The Feynman Lectures on Gravitation

Here is an interesting nugget that I never saw before in any other textbook about how the exchange of virtual particles causes a static force like the the 1/r^2 Coulomb potential.

1.In order to get a force out of an exchange, and not only scattering, it is necessary that the diagram including the exchange should be able to interfere with the diagonal terms in the scattering amplitude, that is, it should add to the amplitude that nothing happens.

2. Thus the possibility of one neutrino exchange is rules out by the fact that a half unit of angular momentum cannot be emitted by an object that remains in the same internal state as it started in. p.24 Lecture 2

Let |0) be the vacuum of the field of force quanta, and let |1) be the Fock state with one force quantum. Note at this state we have to treat the force quantum as "real" on the mass shell pole of the force field propagator. The virtual nature of the force quantum is not apparent until we also put in the absorption of the force quantum by the second sink. This is because , for a single mode, the vacuum already has one zero point virtual force quantum in it. We need orthogonality of |0) and |1) to make the argument and this requires a real force quantum in |1) in addition to the omnipresent virtual force quantum.

So, in general, we have the entangled pair state

Now select out the term where J' = J in the sum over J' which when added to the second term on the RHS gives the coherent unentangled fragment

For this fragment the source and the force field each have their own effective wave function. Let's say it another way using the "mixed" reduced density matrix of the force quanta from the trace over the source states of the total "pure" projection operator |psi)(psi|. The J'= J source contribution makes off-diagonal elements in the reduced density matrix for the force quanta and this, according to Feynman, is required for there to be a static Coulomb force in the classical limit. Therefore, without quantum coherence there would be no classical electrical, and presumably also, no long-range Newtonian gravitational force.

Problem 1. Justify the above remark P1.

Newsgroups: sci.physics

Subject: Re: How do opposite charges attract due to exchange of photons?

In mcirvin@scws5.harvard.edu (Matt McIrvin) writes:

What the charge of an object affects is the sign of the factor that comes into the probability amplitude for interaction,... for like charges the processes with odd and even numbers of photons interfere so that the wave packets move away from each other, and for opposite charges they interfere so that the packets move toward each other.

So this does explain it pretty well using the Feynman pictures.

Yes, Matt you made a valuable contribution. The picture of constructive (repulsion) and destructive (attractive) interference of the two diagrams is pretty.

P2 The 1/r dependence is from the fact that "any radial dependence us determined exclusively by the mass" (p.30)

Problem 2. Justify the above remark P2.

P3 "one consequence of spin 1 is that likes repel and unlikes attract. This is in fact a property of all odd spin theories; conversely, it is also found that even spins lead to attractive forces. "

Problem 3. Justify the above remark. P3

"... the rejection of spin-zero theories of gravitation is made on the basis of the gravitational behavior of binding energies .."

Feynman shows that the spin-zero theory predicts that the gravity attraction between hot regions of gas is smaller than for cool regions. In fact, the opposite is true.

Spin 0 X Lorentz Group Scalar Potential

Spin 1 Au Lorentz Group First Rank Tensor

Spin 2 huv Lorentz Group Second Rank Tensor

Note Spin 1/2 described by a half-tensor or "square root" spinor. So a first rank tensor is a second rank spinor etc. A first rank tensor is also called a four-vector.

The antisymmetric part of huv does not lead to a theory that resembles gravity, "but rather something resembling electromagnetism; the six independent components of the antisymmetric tensor would appear as two space vectors" (p.32)

*Note: Sum over repeated indices. The limits of the net require me not to keep track of upper and lower tensor and spinor indices.

Look at quantum electrodynamics in the Fourier- transformed flat spacetime. The vector four vector potential Au(k) which creates and destroys photons, is related to the source electron four-vector current density ju(k) by

where the relativistic invariant

w is the photon frequency, K is the photon wave number (c and hbar = 1 in dimensionless units etc). A real photon (whose world line is on the light cone) corresponds is the "pole" (i.e., zero of the denominator -- from math of functions of a complex variable). A virtual photon is not a pole. In fact, as Feynman emphasizes, all photons that are emitted and then absorbed are, strictly speaking, "virtual".

Note that the "flat" Fourier transform of the D'Alembertian "wave" operator is

"We compute amplitudes for such processes as a function of the relativistic invariants, and restrict our answer as demanded by rules of momentum and energy conservation. The guts of electromagnetism are contained in the specification of the interaction between a current and a field as juAu" (p.32)

Using eq. (3.2.1) above

which corresponds to the single virtual photon (the solid horizontal line, ignore the dotted horizontal lines- we will fix this with a jpg later) exchange Feynman diagram

Choose a 4D Cartesian frame of reference in which the z-axis is the direction of propagation of a photon in a plane wave mode. Therefore,

where the notational order is from

Therefore, the elementary high school algebra of the Feynman diagram in Fig. 3.2 above is the current-current interaction

Local conservation of electric charge current density is

* Remember this is in 4D k-space. To get conservation of total electric charge Q, one has to Fourier transform ju(k) back to 4D spacetime, select a 3D spacelike surface, and integrate the timelike component j4(x), which is the charge density, over the support of the source field over the spacelike surface. The result is dQ/dt = 0, where t is defined in terms of the normal to the spacelike surface. Relativistic causality, i.e., no Heisenberg uncertainty relations between values of j4(x) at different points on the spacelike surface, plays an essential role. What would happen if there were faster-than-light source tachyons has not been seriously studied by the big shots and is a good research problem.

* Hint:

"Feynman's 'plan of attack' led him to an important discovery in field theory, namely the need for covariant ghosts to maintain unitarity at one-loop." Brian Hatfield p. xxxi

These ghosts violate the usual spin-statistics connection so that scalars and vectors are fermions that obey the Pauli exclusion principle, and spinors are bosons. Gerald Feinberg showed that tachyons violate the spin- statistics connection.

Problem 4 Does this have any connection to the conjectured missing mass or "shadow matter" of the universe and to the exotic matter needed to maintain traversable wormhole star gates for our navy and for possibly alien UFOs?

Part 3 Synopsis of this part.

The amazing counter-intuitive result is that gauge invariance and Lorentz invariance conspire so that the non-radiative near field is instantaneous in every frame of reference -- at least in the case that the Lorentz boost is in the same direction as the virtual plane wave photon propagation direction!

Problem 5 How general is this result? Is it still true if the Lorentz boost is not in the direction of propagation or if the virtual photon is in a localized wave packet?

The local conservation of electric current density (eq. 3.2.6) in the special 4D frame of reference is

Substitute this into the current-current interaction (eq. 3.2.5) for the single-photon exchange Feynman diagram to get

So, we see, that the gauge invariance symmetry associated with local conservation of electric current density separates the Lorentz-invariant (i.e., frame-invariant) photon-electron interaction into a faster-than-light frame- dependent instantaneous static (i.e. w = 0) Coulomb potential

whose Fourier transform is

The transverse term

involving electric current density components in the x-y plane perpendicular to the z direction of virtual photon propagation "represents corrections to the instantaneus Coulomb interaction." (p. 33)

The important point is that the Lorentz frame- invariant electron-photon interaction of special relativity does separate out into a faster-than- light instantaneous non-radiative near-field part and a transverse radiative far-field part in each frame.

*The amazing counter-intuitive result is that gauge invariance and Lorentz invariance conspire so that the non-radiative near field is instantaneous in every frame of reference -- at least in the special frame transformation (i.e., Lorentz boost) considered here.

Proof. Lorentz transformation along the z- axis to an inertial frame moving with speed v. Ju is the current density in the new frame.

j1 = J1 transverse x-linear polarization of radiative far-field

j2 = J2 transverse y-linear polarization of radiative far-field.

j3 = gamma[J3 - vJ4] longitudinal z-polarization of non-radiative near-field.

j4 = gamma[J4 - vJ3] timelike t-polarization of non-radiative near-field

Let the wave vector in the new frame be K* and the frequency in the new frame be w*

Therefore,

so that the transverse radiative part keeps its same form since in this special class of frame transformations it consists of invariants. If we were to transform off the z axis the computation is more complicated.

How do we compute the faster-than-light non- radiative near-field potential in the new frame? We simply rewrite

using the transformed variables Doing that gives

Current density conservation in the new frame is

substituting

so that the instantaneous form of the non- radiative near-field is frame invariant at least for this special class of Lorentz transformations in which the boost direction is the same as the propagation direction. QED

Note that

In doing the dw integration for getting the amplitudes in spacetime, Feynman uses a special contour in the complex w-plane such that the negative energy of the real quantum corresponding to the pole at w = -|K| propagates with an advanced wave backwards in time (-|t|), while the positive energy pole at w = +|K| propagates forwards in time. Since the phase advances by the product wt, this choice of contour, or boundary condition on the propagator, implies that the advanced wave of negative energy backwards in time can be replaced by a retarded wave of positive energy moving forward in time.

Feynman's far-field single-quantum amplitude in (3.38) has a spin 0 polarization part which corresponds to

But the trace of the stress-energy tensor for the free electromagnetic field is zero. Therefore, a scalar gravity will not bend light. Since we are only interested now in reproducing Einstein's classical general relativity from the special relativistic quantum field theory, and since the former bends light, Feynman adds an adhoc compensating term to subtract out the scalar polarization. The result of the subtraction is that the "retarded" far-field term is

Each term is then the product of two kinds of polarizations from each vertex of the Feynman diagram.

One type of polarization state is

the other is (via symmetry of T)

so that the purely spin 2 part of the far-field terms is

which is formally identical to the QED result of (3.2.8). Of course,the physical meaning of the polarization for spin 1 and spin 2 is very different.

*Hard Research Problem. It is not at all obvious that if we make a Lorentz boost that this separation into an instantaneous near field part and a delayed radiative far field part will persist in every local inertial frame the way it does in quantum electrodynamics at least when the boost is parallel to the propagation direction. My intuition was surprised at the QED result. I would have wrongly guessed that the near field would only be instantaneous in one frame though it would still be superluminal in all frames. This provides a clue on how to make Bohm's hidden variable theory fully relativistic since what is good for the goose may also be good for the gander. That is, if the Coulomb potential can be instantaneous in every frame because of a synchronization between the gauge transformation and the Lorentz transformation, can a similar trick work for Bohm's quantum potential in the Hamiltonian- Jacobi equation for the particle?

We have been talking about polarization in terms of the matter source stress-energy tensor T not in terms of the weak graviton field h. But recall (3.3.1) which is

Feynman defines

Feynman rewrites this as

so that Prs,uv is the graviton propagator. Recall that nuv is the diagonal flat Minkowski metric whose only nonzero elements are the diagonal ones:

However, Feynman also gives an alternate version keeping the graviton propagator 1/k^2 and using the pure spin 2 tensor

This huv describes the emission of a single virtual (i.e, k^2 =/ 0) graviton such that the amplitude for absorption is

In contrast, the amplitude to emit a real graviton with polarization euv, assuming

The physical meaning of the two independent spin 2 transverse polarization oscillations in the x- y plane are: for a fixed instant imagine a maximal compression along the y-axis with a maximal stretch along the x-axis. A half-period later there is maximal stretch along the y axis and a maximal compression along the x-axis. This is one of the polarization modes. The second orthogonal mode is the same but relative to a set of axes x' y'rotated by 45 degrees to the first. One can also look at this mode in terms of the original x and y axes. There is an analogy with two opposing force couples. For example, a counterclockwise torque applied on the x-axis with a clockwise torque applied on the y axis at one point in the cycle etc.

If you rotate by pi/2 each polarization returns to itself shifted by a half-cycle.

The fact that the two orthogonal modes are only 45 degrees apart in the transverse plane in space (for graviton propagation K along z-axis) for spin 2 comes from e^iSpinxangle .= e^iS@. When S = 2 and @ = pi/4, S@ = pi/2, so e^iS@ = i which is a rotation of 90 degrees in the complex plane. But recall that the wave function is a complex number, and the e^iS@ operates on the polarization eigenfunctions. The operator S is a rotation about the direction of propagation of the massless quantum. This rule that the two orthogonal transverse polarization eigenfunctions in Hilbert space for massless particles differ by a factor of i works for spin 1/2 and spin 1. Thus for spin 1/2, @ = pi (neutrino), and for spin 1, @ = pi/2 (photon). The rule also works for massive spin 1/2 particles. That massless particles of any spin only have two orthogonal polarizations is a theorem by Wigner.

In general, for massless particles there are two orthogonal linear combinations whose rotational phase change behaves like e^iS@ and e^-iS@. For spin 1/2 two complete rotations in physical space return the polarization eigenfunction into itself. For spin 1 one rotation in physical space returns the eigenfunction to itself. A half rotation also does it for spin 1. For spin 2 a half rotation in physical space also returns the eigenfunction to itself.

Problem. Suppose the spin is 3/2. How many rotations in physical space return the eigenfunction to itself?

Ans 3/2@ = pi/2, so @ = pi/3 or 60 degrees. Therefore, the two polarizations are 60 degrees apart in physical space.

So the two polarizations are 180 degrees apart in physical space for spin 1/2, 90 degrees for spin 1, 60 degrees for spin 3/2 and 45 degrees for spin 3/2. One needs to rotate by 720 degrees in physical space for the spin 1/2 state in Hilbert space to return to itself. A 360 degree rotation returns the spin 1 state to itself. A 180 degree rotation in physical space.returns the state to itself multiplied by a half-wave phase shift (i.e., e^ipi = -1).

The general formula is

So for spin 3/2, a 360 degree rotation in Hilbert space is a 240 degree rotation in physical space. So a 240 degree rotation in physical space returns the spin 3/2 state to itself.

A 180 degree rotation in physical space returns the spin 2 state to itself. A 90 degree rotation in physical space turns the spin-2 state to itself multiplied by a half-wave phase shift.

Problem: What do the two polarization patterns look like for spin 3/2?

Recall that the the basic graviton-matter coupling divides into a non-radiative near-field instantaneous part that depends on the T44, T41 and T42 matter stress-energy tensor components only, and a radiative far-field delayed (and possibly advanced) purely tansverse part depending only on T11, T22 and T12 components. Conservation of stress-energy flow (i.e., eq.3.3.6 p.36) enables us to eliminate any reference to the longitudinal 3 index which, in our special frame, is the z-axis direction of propagation of the plane wave graviton.

Feynman transforms the far-field delayed part into the equivalent form

Note, if P1 and P2 are the gravity spin-2 analogs to the linear polarization basis for spin 1, then (3.4.1) is the analog to the circular polarization or helicity basis. In terms of spherical harmonics YL(M) we have Y2(+2) and Y2(-2). That is

So the far-field part has the form

multiplied by the virtual graviton propagation factor

*Feynman adds in 1963 that unlike the spin 1 case "we have observed neither gravitons, nor even classical gravity waves." The latter is no longer true since gravity waves from pulsars have been observed. For details see Penrose's two books, The Emperor's New Mind and Shadows of the Mind.

Let us look again at the pure spin-2 coupling. The near-field part is fundamentally faster- than-light in every local frame. In the spin 1 case it is even instantaneous in every frame boost parallel to the direction of propagation of the virtual quantum. This is due to a synchronization between the spacetime Lorentz and the internal U(1) gauge transformation induced by local conservation of charge-current. Virtual particles can and do go both slower and faster than light even when their real form is massless. This subtle but important distinction is glossed over in most text books because of the taboo against speaking words like "faster-than-light".

One can write the near-field part as

For v/c << 1 the pure space components like T11, P1 and P2 are of order (v/c). In the limit (v/c) - > 0 and w -> 0 only the T44'T44/K^2 term is important and that is the Fourier transform of Newton's classical law of gravitation. The terms T41 and T42 are analogs to instantaneous near-field magnetism and they also vanish when (v/c) -> 0.

Feynman writes

"We may from the Lagrangians of fields deduce some important properties, for example, we can understand why gravitation is attractive for likes and unlikes, whereas in electricity likes repel and unlikes attract. It can be shown that this property is inherent in the sign of the Lagrangian, so that if we change Action to - Action, the force changes sign. The sign of the coupling constant ... makes no difference, since it appears as a square in any diagram which represents a correction to the energy; always two vertices are involved. We can change the sign of the energy corresponding to a diagram such as Fig. 4.1 only if we can introduce a factor i at each vertex, for example if we are to use fields iphi rather than phi." p.48

Feynman allegedly then treats the spin 0 Lorentz group real uncharged classical scalar field phi and shows that using iphi gives negative energy which he rejects. The conclusion is that the uncharged scalar field must be attractive.

He allegedly then treats the spin 1 field saying

"For electromagnetic waves, it is the components in the transverse direction, perpendicular to the direction of propagation, which are restricted by a similar consideration. A negative sign appears in the associated energy because the energy involves the space indices in the dot product of two vectors (+ ---)..the sign of the Coulomb forces comes from the sign of the time components in the Lagrangian." p. 49

The argument in the book here is too terse and not logically clear to me at least. If we use iA, where A is the four-vector potential, the the time component of the dot products becomes negative and the space component becomes positive. I presume, Feynman means that this means it's OK to use iA since the two parts of the dot product had opposite signs to begin with.

"For the gravity waves, it is again transverse components that are restricted, but in contracting over two indices (or any even number of indices) the signs cancel out, the sign of the time components h44 is the opposite of the electrical case and we have attractions." p.49

I am not convinced by the above remarks written by one of his students who seems to have left out something. If any one can clarify this please do.

Feynman analyzes the precession of the orbit of the planet Mercury in the classical limit of his linearized theory and finds that the prediction is too large by a factor of 4/3. (p.65) Thus, the nonlinear part of Einstein's geometric theory is quite significant experimentally.

5.2 is "Time Dilation in a Gravitational Field".

Again for Feynman's classical linearized approximation to Einstein's more complete theory:

"If we compare the differential equations of motion of particles in electrical and gravitational fields we find that the gravitational equation has a qualitatively distinct new feature; not only the gradients, but also the potentials themselves appear in the equations of motion. ...

Thus, even though the differential equations for the fields themselves are closely parallel, there is a distinction in the interpretation." p.66

This story is repeated in Bohm's theory of the quantum in which these classical equations when put in Hamilton-Jacobi form have another "qualitatively distinct new feature" -- the nonlocal and context-dependent quantum force.

In the case of gravitation, the quantum force comes from the wave function of the universe which obeys the Wheeler-Dewitt equation that is quite literally beyond time.

Wheeler defines a classical superspace in which each point is a possible three-dimensional spatial geometry 3G of our universe. This 3G plays the role of x, the particle position (i.e., the misnamed "hidden variable") in the simplest example of Bohm's theory. This is not the same as the different branches of the wave function of the universe which define different"parallel universes" that are independent directions or axes in Hilbert space. In Bohm's theory there is only one actual material 3G universe that is guided by the cosmic quantum force from one of the possible independent directions or parallel universes in Hilbert space. That is, there is really only one actually existing path through Wheeler superspace. Under rare conditions a different branch or Hilbert space universe parallel to the one we are actually in can coherently interfere and modify the cosmic quantum force on our evolving 3D space geometry. The Bohm picture is very different from Hawking's picture based on Bohr's Copenhagen interpretation in which the particle takes all paths at once. In this case different paths through classical Wheeler superspace are simultaneously existing so this is a different meaning to the term "parallel universe". We shall return to this most important idea later. The basic idea to keep in mind is that in Bohm's theory there is a unique three-dimensional-matter (3G) universe whose large scale evolution in physical time is not only determined by gravitation within spacetime, but is also determined by a cosmic quantum force from Hilbert space which is literally beyond spacetime. The really interesting issue is whether there is a direct backreaction of our 3G evolving universe on the source of the cosmic quantum force. That source is the Mind of God. This backreaction is beyond quantum mechanics as we know it today and it is the mathematics of Abraham's Covenent with God which is the corner stone of Judeo-Christian-Islamic civilization. This may well be what Robert Anton Wilson called "The Cosmic Trigger" , the "Final Secret of the Illuminati" captured by Stanley Kubrick in the film 2001 when one of us monkeys touch the black monolith. The same mythic vision was dramatized by Plato in the"Allegory of the Cave" in The Republic.

Returning to Feynman.

"For example, the the equations do not say the same thing in a region of constant potential and in a region of zero potential."

Note that Wigner defines gauge invariance in electrodynamics by the condition that Maxwell's field equations say the same thing in a region of constant potential as they do in a region of zero potential.

Feynman shows from the action of the linearized theory (eq.5.3.3) p. 67 that "the simple substitution t' = t sqrt(1 + e) " shows

"that the effect of a constant potential is like a change in scale of time to make physical processes run more slowly in regions of lower gravitational potential.

where g44 = 1 + e, gaa = g22 = g33 = -1 and the action for a simple point particle of rest mass mo is

-

But that is not good enough, Feynman then considers a scalar field phi (eq. 5.2.5) in which the action integral is

"Again it happens that for dt' = dtsqrt(1 = e) = dt(1 + e/2) (for e << 1) the action is restored to its previous algebraic form. ... It is possible to show that the time dilation should occur for all interactions, regardless of the exact nature of the total Lagrangian." p.67

Feynman then gives the general proof on p.68.

... any terms in the Lagrangian involving the time gradients carry their own factors sqrt(1 + e) so that the substitution t' = t sqrt(1 + e) exactly reproduces the effect of the constant gravitational field. All physics therefore remains the same except for the time dilation. The gravitational potentials are negative, so that the clocks should run more slowly as they come nearer to a massive object such as a star.

Feynman then discusses the observation of this gravitational redshift over a 24 meter height tower at Harvard by Pound and Rebka using the extremely sharp Mossbauer spectral lines from nuclear transitions in crystals. The entire crystal absorps the recoil needed to conserve momentum when the nuclear photon is emitted. The fractional change in frequency over a 24 meter altitude in the earth's gravity while only one part in 10^15 was readily measurable with the technology to the 1950s to an uncertainty of 10%. The experimental method compensated the gravity red shift by an artificial Doppler shift since a null measurement is easier to make. An excited nucleus in a crystal at the bottom of the tower emits a photon that moves to the top of the tower where it is compared to a photon emitted by a second nucleus in another crystal at that height. The artificial compensating Dopper shift is in the controlled relative motion of the crystals.

When the absorption as a function of the relative velocities of the crystals is used to determine the frequency shift ... the clocks which run more slowly in this case are nuclear mechanisms which produces photons of definite frequencies; the fractional difference in the frequencies of the clocks at the top and bottom is &w/w = e/2 = the difference in gravitational potential divided by c^2. .. The prediction of this frequency shift does not really need the machinery of our theory... since it is implicit in the experimental results of Eotovos, that gravity potentials are proportional to the energy content. ... According to Eotovos, the excited nucleus is heavier by (Eo/c^2)g , if Eo is the excitation energy, since as we know from from nuclear experiments its mass is M + Eo/c^2, if M is the mass in the ground state. When it is raised by a height h it contains an energy Eo + (Eo/c^2)gh + Mgh more than an unexcited nucleus at zero height. If we excite only the lower nucleus, we require only Eo =hbarwo . After the upper nucleus makes a transition, its total energy should exceed that of the lower nucleus only by Mgh . Since the photon frequency is E = hbarw, the frequency of the (top) photon emitted is w = wo(1 + gh/c^2) . It is thus obvious that the frequency shift is required by energy conservation. If this shift were not there, we might yet construct a perpetual-motion machine, using such nuclear transitions. ... it suggests that the frequency shift required by energy conservation be considered a general property of all physical processes, that they run more slowly in lower gravitational potentials.

There is nothing like the "twin paradox" of special relativity here. The man on top of a mountain is living and aging at a faster rate than we; we see him move faster. When he looks back at us, he sees us moving more slowly than he. It is not like the time dilation of high relative velocities when each observer sees the other moving slowly ... however, the rates of aging change very little ... the center of the earth should be a day or two younger than the surface!" p.69

Feynman discusses Mach's Principle that, in contradiction to Newtonian mechanics, absolute acceleration is meaningless. Acceleration, Mach proposed, is relative to the distant stars. This, says Feyman,

"would profoundly alter the laws of mechanics, since the usual mechanics asssumes unaccelerated rectilinear motion to be the 'natural' motion in the absence of forces. When accelerations are defined as accelerations relative to other objects, the path of a particle under 'no acceleration' depends on the distribution of the other objects in space, and the definitions of forces between objects would be altered as we change the distributions of other objects in space." p.71

"the definitions of forces between objects would be altered as we change the distributions of other objects in space."

"It must be the influence of the nebulae which determines the scale of time at each point in space.. the Compton wavelength relative to the size of the universe depends on how many nebulae are in it ...the inertial frame is now also automatically determined from the nebulae, and the phenomena of inertia .... Mach's Principle is equivalent to the statement that the fundamental units of length and time at a point are the result of the influence of the nebulae. p.72

"The absence of acceleration is a consequence of the natural time scale being equal at all points in a region of space. This constancy is understood if the nebulae determine the natural scale..."

5.5 is on the self-energy of the gravitational field. Feynman has only built an incomplete weak-field theory for spin2 in analogy with spin 1 quantum electrodynamics.

"we have written a stress tensor which did not include the energy of the gravitational field itself... our present theory is physically untenable, since the energy of the matter is not conserved ... Theories not coming from some kind of variational principle, such as Least-Action, may be expected to lead to trouble and inconsistencies ... we are attempting to describe a nonlinear effect: the gravitational field is produced by energies, and the energy of the field is a source of more fields ..." p.75

There is a parallel here with Bohm's quantum theory. In general relativity theory matter-energy is a source of the metric gravity field, but in Feynman's perturbative approach, he starts with the matter-energy shaping the metric with no back-reaction of the metric on the matter-energy. This is analogous, in Bohm's derivation of Schrodinger's equation, that the wavefunction acts on the matter-energy but the matter-energy does not react back on the wavefunction.

6.2 is "Formulation of a theory correct to all orders".

"We shall search for a functional which is to be an action to be varied, for empirical reasons ... It is possible that the fundamental truth may be that processes occur according to a priniple of of minimum phase .... There is ... an evident simplest solution (involving the smallest number of derivatives of guv--just two). When this is done, we shall have arrived at a theory which is identical to Einstein's." p.82

"Our problem then is this: To find a form for F a functional of guv such that under infinitesimal transformation.. of the guv to g'uv, the F is unchanged to first order ....

Remember A need not commute with B and also note that

After a difficult analysis Feynman winds up with (6.4.8) Action F proportional to the 4Dspacetime integral of the Lagrangian density g^uvR^t uvtsqrt(-Detg), where R is a 4th-rank Riemann curvature tensor. p.87 Not even Feynman can make this easy to understand. The math of general relativity is much harder than the math of quantum mechanics. Feynman describes Einstein's achievement with awe like Houdini escaping chains under water

"I feel as though he had done it while swimming underwater, blindfolded, and with his hands tied behind his back!"

The theory of gravitation as viewed within the framework of Einstein's ideas is something so beautifully exciting that we shall be sorely tempted to try to make all other fields look like gravity, rather than continue with the Venutian trend of making gravity look like other fields that are familiar to us. We shall resist the temptation. p.89

"it was a puzzling fact about gravity that led to his theory of gravitation when he converted this fact into a physical principle." p.89

"The experiment of Eotvos showed how a centrifugal force added to a gravity force in such a way that the resultant was indistinguishable from a purely gravitational effect. These facts suggested to Einstein that accelerations imitate gravity in all respects." .... "today we are familiar with the weightlessness in satellites, which is a cancellation of gravitational forces by an acceleration. It is this possibility of cancellation which is the core of the principle of equivalence." p. 90

"The second law is given in the spirit of 'cherchez la femme': If we see a force, we are to search for the guilty object which is producing it." p.91

"he is making a physical statement, since he is making a specification on the connection between forces and physical object."

"It is not possible to cancel out gravity effects entirely by uniform accelerations. We imagine a box in orbit about the earth, a satellite. Since the earth's field is not uniform, it is only at one point, near the center of mass of the satellite, that gravitational effects are exactly balanced by the acceleration. As we go far away from the center of mass, the earth's field changes in either strength or direction, so that there will be small uncancelled components of the gravitational forces. If the box is not very large, these small additional forces are very nearly proportional to the distance from the center of the box, and they have a quadrupole character. ... Forces such as these cause tides on the earth.." p.91

"We can get rid of gravity at any one point and at any one time; over a small region about such a point, the residual differences should be proportional to the distance from the point of cancellation .. we will be considering (general coordinate) transformations ...

(gravity)' = (gravity) + (acceleration) (7.1.3)

.. we shall not in any absolute way be able to say that one effect is gravitational and one is inertial; it will not be possible to define a 'true' gravity ... It is true that we cannot imitate gravity with accelerations everywhere, that is, if we consider boxes of large dimensions. However, by considering these transformations (7.1.3) over infinitesimal regions, we expect to learn how to describe the situation in differential form; only then shall we worry about boundary conditions ...In special relativity, extensive use is made of reference frames which are moving with a uniform velocity in a straight line. But, as soon as we allow the presence of gravitating masses anywhere in the universe, the concept of such truly unaccelerated motion becomes impossible. because there will be gravitational fields everywhere...we cannot tell from inside the box whether we are accelerating relative to the nebulae, or whether the forces are due to masses in the neighborhood .. an accelerating box in some gravitational field is indistinguishable from a stationary box in some different gravitational field ... " pp.92 - 93

The principle of equivalence also implies that clock rates change in changing gravitational fields.

"Light which is emitted from the top of the accelerating box will look violet-shifted as we look at it from the bottom... The time that light takes to travel down is to a first approximation c/h, where h is the height of the box. In this time, the bottom of the box has acquired a small additional velocity gh/c. The net effect is that receiver is moving relative to the emitter, so that the frequency is shifted

freceive = femit((1 + v/c) = femit(1 + gh/c^2) (7.2.1)

Note that this conclusion does not depend on E = mc^2 and on the existence of energy levels, which we had to postulate in the argument we have previously given. ...Similarly, we may compute the frequency shift for light emitted by the man living downstairs. Since the receiver is receding from the source in this case, the man downstairs looks redder when viewed from upstairs. ... the time scale is faster at the top; time flows are different in different gravitational potentials ... we compare the time rates with an absolute time separation, defined in terms of the proper times ds. ... suppose that there are two events occurring at the top, which are reported to be a time dt apart; then

phi = gh, ds = dt(1 + 2phi/c^2)^1/2 (7.2.2) p.95

Make a Taylor series expansion for small phi/c^2 and small v/c. Therefore,

Integrate ds from t1 to t2. The result is (7.3.2)

But the nonrelativistic Lagrangian L is precisely

Since t2 - t1 is a fixed boundary condition, the elapsed time recorded by the clock on the path is maximized when the action, which is the time integral of the Lagrangian, is minimized. Therefore, in spacetime, the path of a particle in a gravitational field that minimizes the action, maximizes the elapsed time of a clock on that path. This is compatible with the twin paradox of special relativity. p.97 Novices often get this the wrong way round.

Einstein guessed that the basic principle of classical motion in a gravitational field is that the variation in the integral of the proper time is zero. The next problem is how to compute phi.

"It was Einstein's guess that ... the physics should be independent of the particular way in which we have separated inertial and gravitational effects." p.97

This new diffeomorphic invariance implied by the equivalence principle - or gauge symmetry that the separation into true and pseudo (inertial) gravity fields is, like the Galilean ether of absolute rest, undetectable - means that we must

"study very carefully the way in which the proper time interval ds is expressed in different coordinate systems, as we apply transformations ... (7.1.3) ... "

phi = 0 is the limit of special relativity. In general corrdinates, partial differential calculus implies

summing over repeated indices. The equivalence principle tells us we can locally eliminate gravity (i.e. make phi = 0) to any approximation by making the 4D spacetime region sufficiently small, therefore, with nuv the special relativity metric

Since the metric tensor g'ab is symmetric so that g'ab = g'ba, there are 10 independent components rather than 16 in 4D spacetime. Specifiy these ten functions and then compute which geodesic worldlines make the elapsed proper time a maximum relative to all worldlines connecting the same two spacetime events. Einstein saw from the equivalence principle, first glimpsed by Galileo, that the general transformation of the metric tensor between arbitrarily distorted and relatively accelerating reference frames (with different partitions into inertial and gravity forces) was

The gravitational field cannot be reduced to zero over a finite region because of the presence of mass-energy concentrations. It can be reduced to an arbitrarily small amount as the region shrinks to zero.

7.5 is where Feynman links his field-theoretic approach with Einstein's original geometric way of thinking. For uniform gravity fields, the metric tensor guv "describes how the scale of time is different at different locations in space" (p.99). In general the metric represents how both space and time scales vary from place to place. Consider the analogy with electrodynamics. The second rank metric tensor guv for spin 2 gravity plays the role that the first rank tensor vector potential Au plays for spin 1 electrodynamics. The local space and time scales for guv are analogous to the positions of the single hand of the phase clock at each point in spacetime. Every shift in the pattern of these local phases corresponds to a gauge transformation of the Au that keeps the action invariant. Similarly, every shift in the pattern of these local space and time scales corresponds to a diffeomorphism or local coordinate transformation.

Consider the simpler 2D problem.

Objective physics must be independent of which coordinate system (i.e., frame of reference) we subjectively decide to use. This is the central idea of all modern physics and it has worked very well.

So for example, we could choose a crazy set of coordinates in which

Consider

where

Suppose you are a smart flatlander living on the surface which has this metric. You will find for circles of fixed r, where you move tangentially around a fixed center, that

then if you move out radially from 0 to r say along the y axis

where b^2 = 1/a. Note the physical radius R should not be confused with the radial coordinate r.

Therefore, the ratio of circumference to radius is

Indeed, the surface is that of a 2-sphere.

"Our previous point of view about gravity may be compared to that which might be held by the more conservative of the bugs: The tiles are 'really' square, but the rulers are affected as we move them from place to place, because of a certain field whcih has this effect. Our newer geometrical point of view, will be that we cannot truly define the tiles to be 'really' square; we live in a world which is in general not Euclidean, which has a curvature which is measurable by doing suitable experiments." p.101

The curvatures are diffeomorphic invariants (as is ds) under the general coordinate transformations induced by the equivalence principle which is to spacetime what gauge (i.e. local phase) transformations are to the operator-valued fiber spaces and their associated Hilbert spaces beyond spacetime.

The general metric in 2D has the diffeomorphic-invariant proper time line element ds, where

Only one of the three guv functions is independent. A sufficiently general form in 2D is

"The function f(x,y), in one viewpoint, represents the factor by which rulers are changed as we move about the surface. In the other viewpoint, it evidently determines the curvature of the surface." p.102

The local curvature of a point is defined as a limit "of measurements made on smaller and smaller objects." Take the limit of smaller and smaller circles, the "intrinsic curvature" (i.e. Gaussian mean-square curvature) is defined in terms of the ratio of the area of the circles to the Euclidean value of pi R^2.

A "rolled" cylindrical surface has zero intrinsic curvature. It can be rolled out into a plane by cutting without stretching. This is a question of global topology. We can imagine a cylindrical flat spacetime in which the time dimension is a circle rather than a line.

"... if the surfaces are smooth. they must look like either paraboloids or hyperbolic paraboloids ober the infinitesimal regions in which we define the intrinsic curvature. These surfaces are described by two length parameters, the radii of curvature in two perpendicular planes... the intrinsic curvature is given by 1/R1R2. It is positive if the surface is paraboloidal, or negative if the surface is hyperbolic paraboloidal." p.103

For the cylinder, one radius is infinite, so the curvature is zero.

Sarfatti digression: The hyperbolic case is satisfied if both R1 and R1 are pure imaginary since i^2 = -1. We recall that surfaces of dimension n can be defined in terms of the roots of a polynomial of power n. If we have the two roots iR1 and iR2, the polynomial is

This polynomial is the secular determinant Det(M - xI) = 0 of the matrix M where I is the unit matrix. The root principal radii of curvature are the eigenvalues of this matrix which is determined by the metric guv.

Consider the function

Research Problem: So for the hyperbolic case this is a complex function of a real variable. Imagine a particle in one dimension. The space part of the quantum wavefunction is some complex function of the real variable x. If we can approximate that wavefunction as a polynomial, we can imagine an abstract metric space representation of that quantum wavefunction. How far can we take this wild idea? We locally approximate any function by polynomials in numerical analysis in practical computer calculations.

A curved-three dimensional space can be thought of as embedded in a 6 dimensional space, one dimension for each independent component of the symmetric metric tensor. Curved 4-D spacetime needs a 10-D embedding space. You can measure curvature intrinsically without ever having to go outside 4D conceptually. 4D metric space has 20 real numbers at each spacetime event that describe the curvature. We cannot generally eliminate gravity by going to a special frame, except at a single point because of the tidal forces caused by variable curvature of spacetime induced by inhomogenous distributions of matter, radiation, and, possibly a cosmological constant determined by the structure of the quantum vacuum.

I go into the following in detail since it is fundamental and it is more clear the way I do it then the way Feynman's student copied the words and blackboard of his master. There exists a tangent hyperplane or Riemannian normal coordinates in the neighborhood of each spacetime event such that the spacetime is flat in first order with second order corrections depending on the distance from the event. Equation (7.7.1) is a Taylor series expansion of the metric tensor g'ab(x) about event xo out to second order. Then make the tensor transformation which connects the 10 g'ab to guv (7.7.2) of this Taylor expansion of g'ab. So we have also 40 first g'ab,c and 100 second order g'ab,cd partial derivatives of g'ab. There are 16 first order partial derivatives &xu/&x'a at xo, 40 second order &^2xu/&x'a&x'b, 80 third order &^3xu/&x'a&x'b&x'c. We must do the Taylor expansion of both g'ab and guv and equate coefficients of like order (0,1,2) in powers of (xc - xoc) on both sides of the tensor transformation connecting the two Taylor series expansions of g' and g, not going beyond second order (xc - xoc)(xd - xod) terms. This is tedious and Feynman gives

So, we have 10 components g'uv(xo) and force it to be the flat Lorentz metric nuv of special relativity which defines the tangent hyperplane at event xo. That is,

We have 16 free parameters &xu/&x'a to obey this diffeomorphism requirement to get to Riemann normal or "geodesic" coordinates. The 6 parameters that remain specify the 6 generators of the Lorentz transformations in the tangent hyperplane of special relativity. So we see in this split of 16 into 10 and 6 how the 10 parameter zero order diffeomorphism from the equivalence principle and the 6 parameter Lorentz group (3 rotation, 3 boost) relate to each other.

Next make all 40 first derivatives g'ab,c(xo) vanish exactly using the matching 40 second partial derivatives. Physically, we are choosing a Local Inertial Frame at the single point xo in which gravity is momentarily cancelled out. This is the nitty gritty of the equivalence principle first glimpsed by Galileo and then elevated 300 years later into a profoundly beautiful exciting structure of the universe by Einstein's genius.

"That they can be made to vanish at a point means that all gravity forces can be removed at any one point and time by suitable accelerations."

Research Problem. Notice the coincidence that there are exactly 40 second partial derivatives in the tensor transformation to match the 40 first partial derivatives of the metric tensor. It is this seeming mathematical coincidence which permits the equivalence principle to make its most important mark. Is this unique to 4D spacetime? Can one have a consistent theory of gravitation in a 2D, 3D, 5D etc spacetime? Is this why we have 4 dimensions?

*John Baez says there is a large literature of general relativity in other than 4D so my idea about the significance of the coincidence of 40 may be wrong. I am still not sure. Could it be that the attempts to do GR in other than 4D violate the equivalence principle? John Baez is probably right on this, and I am probably wrong -- but it's worth thinking about more.

Thus on p.105 Feynman says

we can make all 40 g'uv,c (xo) vanish exactly by our 40 [&^2x^u/&x'^a&x'^b]|xo

It is the 100 - 80 = 20 independent linear combinations of the second derivatives g'ab,cd which describe the irremovable tidal forces represented by the diffeomorphic-invariant local curvature of spacetime at each event xo.

The accelerated, twisted and stretched coordinate frames used in general relativity require a more careful book keeping of the tensor indices. There are more modern elegant and abstract mathematical notations that Feynman did not use back in 1963. I first met him there at Cal Tech when I was working at Ford Aeronutronics in Newport Beach. I took Feynman for a ride in my black 1959 Jaguar convertible with white leather seats and real wire wheels to pick up his car that was being repaired.

Contravariant components are written as superscripts ^u. The infinitesimal coordinate displacements dx^u are contravariant components of a vector. The diffeomorphism is

summing over repeated tensor indices. Any contravariant first rank tensor A^u(x) transforms the same way. That is

A second rank tensor transforms under the diffeomoephism as an outer product of two vectors

Be careful of the notation in the book, it's a bit sloppy about the primes on p.108-- poor proof reading from Addison-Wesley.

Note guv A^u B^v is a scalar invariant under diffeomorphisms.

The covariant vector is

so AuB^u is also a scalar invariant under diffeomorphisms.

Note

The distinction between covariant and contravariant components of a tensor are only important when the coordinate axes are not orthogonal. In that case, the covariant projection on a given axis is perpendicular to that axis. The contravariant projection is parallel to the other axes. This is easy to picture in 2 dimensions. (p.110)

"There is no fundamental physical distinction between the covariant and contravariant components ... they have the same physical content, and it is only the representation which is changed... the tensor components guv describe the lack of orthogonality of the coordinates at a given point."

Research Problem. In quantum mechanics it can be proved that Hermitian operators on Hilbert space have orthogonal eigenfuctions. The Hermitian operator for the total energy infinitesimally generates a unitary transformation of the state vector through time which conserves total probability. Suppose the energy operator was not Hermitian. In that case total probability is not conserved. This is in fact what happens for an open system far from thermal equilibrium. Unitary evolution is only observed in the idealized case of an isolated system that is not being measured by an external probe. Must the eigenfunctions of a non-Hermitian operator be orthogonal? If not, can we construct a metric Guv in Hilbert space so that the non-Hermiticity of the open observables is a kind of curvature of the Hilbert space of infinite dimensions? Asher Peres, in his recent textbook on quantum mechanics (a la Bohr) shows that such a non-unitary theory permits communication using Einstein-Podolsky-Rosen quantum correlations, and also allows a Maxwell Demon to win, violating the classical limit of the second law of thermodynamics.

Make a small change in coordinates,

Make the diffeomorphic transformation of the metric tensor keep in only zero and first order terms treating h as small.

Note the Taylor series expansion to first order in small h

Therefore,

"The new g'uv equals the old guv plus some terms of order h ... which functions of guv are allowed, if we insist that their form remain invariant ... there is thus more than one physical point of view which produces the same equation, and which has the same content... What will be the physical significance of the invariants of guv?" p.111

In the case of gravity ,,, we know of no scale that would be unaffected--there is no "light" unaffected by gravity with which we might define a Galilean coordinate system. Thus, all coordinate systems are equivalent, and they differ only in that different values for the fields are necessary for the description of clock rates or length scales... There is one case in which there is a significance in a Galilean .. system, the limiting case of zero gravity.. Here the physical and Euclidean distances follow the same geometry. If we strted from a curved labelling of positions, we would find that a certain coordinate transformation enabled us to describe measurements without the use of a field... If the forces are not zero everywhere, there is no possibility of defining the "nicest" coordinate system. It is, however, possible to make them locally zero (principle of equivalence)...

the fact is that a spin-two field has this geometrical interpretation; this is not something readily explainable--it is just marvelous ... It might be that the whole coincidence might be understood as representing some kind of gauge invariance ... in attempting to understand how gravity can be both geometry and a field. p.113

The classical Maxwell field equations are invariant under the above gauge transformation of the vector potential. If A is a four-vector, then the grad operator is understood in the 4-vector sense.

Consider a quantum wave function psi. If we multiply psi by a constant phase factor, there should be no physical effect because technically the quantum wavefunction, or its generalization to a quantum amplitude that can depend on a spacetime path, is a projective ray defined modulo any constant complex number z. That is zpsi and psi are the same quantum amplitude. Note, this is not the same thing as splitting psi into the sum of psi1 + psi2 and multiplying psi2 by a unimodular phase factor. That will make a physical difference -- a fringe shift in the corresponding interferometer experiment.

So far this is called "global phase invariance" of quantum mechanics. Now we come to the new idea of "local phase invariance". Now consider the local phase transform of psi which is

Where X is an arbitrary function of spacetime as in (8.4.1) above.

But now consider

Problem: Check to see if

which is required if the non-relativistic Schrodinger equation is to be gauge invariant.

The idea of local gauge invariance, therefore, is to combine the original classical gauge transformation with the quantum phase transformation using the same function X in both cases. The resulting Schrodinger equation plus classical Maxwell field equations are both left invariant. This is semi-classical radiation quantum mechanics which works very well in many practical cases in atomic physics. The idea carries through in quantum electrodynamics when the Maxwell equations are also quantized.

In terms of group theory, the above electromagnetic gauge invariance is a U(1) internal symmetry transformation. Yang and Mills introduced an SU(2) internal symmetry which explained the the isotopic spin of the strong nuclear force. The same math was later found to be useful to explain the weak force if one also added spontaneous symmetry breakdown of the vacuum with the Higgs mechanism and nonconservation of parity. These theories are renormalizable meaning we can handle the short wave divergences with a small number of fudge factors called mass, charge and wavefunction renormalizations in the case of quantum electrodynamics. Naive attempts to quantize gravity are not renormalizable. Weinberg thinks renormalizability is a law of nature. Feynman always thought it was a "shell game" and that it was a scandal that physicists could not do better.

Isotopic spin is a broken internal symmetry of the nucleon. Neglecting electric charge (which has a superselection rule) one can imagine "an object which is partly a proton and partly a neutron" (p.114)

the invariance of the nuclear forces with respect to changes in isotopic spin means that the new object psi' acts in all nuclear respects like psi. The proposal of Yang and Mills is that a field should be added to the Lagrangian in such a way that a space-dependent phase change makes no difference in the equations ...

How can such ideas be connected to gravity? The equations of physics are invariant when we make coordinate displacements by any constant amount a^u.

x'^u = x^u + a^u (8.4.5)

The idea, carried through by Kibble at Imperial College, is to replace the constant dispacement a^u by an arbitrary function of spacetime. This is the diffeomorphism. Thus Einstein's classical gravity is the local gauge theory that results when we make different displacements at different spacetime events and require that the action is invariant.

For comparison, the Yang-Mills SU(2) phase transformation is

where the rotation angles are in an internal fiber space attached to each spacetime event. This 3D- fiber space is not infinite-dimensional Hilbert space, however, the latter is "associated" with it. The taus have the form of the Pauli matrices for isotopic spin 1/2 which form the smallest non-trivial irreducible representation in the Hilbert space. For higher isotopic spins the matrices are formally like the orbital angular momentum matrices. But their eigenfunctions are not in physical space as for the rotation group. The SU(3) theory has eight terms instead of three with an 8D fiber space. Modern physics uses many different kinds of spaces.

Note also that the gauge invariance ideas above are at the classical level. We have used gauge invariance for the quantum Schrodinger equation where the wave function is a complex-valued c-number. If we do non-relativistic many-body theory in second quantization, psi becomes a field operator PSI that creates and destroys particles. The operator PSI acts on a Fock space (a version of Hilbert space) whose different dimensions or base vectors correspond to different numbers of particles in different possible modes. The modes can correspond to plane wave states of different momenta, or to delta functions in position space, or to wavelets.

Let us review the formal meaning of the equivalence principle. Given any curved spacetime metric guv one can make it equal the flat Lorentz metric nuv exactly at a single point event xo. We are doing classical general relativity so the scale is always much larger than the Planck scale of 10^-33 cm. We can satisfy the equation

by adjusting the 16 first-order partial diffeomorphic derivatives &x^a/&x/^v and the 40 second-order diffeomorphic partial derivatives &^2x^a/&x'^u&x'^v to make all the 40 first-order partial derivatives guv,c of the metric guv vanish exactly at the single Taylor expansion starting point xo.

That is, the equivalence principle means that one can find a class of local inertial frames of reference (i.e. "Riemann normal" or "geodesic" coordinates) defined by the condition that the metric has a critical point at the given spacetime event. The unerasable local curvature is in the nonvanishing second-order partial derivatives. This match between the 40 second-order diffeomorphic derivatives and the 40 first-order metric derivative seems to be a meaningful coincidence that only happens for a 4D spacetime. If that is true, then the ability to obey the equivalence principle selects out 4D spacetime. There are 80 adjustable third-order diffeomorphic derivatives and 100 second derivatives of the metric tensor. This leaves 20 independent conditions on the second derivatives of the metric tensor which describe the actual local curvature at the point xo as measured by the tidal forces that are due to real inhomogeneities in the metric.

It is the 20 linear combinations of these second [metric] derivatives which are things that may have a geometrical definition. They cannot be reduced away by transformations of the coordinates. What we are looking for is an expression for 20 such quantities in terms of the initial guv. p.115

The curvatures are precisely a measure of the local mismatch between the [spacetime] and the tangent plane. They are a description of the essential charcter of the space at the point.

Feynman then derives (pp116-117) the 20 linear combinations using Taylor series, the law of tensor transformation, symmetry of the indices to get

where

This quantity is not a tensor; it is not general enough;it is defined only in a place having zero net fields. These are ... the irreducible pieces of the gravitational tensor, those which cannot be removed by a [diffeomorphic] transformation. They represent the purely tidal forces.

There is an analogy here with the Bohm-Aharonov effect. The metric tensor guv is the analog of the electromagnetic vector potential Au. The equivalence principle says that the first order derivatives are not physical because they can be transformed away. Only the invariant combinations of the second derivatives are physical. However, in quantum mechanics Au makes a fringe shift even when the electromagnetic field tensor Fuv is zero. Similarly, we expect a gravitational version in which guv causes a fringe shift even when the local curvature is zero along the alternative interfering paths of a single quantum provided that the loop formed by the paths encloses a region of finite curvature. This implies that one can use a SQUID to measure gravitational curvature.