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A. Graviton action

The source of a massless, helicity-2 graviton is a conserved, symmetrical stress-tensor,

µν µν νµ ∂µT =0,T = T , (3.1) from which the generating function for all the Green’s functions, the vacuum persistence amplitude, follows:

T iW [T ] 1 ′ µν ′ ′ 1 ′ ′ 0+ 0− = e , W [T ]= (dx)(dx ) T (x)D+(x x )Tµν (x ) T (x)D+(x x )T (x ) , (3.2) h | i 2 Z  − − 2 −  which describes the free propagation of gravitons between sources. Here appears the causal or Feynman massless propagator,

′ ip(x−x ) ′ (dp) e D+(x x )= , (3.3) − Z (2π)4 p2 iǫ − 2 0 2 µ where p = (p ) + p p and T is the trace of the graviton source tensor, T (x)= T µ(x). The above is expressed in natural units;− to connect· to the real world, we rescale the source:

T µν = √κtµν , κ =8πG, (3.4) where G is Newton’s constant. Because gravity is of infinite range, the graviton should be massless. However, Schwinger showed how you can start with a massive spin-2 particle, of m, which has 5 helicity states. Then provided we define

µν m ν µ 1 ∂µT = J , ∂µJ = m √3K T , (3.5) √2  − √2  µν µ in the limit m 0, ∂µT = 0, ∂µJ = 0, and the action decouples into independent helicity 2, 1, and 0 components, represented by→ a tensor source T µν , a vector source J µ, and a scalar source K. As explained, for example, in Ref. [8], the coefficients in Eq. (3.5) allow for the decomposition into the three helicity components, and the count of states between those for the massive tensor description, and the massless helicity 2, 1, and 0 description is 5 = 2 + 2 + 1, suggesting a correspondence with the graviton, the photon, and a massless scalar.

B. Tests of

In the paper [7] on the source theory of gravity, and in his more detailed discussion in Particles, Sources, and Fields [8], Schwinger rederives, simply, just starting from the expression for the graviton action given in Eq. (3.2), the standard tests of general relativity: • The gravitational red shift, • the light deflection by the sun, • the delay in radar echos from planets, • the precession of Mercury’s perihelion. These arguments, or ones very similar thereto, of course had been supplied earlier by others. For a bibliography of tests of gravity, see Ref. [11]. In a couple of short articles in the American Journal of Physics a few years later (1974) [9, 10], Schwinger extended these elementary derivations to precession tests, which had not been performed up to that time: The Thirring effect, the precessional angular velocity associated with a rotating shell of radius R, mass M, and angular velocity ω: 4 GM ω = ω, (3.6) prec 3 R 3 and the Schiff effect, the precession (precessional velocity Ω) of a gyroscope in a satellite in orbit around a planet:

3 GM GI Ω = r v + (3rω r ω r2). (3.7) 2 r3 × r5 p · − p

Here r is the radius of the orbit, v is the velocity of the satellite, and ωp is the angular velocity of the planet, which has moment of inertia I. I followed these papers up with a simple source theory rederivation of the Lense-Thirring effect [12], the effect of the spin of the sun (mass M) on the motion of a planet (mass m) orbiting it, in terms of the precession of its axial vector1 r M + m A = p L, (3.8) r − GM 2m2 × where r is the position of the planet relative to the sun, p is the relative momentum, and L is the orbital angular momentum. The precession equation is dA = Ω A, (3.9) dt A × where 2G 3ω (ω S) Ω S p p A = 3 2 · , (3.10) r  − ωp  where S is the spin of the sun, and ωp is the angular velocity of the planet. These “frame-dragging” and “geodetic” effects have now been confirmed by Gravity Probe B [15], although the Lense-Thirring effect was earlier seen by the LAGEOS experiment [16].

C. But Einstein’s General Relativity emerges

The theory proposed by Schwinger to this point is nothing but linearized gravity, with some extrapolation to include gravity itself as a source of . The symmetric tensor gravitational field hµν is related to the metric tensor gµν of general relativity and the flat- Minkowski metric η = diag( 1, 1, 1, 1) by µν −

gµν = ηµν +2hµν , (3.11) where hµν is regarded as a small perturbation. (Schwinger inserted the factor of 2 to simplify the equations of motion for hµν .) But Schwinger was quite aware this was inadequate. In his paper [7], and especially in the last section of Vol. I of his book [8], he recognizes that gravitational gauge invariance,

h h + ∂ ξ + ∂ ξ , (3.12) µν → µν µ ν ν µ where the vector field ξµ is arbitrary, is necessarily generalized to general coordinate invariance, and in that way he was led inexorably to Einstein’s equations,

1 R (x)= κ t (x) g t(x) , (3.13) µν  µν − 2 µν  in terms of the usual Riemann curvature tensor, and to the Einstein-Hilbert action,

µν W = (dx)[Lm + Lg], 2κLg = g(x)g (x)Rµν (x), (3.14) Z p− where Lm is the matter Lagrangian.

1 Often referred to as the Laplace-Runge-Lenz vector, or by some subset of those names, but actually first discovered by Hermann in 1710 [13], and generalized by Bernoulli in the same year [14]. 4

D. Scalar-tensor gravity

He did go on to notice that conformal symmetry is spoiled by this theory, which motivated him to develop his own version of scalar-tensor gravity, which for weak coupling agrees with Brans-Dicke theory [17], but which could have different consequences in the cosmological domain [18].

IV. AND

Later in the 1970s, Schwinger was chagrined that he had not come up with the idea of supersymmetry [19], since he had developed the multispinor formalism that naturally allowed the treatment of all spins on the same footing [8]. He invited his former student, , for a private audience with Schwinger’s group of students, postdocs, and faculty, and we were initiated into the mysteries of supersymmetry and supergravity [20]. A paper by Schwinger giving a simple rederivation of supersymmetric ideas followed [21]. Bob Finkelstein, Luis Urrutia, and I immediately continued with a paper in which we showed that supergravity emerged by requiring solely invariance under local supersymmetry [22].

V. SCHWINGER’S ATTITUDE TOWARD GENERAL RELATIVITY

Schwinger certainly had no quibble with general relativity, and provided an alternative derivation of Einstein’s equations. In this he behaved analogously to Richard Feynman who, in his Polish lecture, also provided a derivation of general relativity [23]. In fact, Schwinger privately stated that he thought he would have discovered general relativity had he been in Einstein’s place. It is true that Schwinger’s approach was always algebraic, and consequently he had little use for the geometric interpretation of curved space. Yet this is merely an interpretation of the theory, and had no effect on testable consequences. Schwinger did not believe in a massive graviton, and certainly not in tachyons.

VI. SCHWINGER’S ATTITUDE TOWARD THE UNSEEN UNIVERSE

Schwinger never expressed an opinion on , to my knowledge. In those days, dark matter, manifested by galactic rotation curves, was largely only of interest to astronomers, and not to physicists in general [24]. Of course, he didn’t know about “cosmic acceleration” [25], so he likely thought that the was zero. Since he became very fascinated with the Casimir effect [26], I’d like to imagine he would have thought that dark energy originated from a nonzero cosmological constant, arising from quantum fluctuations. Such ideas go back at least to Pauli [27], and for a history of the connection of the ideas of zero-point energy and the cosmological constant see Kragh [28]. Particularly noteworthy are the contributions of Gliner [29], Zel’dovich [30], and Sakharov [31]. See also Ref. [32]. An idea along these lines was proposed some time ago [33]. For example, quantum fluctuations of fields in “large” compactified dimensions would give rise to a cosmological constant: (a = size of compact space of dimension d) Λ T µν = ugµν = gµν . (6.1) h i − −8πG Roughly speaking, the quantum vacuum energy of the fluctuations must have the form γ α ln a/L u = d , d odd, u = d Pl , d even. (6.2) a4 a4 for odd and even compactified dimensions d, respectively, where LPl is the Planck length. The coefficients αd and γd depend on the fields compactified [34]. We must require that the density of dark energy be less than the critical density that would close the universe, which for a reduced Hubble constant of h0 =0.7 corresponds to a length scale of 80 µm, which leads to a γ1/480 µm, a [α ln(a/L )]1/480 µm, (6.3) ≥ ≥ Pl which are nearly excluded by the E¨otWash experiment [35], a 44µm. (6.4) ≤ See Table I, which is reproduced from Ref. [33]. These ideas have beeb more recently elaborated, for example, in Refs. [36] and [37]. 5

S Gravity Scalar Fermion Vector S1 (u) * * 9.5 µm — S1 (t) 9.9 µm 6.6 µm * — S2 84 µm* * * S3 — 7.5 µm 9.5 µm — S4 * * * 77 µm S5 — 11.5 µm* — S6 350 µm * * 110 µm S7 — 13.5 µm 7.0 µm —

TABLE I: The lower limit to the radius of the compact dimensions deduced from the requirement that the Casimir energy not exceed the critical density. The numbers shown are for a single species of the field type indicated. The dashes indicate cases where the Casimir energy has not been calculated, while asterisks indicate (phenomenologically excluded) cases where the Casimir energy is negative.

VII. CONCLUSIONS

Although Schwinger approached gravity from a particle-physics viewpoint, he never expressed any doubt about the validity of general relativity. He thought algebraically, not geometrically, so he didn’t find the notions of curved space useful. Many of his students have made major contributions to the quantum theory of gravity (for example, beside Arnowitt and Deser mentioned above, one cannot forget the contributions of Bryce DeWitt [38] or of David Boulware [39]), which, although still not fully developed [40], must reduce to Einstein’s theory under most circumstances.

Acknowledgments

I thank the Laboratoire Kastler Brossel for their hospitality, particularly Astrid Lambrecht and Serge Reynaud. CNRS is thanked for their support. This work was further supported in part by a grant from the Simons Foundation; earlier work summarized here was supported by grants from the US Department of Energy. I thank Ron Kantowski and Yun Wang for earlier collaborations on the dark energy scheme. I thanks Stanley Deser for critical comments on the first version of this paper.

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