2QFT in one dimension (i.e. QM)

III Advanced Quantum Field Theory

2.4 Quantum gravity in one dimension

In quantum gravity, we also include a (path) integral over all metrics on our

spacetime, up to diffeomorphism (isometric) invariance. We also sum over all

possible topologies of

M

. In

d

= 1 (and

d

= 2 for string theory), we can just do

this.

In

d

= 1, a metric

g

only has one component

g

tt

(

t

) =

e

(

t

). There is no

curvature, and the only diffeomorphism invariant of this metric is the total

length

T =

I

e(t) dt.

So the instruction to integrate over all metrics modulo diffeomorphism is just

the instruction to integrate over all possible lengths of the worldline

T ∈

(0

, ∞

),

which is easy. Let’s look at that.

The path integral is given by

Z

T

dT

Z

C

[0,T ]

[y,x]

Dx e

−S[x]

.

where as usual

S[x] =

1

2

Z

T

0

˙x

2

dt.

Just for fun, we will include a “cosmological constant” term into our action, so

that we instead have

S[x] =

1

2

Z

T

0

˙x

2

+

m

2

2

dt.

The reason for this will be revealed soon.

We can think of the path integral as the heat kernel, so we can write it as

Z

∞

0

dT hy|e

−HT

|xi =

Z

∞

0

dT

d

n

p d

n

q

(2π)

n

hy|qihq|e

−HT

|pihp|xi

=

Z

∞

0

dT

d

n

p d

n

q

(2π)

n

e

ip·x−iq·y

e

−T (p

2

+m

2

)/2

δ

n

(p − q)

=

Z

∞

0

dT

d

n

p

(2π)

n

e

ip·(x−y)

e

−T (p

2

+m

2

)/2

= 2

Z

d

n

p

(2π)

n

e

ip·(x−y)

p

2

+ m

2

= 2D(x, y),

where

D

(

x, y

) is the propagator for a scalar field on the target space

R

n

with

action

S[Φ] =

Z

d

n

x

1

2

(∂Φ)

2

+

m

2

2

Φ

2

.

So a 1-dimensional quantum gravity theory with values in

R

n

is equivalent to

(or at least has deep connections to) a scalar field theory on R

n

.

How about interactions? So far, we have been taking rather unexciting

1-dimensional manifolds as our universe, and there are only two possible choices.

If we allow singularities in our manifolds, then we would allow graphs instead of

just a line and a circle. Quantum gravity then says we should not only integrate

over all possible lengths, but also all possible graphs.

For example, to compute correlation functions such as

h

Φ(

x

1

)

···

Φ(

x

n

)

i

in

Φ

4

theory on

R

n

, say, we consider all 4-valent with

n

external legs with one

endpoint at each of the x

i

, and then we proceed just as in quantum gravity.

For example, we get a contribution to hΦ(x)Φ(y)i from the graph

x x

The contribution to the quantum gravity expression is

Z

z∈R

n

d

n

z

Z

[0,∞)

3

dT

1

dT

2

dT

3

Z

C

T

1

[z,x]

Dx e

−S

T

1

[x]

Z

C

T

2

[z,z]

Dx e

−S

T

2

[x]

Z

C

T

3

[y,z]

Dx e

−S

T

3

[x]

,

where

S

T

[x] =

1

2

Z

T

0

x

2

dt +

m

2

2

Z

T

0

dt.

We should think of the second term as the “cosmological constant”, while the

1D integrals over

T

i

’s are the “1d quantum gravity” part of the path integral

(also known as the Schwinger parameters for the graph).

We can write this as

Z

d

n

z dT

1

dT

2

dT

3

hz|e

−HT

1

|xihz|e

−HT

2

|zihy|e

−HT

3

|zi.

Inserting a complete set of eigenstates between the position states and the time

evolution operators, we get

=

Z

d

n

p d

n

` d

n

q

(2π)

3n

e

ip·(x−z)

p

2

+ m

2

e

iq·(y−z)

q

2

+ m

2

e

i`·(z−z)

`

2

+ m

2

=

Z

d

n

p d

n

`

(2π)

2n

e

ip·(x−y)

(p

2

+ m

2

)

2

1

`

2

+ m

2

.

This is exactly what we would have expected if we viewed the above diagram as

a Feynman diagram:

x

y

p p

`

This is the worldline perspective to QFT, and it was indeed Feynman’s original

approach to doing QFT.