), (19.372) h i ≡ M 0 − f ′ ′ f ′ with ∆0(τ τ ) = ǫ(τ τ )/2. In this normalization, G (τ, τ ) coincides, up to a sign, with the− first term− in the derivative ˙G(τ, τ ′) of the bosonic correlation function [recall (19.232) and the first term in (19.234)]. Let us apply to the variables the infinitesimal transformations

δxµ(τ)= iαθµ(τ), δθµ(τ)= αx˙ µ(τ). (19.373) 1434 19 Relativistic Particle Orbits where α is an arbitrary Grassmann parameter. For the free terms this is obvious. The interacting terms change by e iα d4x θ˙µA + F x˙ µθν . (19.374) − c µ µν Z Inserting F x˙ µ(τ)= dA (x(τ))/dτ ∂ [Aµ(x(τ))x ˙ µ(τ)], the first term cancels and µν ν − ν the second is a pure surface term, such that the action is indeed invariant. Supersymmetric theories have a compact representation in an extended space called superspace. This space is formed by pairs (τ,ζ), where ζ is a Grassmann variable playing the role of a supersymmetric partner of the time parameter τ. The coordinates xµ(τ) are extended likewise by defining Xµ(τ) xµ(τ)+ iζθµ(τ). (19.375) ≡ A supersymmetric derivative is defined by ∂ ∂ DXµ(τ) + iζ Xµ(τ)= iθµ(τ)+ iζx˙ µ(τ). (19.376) ≡ ∂ζ ∂τ ! If we now form the integral, using the Grassmann formula (7.379), dζ dζ dτ iX˙ (τ)DXµ(τ)= dτ i x˙(τ)+iζθ˙µ(τ) [iθµ(τ)+ iζx˙ µ(τ)] , (19.377) 2π µ 2π Z Z h i we find dτ x˙ 2 + iθ θ˙µ , (19.378) − µ Z which proportional to the free part of the action (19.347). As a curious property of differentiations in superspace we note that D2Xµ(τ)= ix˙ µ(τ) ζθ˙µ(τ),D3Xµ(τ)= θ˙µ(τ) ζx¨(τ), (19.379) − − − such that the kinetic term (19.377) can also be written as

dζ 3 µ dτ Xµ(τ)D X (τ). (19.380) − Z 2π The interaction is found from the integral in superspace dζ i dτ Aµ(X(τ))DX(τ) Z 2π dζ µ µ ν µ µ = i dτ [A (x(τ)) + i∂ν A (x(τ))θ (τ)] [iθ (τ)+ iζx˙ (τ)] , (19.381) Z 2π which is equal to

µ i µ ν dτ A (τ)x ˙(τ)+ Fµν θ (τ)θ (τ) , − Z 2 thus reproducing the interaction in (19.347). The action in superspace can therefore be written in the simple form

dζ M 3 µ e µ [X]= i dτ Xµ(τ)D X (τ)+ A (X(τ))DX(τ) . (19.382) A Z 2π − 2 c

H. Kleinert, PATH INTEGRALS 19.6 Supersymmetry 1435

19.6.2 Local Invariance A larger class of supersymmetry transformations exists for the action without gauge ﬁxing which is the sum of the free part (19.345) and the interacting part (19.346). Absorbing again the factor h/¯ 2M into the Grassmann variable θµ(τ), and rescaling in addition h(τ) by a factorq 1/c, the reparametrization-invariant action reads

h(τ) e 2 ¯[x,p,θ,h]= dτ px˙ + p A M 2c2 A Z (− 2M " − c − # M ˙µ e µ ν + iθµ(τ)θ (τ) ih(τ) Fµν(x(τ))θ (τ)θ (τ) . (19.383) 2 − c Let us now compose the action from invariant building blocks. For simplicity, we ignore the electromagnetic interaction. In a ﬁrst step we also omit the mass term. The extra variable h(τ) requires an extra Grassmann partner χ(τ) for symmetry, and we form the action

h(τ) 2 M µ i µ ¯1[x, p, θ, h, χ]= dτ px˙ + p + iθµ(τ)θ˙ (τ)+ χ(τ)θ (τ)pµ(τ) . (19.384) A Z (− 2M 2 2 ) This action possesses a local supersymmetry. If we now perform τ-dependent versions of the supersymmetry transformations (19.373)

δxµ = iα(τ)θµ, δθµ = α(τ)p, δp =0, δh = iα(τ)χ, δχ = 2α ˙ (τ). (19.385)

If we integrate out the momenta in the path integral, the action (19.384) goes over into 2 x˙ M µ i µ ¯1[x, θ, h, χ]= dτ + iθµ(τ)θ˙ (τ)+ χ(τ)θ (τ)x ˙ µ(τ) , (19.386) A Z (−2h(τ) 2 2h(τ) ) where a term proportional to χ2(τ) has been omitted since it vanishes due to the nilpotency (7.375). This action is locally supersymmetric under the transformations α(τ) i δxµ = iα(τ)θµ, δθµ = x˙ χ θµ , h(τ) − 2 δh = iα(τ)χ, δχ = 2α ˙ (τ). (19.387)

We now add the mass term

1 2 M = dτ h(τ)Mc . (19.388) A −2 Z This needs a supersymmetric partner to compensate the variation of under Am (19.387). i = dτ θ (τ)θ˙ (τ)+ Mcχ(τ)θ (τ) . (19.389) A5 2 5 5 5 Z h i 1436 19 Relativistic Particle Orbits

Indeed, add to (19.387) the transformation

δθ5 = Mcα(τ), (19.390) we see that the sum + is invariant. Adding this to (19.384), we obtain the AM A5 locally invariant canonical action

h(τ) 2 h(τ) M ˙µ ˙ ¯[x,p,θ,θ5, h, χ]= dτ px˙ + p Mc + i θµ(τ)θ (τ)+ θ5(τ)θ5(τ) A (− 2M − 2 2 Z h i i µ + χ(τ)[θ (τ)pµ(τ)+ Mcθ5(τ)] . (19.391) 2 Appendix 19A Proof of Same Quantum Physics of Modiﬁed Action

Consider the sliced path integral for a relativistic point particle associated with the original action (19.12). If we set the initial and ﬁnal paramters λa and λb equal to λ0 and λN+1, and slice the λ-axis at the places λn (n =1, 2,...,N), the action becomes

N+1 = Mc x x , (19A.1) Acl,e | n − n−1| n=1 X 2 where xn xn−1 = (xn xn−1) are the Euclidean distances [recall (19.2)]. The Euclidean amplitude| − for the| particle to− run from x = x to x = x is therefore given by the product of p a 0 b N+1 integrals in D spacetime dimensions

N N+1 D −Mc |xn−xn−1|/h¯ (x x )= d x e n=1 . (19A.2) b| a N n n=1 Y Z P As a consequence of the reparamterization invariance of (19.12), this expression is independent of the thickness λn λn−1 of the slices, which we shall denote by rename as hnǫ, where ǫ is some ﬁxed small number.− ≡ We now factorize the exponential of the sum into a product of N +1 exponentials and represent each factor as an integral [using Formulas (1.347) and (1.349)]

∞ ǫMc 2 e−Mc|xn−xn−1|/h¯ = dh h−1/2e−ǫhnMc/2¯h−Mc(xn−xn−1) /2hnǫh¯ . (19A.3) 2π¯h n n r Z0 Absorbing constants in the normalization factor , we arrive at N N+1 ∞ N+1 ( 1) 2 Σ n 2 (x x ) = dh h D− / e−Mcǫ n=1 h / b| a N n n n=1 0 Y Z N D N+1 2 1 d xn Σ ( n n− ) 2 n e−Mc n=1 x −x 1 / h ǫ, (19A.4) × D D 2πǫh /Mc n=1 " 2πǫh /Mc # N+1 Y Z n

The second line containsp only harmonic integralsp over xn, which can all be done with the help of the formulas of Appendix 2B, with the result

2 1 −Mc(xb−xa) /2Sh¯ (xbS xa0) = D e , (19A.5) | 2πS¯h/Mc p H. Kleinert, PATH INTEGRALS Appendix 19A Proof of Same Quantum Physics of Modiﬁed Action 1437 where S is the total parameter length

N+1 S ǫh , (19A.6) ≡ n n=1 X Replacing (19A.5) by its Fourier representation [compare with (1.333) and (1.341)], we can rewrite (19A.4) as

N+1 ∞ D d p 2 (x x ) = dh h(D−1)/2 e−McS/2¯h e−Sp /2Mc+ip(xb−xa)/h¯ . (19A.7) b| a N n n (2π¯h)D n=1 0 Y Z Z Thus we are left with the product of integrals over hn. Before we can perform these, we must make sure to respect the sum (19A.6). This is done by inserting an auxiliary unit integral that separates out an integral over the total length S:

∞ ∞ ∞ N+1 N+1 dS dσ (Σ n ) 2 C 1= dS δ(Σ ǫh S)= e−σ n=1 ǫh −S / λ , (19A.8) n=1 n − 2λ 2πi Z0 Z0 C Z−i∞ where λC is the Compton wavelength (19.31). Then the product of integrals over hn in the brackets of (19A.7) can be rewritten as follows:

N+1 ∞ dS i∞ dσ ∞ eσS/2λC dh h(D−1)/2e−σǫhn/2λC e−S/2λC (19A.9) 2λ 2πi n n 0 C ( −i∞ n=1 0 ) Z Z Y Z 2 Setting hn = rn, we treat the curly backets as

i∞ N+1 ∞ dσ 2 eσS/2λC dr rDe−σǫrn/2λC 2πi n n −i∞ n=1 −∞ Z Y Z 2πλ N+1 i∞ dσ = C eσS/2λC σ−(N+1)(D+1)/2. (19A.10) Γ(D+1)ǫ 2πi Z−i∞ For large N, the integral over σ can be approximated by the Gaussian integral around the neigh- borhood of the saddle point at σ = (N + 1)(D + 1)λC /S:

i∞ dσ 1 S (N+1)(D+1)/2 eσS/2λC σ−(N+1)(D+1)/2 . (19A.11) 2πi large≈ N √2π (N + 1)(D + 1)λC Z−i∞ While letting N tend to inﬁnity, we keep S/(N + 1) ¯ǫ ﬁxed. Then we may write the right-hand side as an exponential ≡

1 (D + 1)λ −S(D+1)/2¯ǫ 1 C = e−zS/2λC , (19A.12) √2π ǫ¯ √2π where

z ν log ν with ν (D + 1)λ /ǫ¯ (19A.13) ≡ ≡ C is a large number. Inserting (20.246) back into the curly brackets of (19A.9), the constant z can be absorbed into the mass of the particle by replacing M by the renormalized quantity M1 = M(1+z). With this, (19A.9) becomes

∞ dS e−SM1c/2¯h, (19A.14) 2λ Z0 C 1438 19 Relativistic Particle Orbits and the path integral (19A.4) reduces to

∞ D dS d p 2 (x x ) = ′′ e−M1cS/2¯h e−Sp /2Mc+ip(xb−xa)/h¯ , (19A.15) b| a N 2λ (2π¯h)D Z0 C Z where all irrelevant factors are contained in the normalization factor ′′. The remaining integral over S can now be done and yields N

D ′′ d k 1 ik(xb−xa) (xb xa)= e , | N (2π)D k2 + M 2 c2/¯h2 Z R where MR is the renomalized mass

1/2 MR = M(1 + z) . (19A.16)

If we choose the normalization factor ′′ = 1, this is exactly the same result as that obtained before in Eq. (19.30) from the modiﬁedN action (19.10), if we use in the original action (19.12) the mass M/(1 + z)1/2 rather than M for the calculation.

Notes and References

Relativistic quantum mechanics is described in detail in J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York, 1964, relativistic quantum field theory in S.S. Schweber, Introduction to Relativistic Quantum Field Theory, Harper and Row, New York, 1962; J.D. Bjorken and S.D. Drell, Relativistic Quantum Fields, McGraw-Hill, New York, 1965; C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, New York, 1985. The individual citations refer to [1] For the development and many applications see the textbook H. Kleinert, Gauge Fields in Condensed Matter, World Scientific, Singapore, 1989; Vol. I, Superflow and Vortex Lines (Disorder Fields, Phase Transitions); Vol. II, Stresses and Defects (Differential Geometry, Crystal Melting) (wwwK/b1, where wwwK is short for (http://www.physik.fu-berlin.de/~kleinert). [2] See Vol. I of the textbook [1] and the original paper H. Kleinert, Lett. Nuovo Cimento 35, 405 (1982) (ibid.http/97). The theoretical prediction of this paper was confirmed only 20 years later in S. Mo, J. Hove, A. Sudbo, Phys. Rev. B 65, 104501 (2002) (cond-mat/0109260); Phys. Rev. B 66, 064524 (2002) (cond-mat/0202215). [3] R.P. Feynman, Phys. Rev. 80, 440 (1950). [4] There are basically two types of approach towards a worldline formulation of spinning particles: one employs auxiliary Bose variables: R.P. Feynman, Phys. Rev. 84, 108 (1989); A.O. Barut and I.H. Duru, Phys. Rep. 172, 1 (1989); the other anticommuting Grassmann variables: E.S. Fradkin, Nucl. Phys. 76, 588 (1966); R. Casalbuoni, Nuov. Cim. A 33, 389 (1976); Phys. Lett. B 62, 49 (1976); F.A. Berezin and M.S. Marinov, Ann. Phys. 104, 336 (1977); L. Brink, S. Deser, B. Zumino, P. DiVecchia, and P.S. Howe, Phys. Lett. B 64, 435 (1976); L. Brink, P. DiVecchia, and P.S. Howe, Nucl. Phys. B 118, 76 (1976). These worldline formulations were used to recalculate processes of electromagnetic and

H. Kleinert, PATH INTEGRALS Notes and References 1439

strong interactions by M.B. Halpern, A. Jevicki, and P. Senjanovic, Phys. Rev. D 16, 2476 (1977); M.B. Halpern and W. Siegel, Phys. Rev. D 16, 2486 (1977); Z. Bern and D.A. Kosower, Nucl. Phys. B 362, 389 (1991); 379, 451 (1992); M. Strassler, Nucl. Phys. B 385, 145 (1992); M.G. Schmidt and C. Schubert, Phys. Lett. B 331, 69 (1994); Nucl. Phys. Proc. Suppl. B,C 39, 306 (1995); Phys. Rev. D 53, 2150 (1996) (hep-th/9410100). For many more references see the review article in Ref. [5]. [5] C. Schubert, Phys. Rep. 355, 73 (2001); G.V. Dunne, Phys. Rep. 355, 73 (2002); [6] As a curiosity of history, Schrödinger invented first the relativistic Klein-Gordon equation and extracted the Schrödinger equation from this by taking its nonrelativistic limit similar to Eqs. (19.34) and (19.35). [7] In particle physics, the Chern-Simons theory of ribbons explained in Section 16.7 was used to construct path integrals over fluctuating fermion orbits: A.M. Polyakov, Mod. Phys. Lett. A 3, 325 (1988). For more details see C.H. Tze, Int. J. Mod. Phys. A 3, 1959 (1988). [8] W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936). English translation available at wwwK/files/heisenberg-euler.pdf. J. Schwinger, Phys. Rev. 84, 664 (1936); 93, 615; 94, 1362 (1954). [9] E. Lundström, G. Brodin, J. Lundin, M. Marklund, R. Bingham, J. Collier, J.T. Mendonca, and P. Norreys, Phys. Rev. Lett. 96, 083602 (2006). [10] A. Vilenkin, Phys. Rev. D 27, 2848 (1983). [11] See the internet page http://super.colorado.edu/~michaele/Lambda/links.html. [12] The path integral of the relativistic Coulomb system was solved by H. Kleinert, Phys. Lett. A 212, 15 (1996) (hep-th/9504024). The solution method possesses an inherent supersymmetry as shown by K. Fujikawa, Nucl. Phys. B 468, 355 (1996). [13] E.A. Uehling, Phys. Rev. 49, 55 (1935). [14] T. Kinoshita (ed.), Quantum Electrodynamics, World Scientific, Singapore, 1990. [15] For a first attempt see H. Kleinert, Lettere Nuovo Cimento 4, 285 (1970) (wwwK/24). New developments can be traced back from the recent papers I.I. Kogan and D. Polyakov, Int. J. Mod. Phys. A 18, 1827 (2003) (hep-th/0208036); D. Juriev, Alg. Groups Geom. 11, 145 (1994); R. Dijkgraaf, G. Moore, E. Verlinde, and H. Verlinde, Comm. Math. Phys. 185, 197 (1997).