Zee A. Quantum Field Theory in a Nutshell

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442 | VIII. Gravity and Beyond

the small q behavior of the scattering amplitude, we need only evaluate these terms in the

limit q → 0. We can throw almost everything away! For example,

→ k

= 0, k

= k

+ k

− p

) → k

Just imagining contracting all those indices in our heads is good enough: We obtain the

amplitude ∼ (k

)(p

)/q

If k

and k

point in the same direction, k

∝k

=0. Two photons moving in the

same direction do not interact gravitationally.

Of course, this result is not of any practical importance since electromagnetic effects are

far more important, but this is not an engineering text. In appendix 1 I give an alternative

derivation of this amusing result.

Kaluza-Klein compactification

You have probably read about how excited Einstein was when he heard of the proposal of

Kaluza and of Klein to extend the dimension of spacetime to 5 and thus unify electromag-

netism and gravity. The 5th dimension is supposed to be compactified into a tiny circle of

radius a far smaller than what experimentalists can see; in other words, x

is an angular

variable with x

+2πa. You have surely heard that string theory, at least in some ver-

sion, is based on the Kaluza-Klein idea. Strings live in 10 -dimensional spacetime, with 6

of the dimensions compactified.

I can now show you how the Kaluza-Klein mechanism works. Start with the action

S =

16πG





−g

(15)

in 5-dimensional spacetime. The subscript 5 serves to indicate the 5-dimensional quanti-

ties. We denote the 5-dimensional metric by g

with the indices A and B running over

0, 1, 2, 3, 5.

Assume that g

does not depend on x

. Plug into S , integrate over x

, and compute

the effective 4-dimensional action. Since R

and the 4-dimensional scalar curvature R

both involve two powers of ∂ and g

contains g

μν

, we must have (exercise VIII.1.5)

= R +

...

. Thus, (15) contains the Einstein-Hilbert action with Newton’s gravitational

constant G ∼ G

/a.

What else do we get? We don’t even have to work through the arithmetic. We can

argue by symmetry. Under the 5-dimensional coordinate transformation x

→ x

A

+ε

(x), we have [see (8)] h



−∂

. Let us choose ε

=0 and ε

(x) to

be independent of x

: We go around and rotate each of the tiny circles attached to every point

in our spacetime a tiny bit. Well, we have h



μν

= h

μν

and h



= h

, but h



μ5

= h

μ5

− ∂

But if we give the Lorentz 4-vector h

μ5

and 4-scalar ε

new names, call them A

and ,

this just says A



= A

− ∂

, the usual electromagnetic gauge transformation!

Since we know that the 5-dimensional action (15) is invariant under x

→ x

A

= x

(x), the resulting 4-dimensional action must be invariant under A

→A



−∂



VIII.1. Gravity as a Field Theory | 443

and hence must contain the Maxwell action. Note once again the power of symmetry

considerations. No need to do tedious calculations.

Electromagnetism comes out of gravity!

Differential geometry of Riemannian manifolds

I hinted earlier at a deep connection between general coordinate transformation and gauge

transformation. Let us flesh this out by looking at differential geometry and gravity. For

this sketch we will consider locally Euclidean (rather than Minkowskian) spaces.

The differential geometry of Riemannian manifolds can be elegantly summarized in

the language of differential forms. Consider a Riemannian manifold (such as a sphere)

with the metric g

μν

(x). Locally, the manifold is Euclidean by definition, which means

μν

(x) = e

(x)δ

(x) (16)

where the matrix e(x) may be thought of as a similarity transformation that diagonalizes

μν

and scales it to the unit matrix. Thus, for a D-dimensional manifold there exist D

“world vectors” e

(x) obviously dependent on x and labeled by the index a =1, 2,

...

, D.

The functions e

(x) are known as “vielbeins” (meaning “many legs” in German, vierbeins

= four legs for D = 4, dreibeins = three legs for D = 3, and so on.) In some sense, the

vielbeins can be thought as the “square root” of the metric.

Let us clarify by a simple example. The familiar two-sphere (of unit radius) has the line

element

= dθ

+ sin

θdϕ

. From the metric (g

θθ

= 1, g

ϕϕ

= sin

θ) we can read off

= 1 and e

= sin θ (all other components are zero). We are invited to define D 1-forms

= e

. (In our example, e

= dθ, e

= sin θdϕ.)

On a curved manifold, when we parallel transport a vector, the vector changes when

expressed in terms of the locally Euclidean coordinate frame. (This is just the familiar

statement that on a curved manifold such as the surface of the earth the notion of a vector

pointing straight north is a local concept: When we move infinitesimally away keeping our

“north vector” pointing in the same direction, it will end up being infinitesimally rotated

away from the “north vector” defined at the point we have just moved to.) This infinitesimal

rotation of the vielbeins is described by

=−ω

(17)

Note that since ω generates an infinitesimal rotation it is an antisymmetric matrix: ω

−ω

. Since de

is a 2-form, ω is a 1-form, known as the connection: It “connects” the

locally Euclidean frames at nearby points. (Since the indices a, b, etc. are associated with

the Euclidean metric δ

we do not have to distinguish between upper and lower indices.

When we do write upper or lower indices a, b, etc. it is for typographical convenience.) In

Note that this represents the square of an infinitesimal distance element and not an area element, and so a

quantity such as dθ

is literally the square of dθ and not the wedge product dθdθ (of chapter IV.4), which would

have been identically zero.

444 | VIII. Gravity and Beyond

the simple example of the sphere, de

= 0 and de

= cos θdθdϕand so the connection

has only one nonvanishing component ω

=−ω

=−cos θdϕ.

At any point, we are free to rotate the vielbeins: If you use the vielbeins e

I am free to use

some other vielbeins e

a

instead, as long as mine are related to yours by a rotation e

(x) =

(x)e

b

(x). [You can check that g

μν

(x) = e

(x)δ

(x) = e

a

(x)δ

b

(x) if O

O = 1.]

The connection ω



is defined by de

a

=−ω

ab

b

. You can readily work out that (suppressing

indices)

ω = Oω



− (dO)O

(18)

The local curvature of the manifold is a measure of how the connection varies from

point to point. We would like the curvature to be invariant under the local rotation O (or at

least to transform as a tensor so that by contracting it with vectors we can form a scalar).

The desired object is the 2-form R

= dω

+ ω

. You can check that R = OR



(For the sphere, R

= dω

+ ω

= sin θdθdϕ.) Written out in components, R

μν

. I leave it to you to verify that R

μν

is the usual Riemann curvature tensor

λσ

μν

, where e

is the inverse of the matrix e

. In particular, R

μν

is the scalar curvature,

which in our convention works out to be +1 for the sphere.

Thus, Riemannian geometry can be elegantly summarized by the two statements (again

suppressing indices)

de + ωe = 0 (19)

and

R = dω +ω

(20)

Look familiar? You should be struck by the similarity between (20) and the expression

for the field strength in nonabelian gauge theories F = dA +A

. Note ω transforms [see

(18)] exactly the same way as the gauge potential A. But one nagging difference, namely

the lack of an analog of e in gauge theory, has long bothered some theoretical physicists

(but is shrugged off by most as inconsequential). Also, note that Einstein theory is linear

in R while Yang-Mills theory is quadratic in F .

Gravity and Yang-Mills

We can make the connection between gravity and Yang-Mills theory more explicit by

looking at the derivative of a vector field. Yang-Mills theory was born of the requirement

that a field ϕ and its derivative ∂

ϕ transform in the same way under a spacetime-

dependent internal symmetry transformation (IV.5.1). In Einstein gravity a vector field

(x) transforms as W

μ



) = S

(x)W

(x) with S

(x) = ∂x

μ

/∂x

. Since the matrix S

depends on the spacetime coordinate x, we see that ∂

could not possibly transform

like a tensor with one upper and one lower index, as we would like naively just by looking

at indices. We would have to introduce a covariant derivative. Not surprisingly, this closely

VIII.1. Gravity as a Field Theory | 445

parallels the discussion in chapter IV.5. Historically, Yang and Mills were inspired by

Einstein gravity.

Using the chain rule and the product rule, we have

∂



μ



) =

∂W

μ



)

∂x

λ

∂x

λ

∂

∂x

(x)W

(x)] =(S

−1

)

∂

+ [(S

−1

)

∂

(21)

Were the second term in (21), which comes from differentiating S, not there, the naive

guess, that ∂

transforms like a tensor, would be valid. The fact that the transformation

S varies from place to place has negated the naive guess.

What is happening is quite clear: as the vector W varies from a given point to a neighbor-

ing point, the coordinate axes that define the components of W also change. This suggests

that we could define a more suitable derivative, called the covariant derivative and written

as D

, to take this effect into account, so that D

would indeed transform like a

tensor. Exactly as in Yang-Mills theory (IV.5.1), we have to add an extra term to knock out

the second term in (21).

Just the way the indices hang together immediately suggests the correct construction.

The factor (S

−1

)

∂

in the unwanted second term in (21) has one upper index and two

lower indices, so we need an object with this set of indices. Lo, the Riemann-Christoffel

symbol 

λν

in (3) (and introduced in chapter I.11) fits the bill perfectly. I will let you have

the fun of verifying that the covariant derivative defined by

≡ ∂

+ 

λν

(22)

indeed transforms like a tensor (note that  was normalized correctly for this purpose).

I end with a technical remark about the coupling of gravity to spin

fields. First,

we of course have to Wick rotate so that the vierbein e

erects a locally Minkowskian

rather than a Euclidean coordinate frame. The indices a, b, etc. are now contracted with

the Minkowskian metric η

. The slight subtlety is that the Dirac gamma matrices γ

are associated with the Lorentz rotation of the vierbein e

(x) = O

(x)e

b



) and thus

carry the Lorentz index a rather than the “world” index μ. Similarly, the Dirac spinor

ψ(x) is defined relative to the local Lorentz frame specified by the vierbein, and thus

its covariant derivative has to be defined in terms of the connection ω rather than the

symbol . Hence the flat space Dirac action



ψ(iγ

∂

− m)ψ must be general-

ized to



√

−g

ψ(iγ

bμ

− m)ψ, where the covariant derivative D

ψ =∂

ψ −

μab

ψ expresses the rotation of the local Lorentz frame as we move from a point x to

a neighboring point. In contrast to the action for integer spin fields in curved spacetime

(see chapter I.11), the Dirac action in curved spacetime involves the vierbein explicitly.

Appendix 1: Light on light again

The stress-energy tensor T

μν

of a light beam moving in the x-direction has four nonzero components: the

energy density T

of course, then T

= T

since photons carry the same energy and momentum, next

= T

by symmetry, and finally T

= T

since the stress-energy tensor of the electromagnetic field is

traceless (chapter I.11). Without having to solve Einstein’s equations in the weak field limit (13) explicitly we

know immediately that h

= h

≡ h. The metric around the light beam is given by g

= 1 + h,

446 | VIII. Gravity and Beyond

= g

=−h, and g

=−1 + h (and of course g

= g

=−1 plus a bunch of vanishing components).

Consider a photon moving parallel to the light beam. Its worldline is determined by (recall chapter I.11)

dζ

=−

μν

dζ

Let’s calculate d

y/dζ

and d

z/dζ

with (dy/dζ ), (dz/dζ )  (dt/dζ ), (dx/dζ ). Using (3) we find (with μ, ν

restricted to 0, x)

dζ

(∂

yμ

+ ∂

yν

− ∂

μν

)

dζ

=−

(∂



(

dζ

)

+ (

dζ

)

− 2

dζ

=−

(∂

h)(

dζ

−

dζ

)

For a photon moving in the same direction as the light beam dt = dx and d

y/dζ

z/dζ

=0. We have once

again derived the Tolman-Ehrenfest-Podolsky effect. Note we never had to solve for h.

Incidentally, if you are a bit unsure of dt = dx, the condition ds = 0 for a light beam moving in the x-direction

amounts to (1 +h)dt

−2hdtdx −(1 −h)dx

=0. Upon division by dt

we obtain −(1 +h) + 2hv +(1 −h)v

0, with v ≡ dx/dt. The quadratic equation has two roots v =∓(1 ± h)/(1 − h). The negative root gives v = 1,

and thus for a photon moving in the same direction as the light beam dx/dt = 1. In contrast, the positive root

v =−(1 + h)/(1 − h) describes a photon moving in the opposite direction.

Appendix 2: The helicity structure of gravity

To gain a deeper understanding of the difference between Einstein’s and Dr. G’s theories let us look at the

helicity structure of the interaction in the two cases. To warm up, consider the interaction between two conserved

currents due to the exchange of a spin 1 particle of momentum k and mass m : J

(1)

(2)μ

(1)

(2)

−J

(1)

(2)

. Use

current conservation k

= 0 to eliminate J

= k

/ω (with ω ≡ k

). We obtain (k

/ω

− δ

(1)

(2)

. Let



point in the 3rd direction and use



= ω

− m

to write this as −[(m

/ω

(1)

(2)

+ J

(1)

(2)

+ J

(1)

(2)

]. We see

that as m → 0 the longitudinal component of the current J

indeed decouples as explained in chapter II.7 and

we obtain −

1+i2

(1)

1−i2

(2)

+ J

1−i2

(1)

1+i2

(2)

), showing explicitly that the photon has helicity ±1. (Obvious notation:

1+i2

≡ J

+ iJ

etc.)

Onward to gravity. Consider the interaction T

μν

(1)

(2)μν

− ξT

(1)

(2)

, where ξ =

for Einstein and

for Dr. G.

For ease of writing I will now omit the subscripts (1) and (2). Conservation k

μν

= 0 allows us to eliminate

= k

/ω and T

= k

/ω

. Again taking



k to point in the 3rd direction we obtain the mess





+ 2





+ T

) + T

+ T

+ 2T

− ξ







+ T







+ T



which simplifies in the limit m → 0to

+ T

+ 2T

− ξ(T

+ T

)(T

+ T

)

In Einstein’s theory, ξ =

and this becomes

− T

)(T

− T

) + 2T

which lo and behold is equal to

1+i2,1+i2

1−i2,1−i2

1+i2,1+i2

), showing that indeed the graviton

carries helicity ±2. In Dr. G’s theory, this would not be the case.

VIII.1. Gravity as a Field Theory | 447

Exercises

VIII.1.1 Work out T

μν

for a scalar field. Draw the Feynman diagram for the contribution of one-graviton exchange

to the scattering of two scalar mesons. Calculate the amplitude and extract the interaction energy between

two mesons sitting at rest, thus deriving Newton’s law of gravity.

VIII.1.2 Work out T

μν

for the Yang-Mills field.

VIII.1.3 Show that if h

μν

does not satisfy the harmonic gauge, we can always make a gauge transformation with

determined by ∂

= ∂

−

∂

so that it does. All of this should be conceptually familiar from

your study of electromagnetism.

VIII.1.4 Count the number of degrees of polarization of a graviton. [Hint: Consider a plane wave h

μν

(x) =

μν

(k)e

ikx

just because it is a bit easier to work in momentum space. A symmetric tensor has 10

components and the harmonic gauge k

imposes 4 conditions. Oops, we are left with 6

degrees of freedom. What is going on?] [Hint: You can make a further gauge transformation and still

stay in the harmonic gauge. The graviton should have only 2 degrees of polarization.]

VIII.1.5 The Kaluza-Klein result that we argued by symmetry considerations can of course be derived explicitly.

Let me sketch the calculation for you. Consider the metric

= g

μν

− a

[dθ + A

(x)dx

]

where θ denotes an angular variable 0 ≤ θ<2π. With A

=0, this is just the metric of a curved spacetime,

which has a circle of radius a attached at every point. The transformation θ →θ +(x) leaves ds invariant

provided that we also transform A

(x) → A

(x) − ∂

(x). Calculate the 5-dimensional scalar curvature

and show that R

= R

−

μν

. Except for the precise coefficient

this result follows entirely

from symmetry considerations and from the fact that R

involves two derivatives on the 5-dimensional

metric, as explained in the text. After some suitable rescaling this is the usual action for gravity plus

electromagnetism. Note that the 5-dimensional metric has the explicit form



μν

− a

−a



(23)

VIII.1.6 Generalize the Kaluza-Klein construction by replacing the circles by higher dimensional spheres. Show

that Yang-Mills fields emerge.

VIII.1.7 Starting with the connection 1-form ω

=−cos θdϕ for the sphere, show that the scalar curvature is a

constant independent of θ and ϕ.

VIII.1.8 The vielbeins for a spacetime with Minkowski metric is defined by g

μν

(x) = e

(x)η

(x), where the

Minkowski metric η

replaces the Euclidean metric δ

. The indices a and b are to be contracted with

. For example, R

= dω

+ ω

. Show that everything goes through as expected.

VIII.2

The Cosmological Constant Problem

and the Cosmic Coincidence Problems

The force that knows too much

The word paradox has been debased by loose usage in the physics literature. A real paradox

should involve a major and clear-cut discrepancy between theoretical expectation and

experimental measurement. The ultraviolet catastrophe, for example, is a paradox, the

resolution of which around the dawn of the twentieth century ushered in quantum physics.

I now come to the most egregious paradox of present day physics.

The electromagnetic force knows about the particles carrying charge, and the strong

force knows about the particles carrying color. And the gravitational force? It knows

everybody! More precisely, anybody carrying energy and momentum.

Within a particle physics frame of mind, which is the only frame of mind we have

in exploring the fundamental structure of physics, the graviton can be regarded as just

another particle. Indeed, given that a massless spin 2 particle couples to the stress-energy

tensor, one can reconstruct Einstein’s theory.

Nevertheless, there is an uncomfortable feel to this whole picture. Gravity has to do with

the curvature of spacetime, the arena in which all fields and particles live. The graviton is

not just another particle.

This in essence is the root origin

of the paradox of the cosmological constant. The

graviton is not just another particle—it knows too much!

The cosmological constant

In the absence of gravity, the addition of a constant  to the Lagrangian L → L −  has

no effect whatsoever. In classical physics the Euler-Lagrange equations of motion depend

For more along this line, see A. Zee, hep-th/0805.2183 in Proceedings of the Conference in Honor of C. N. Yang’s

85th Birthday, World Scientific, Singapore 2008, p. 131.

VIII.2. Cosmic Coincidence Problem | 449

only on the variation of the Lagrangian. In quantum field theory we have to evaluate the

functional integral Z =



Dϕe



xL(x)

, which upon the inclusion of  merely acquires

a multiplicative factor. As we have seen repeatedly, a multiplicative factor in Z does not

enter into the calculation of Green’s function and scattering amplitudes.

Gravity, however, knows about . Physically, the inclusion of  corresponds to a shift

in the Hamiltonian H → H +



x. Thus, the “cosmological constant”  describes a

constant energy or mass per unit volume permeating the universe, and of course gravity

knows about it.

More technically, the term in the action −



x is not invariant under a coordinate

transformation x → x



(x). In the presence of gravity, general coordinate invariance re-

quires that the term −



x in the action S be modified to −



√

g, as I explained

way back in chapter I.11. Thus, the gravitational field g

μν

knows about , the infamous

cosmological constant introduced by Einstein and lamented by him as his biggest mistake.

This often quoted lament is itself a mistake. The introduction of the cosmological constant

is not a mistake: It should be there.

Symmetry breaking generates vacuum energy

In our discussion on spontaneous symmetry breaking, we repeatedly ignored an additive

term μ

/4λ that appears in L.

Particle physics is built on a series of spontaneous symmetry breaking. As the universe

cools, grand unified symmetry is spontaneously broken, followed by electroweak symme-

try breaking, then chiral symmetry breaking, just to mention a few that we have discussed.

At every stage a term like μ

/4λ appears in the Lagrangian, and gravity duly takes note.

How large do we expect the cosmological constant  to be? As we will see, for our

purposes the roughest order of magnitude estimate suffices. Let us take λ to be of order

1. As for μ, for the three kinds of symmetry breaking I just mentioned, μ is of order 10

, and 1 Gev, respectively. We thus expect the cosmological constant  to be roughly

= μ/(μ

−1

)

, where the last form of writing μ

reminds us that  is a mass or energy

density: An energy of order μ packed into a cube of size μ

−1

. But this is outrageous even if

we take the smallest value for μ: We know that the universe is not permeated with a mass

density of the order of 1 Gev in every cube of size 1 (Gev)

−1

We don’t have to put in actual numbers to see that there is a humongous discrepancy be-

tween theoretical expectation and observational reality. If you want numbers, the current

observational bound on the cosmological constant is

∼

(10

−3

ev)

. With the grand unifi-

cation energy scale, we are off by (17 +9 +3)×4 = 116 orders of magnitude. This is the

mother of all discrepancies!

With the Planck mass M

∼ 10

Gev the natural scale of gravity, we would expect

 ∼ M

if it is of gravitational origin. We are then off by 124 orders of magnitude. We

are not talking about the crummy calculation of some pitiful theorist not fitting some

experimental curve by a factor of 2.

450 | VIII. Gravity and Beyond

We can imagine the universe starting out with a negative cosmological constant, fined

tuned to cancel the cosmological constant generated by the various episodes of sponta-

neous symmetry breaking. Or there must be a dynamical mechanism that adjusts the

cosmological constant to zero.

Notice I say zero, because the cosmological constant problem is basically an enormous

mismatch between the units natural to particle physics and natural to cosmology. Measured

in units of Gev

the cosmological constant is so incredibly tiny that particle physicists

have traditionally assumed that it must be zero and have looked in vain for a plausible

mechanism to drive it to zero. One of the disappointments of string theory is its inability

to resolve the cosmological constant problem. As of the writing of this chapter around the

turn of the millennium, the brane world scenarios (chapter I.6) have generated a great deal

of excitement by offering a glimmer of a hope. Roughly, the idea is that the gravitational

dynamics of the larger space that our universe is embedded in may cancel the effect of the

cosmological constant.

Cosmic coincidence

But Nature has a big surprise for us. While theorists racked their brains trying to come up

with a convincing argument that  = 0, observational cosmologists steadily refined their

measurements and discovered dark energy. The “cleanest” explanation of dark energy by

far is that it represents the cosmological constant. Assuming that this is the case (and

who knows?), the upper bound on the cosmological constant would be changed to an

approximate equality

 ∼ (10

−3

ev)

!!! (1)

The cosmological constant paradox deepens. Theoretically, it is easier to explain why some

quantity is mathematically 0 than why it happens to be ∼ 10

−124

in the units natural (?) to

the problem.

To make things worse, (10

−3

ev)

happens to be the same order of magnitude as the

present matter density of the universe ρ

. More precisely, dark energy accounts for ∼74%

of the mass content of the universe, dark matter for ∼ 22%, and ordinary matter for ∼ 4%.

First, the ordinary matter we know and love is reduced to an almost negligibly small

component of the universe. Second, why should ρ

be comparable to  to within a factor

of 3? This is sometimes referred to as the cosmic coincidence problem.

Now the cosmological constant  is, within our present understanding, a parameter in

the Lagrangian. On the other hand, since most of the mass density of the universe resides

in rest mass, as the universe expands ρ

(t) decreases as [1/R(t)]

, where R(t) denotes

the scale size of the universe.

In the far past, ρ

was much larger than , and in the

For an easy introduction to cosmology, see A. Zee, Unity of Forces in the Universe, vol. II, chap. 10.

VIII.2. Cosmic Coincidence Problem | 451

far future, it will be much smaller. It just so happens that, in this particular epoch of the

universe, when you and I are around, ρ

∼ . Or to be less anthropocentric, the epoch

when ρ

∼  happens to be when galaxy formation has been largely completed.

Very bizarre!

In their desperation, some theorists have even been driven to invoke anthropic

selection.

For a recent review, see A. Vilenkin, hep-th/0106083.