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

III Advanced Quantum Field Theory

2.3 Effective quantum field theory

We now see what happens when we try to obtain effective field theories in 1

dimension. Suppose we have two real-valued fields

x, y

:

S

1

→ R

. We pick the

circle as our universe so that we won’t have to worry about boundary conditions.

We pick the action

S[x, y] =

Z

S

1

1

2

˙x

2

+

1

2

˙y

2

+

1

2

m

2

x

2

+

1

2

M

2

y

2

+

λ

4

x

2

y

2

dt.

As in the zero-dimensional case, we have Feynman rules

1/(k

2

+ m

2

) 1/(k

2

+ M

2

)

−λ

As in the case of zero-dimensional QFT, if we are only interested in the correla-

tions involving

x

(

t

), then we can integrate out the field

y

(

t

) first. The effective

potential can be written as

Z

Dy exp

−

1

2

Z

S

1

y

−

d

2

dt

2

+ M

2

+

λx

2

2

y dt

,

where we integrated by parts to turn ˙y

2

to y¨y.

We start doing dubious things. Recall that we previously found that for a

bilinear operator M : R

n

× R

n

→ R, we have

Z

R

n

d

n

x exp

−

1

2

M(x, x)

=

(2π)

n/2

√

det M

.

Now, we can view our previous integral just as a Gaussian integral over the

operator

(y, ˜y) 7→

Z

S

1

y

−

d

2

dt

2

+ M

2

+

λx

2

2

˜y dt (∗)

on the vector space of fields. Thus, (ignoring the factors of (2

π

)

n/2

) we can

formally write the integral as

det

−

d

2

dt

2

+ M

2

+

λx

2

2

−1/2

.

S

eff

[x] thus looks like

S

eff

[x] =

Z

S

1

1

2

( ˙x

2

+ m

2

x

2

) dt +

1

2

log det

−

d

2

dt

2

+ M

2

+

λx

2

2

We now continue with our formal manipulations. Note that

log det

=

tr log

,

since

det

is the product of eigenvalues and

tr

is the sum of them. Then if we

factor our operators as

−

d

2

dt

2

+ M

2

+

λx

2

2

=

−

d

2

dt

2

+ M

2

1 − λ

−

d

2

dt

2

+ M

2

−1

x

2

2

!

,

then we can write the last term in the effective potential as

1

2

tr log

−

d

2

dt

2

+ M

2

+

1

2

tr log

1 − λ

d

2

dt

2

− M

2

−1

x

2

2

!

The first term is field independent, so we might as well drop it. We now look

carefully at the second term. The next dodgy step to take is to realize we know

what the inverse of the differential operator

d

2

dt

2

− M

2

is. It is just the Green’s function! More precisely, it is the convolution with the

Green’s function. In other words, it is given by the function G(t, t

0

) such that

d

2

dt

2

− M

2

G(t, t

0

) = δ(t − t

0

).

Equivalently, this is the propagator of the

y

field. If we actually try to solve this,

we find that we have

G(t, t

0

) =

1

2M

X

n∈Z

exp

−M

t − t

0

+

k

T

.

We don’t actually need this formula. The part that will be important is that it

is ∼

1

M

.

We now try to evaluate the effective potential. When we expand

log

1 − λG(t, t

0

)

x

2

2

,

the first term in the expansion is

−λG(t, t

0

)

x

2

2

.

What does it mean to take the trace of this? We pick a basis for the space we

are working on, say {δ(t − t

0

) : t

0

∈ S

1

}. Then the trace is given by

Z

t

0

∈S

1

dt

0

Z

t∈S

1

dt δ(t − t

0

)

Z

t

0

∈S

1

dt

0

(−λ)G(t, t

0

)

x

2

(t

0

)

2

δ(t

0

− t

0

)

.

We can dissect this slowly. The rightmost integral is nothing but the definition

of how

G

acts by convolution. Then the next

t

integral is the definition of how

bilinear forms act, as in (

∗

). Finally, the integral over

t

0

is summing over all

basis vectors, which is what gives us the trace. This simplifies rather significantly

to

−

λ

2

Z

t∈S

1

G(t, t)x

2

(t) dt.

In general, we find that we obtain

tr log

1 − λG(t, t

0

)

x

2

2

= −

λ

2

Z

S

1

G(t, t)x

2

(t) dt −

λ

2

8

Z

S

1

×S

1

dt dt

0

G(t

0

, t)x

2

(t)G(t, t

0

)x

2

(t

0

) ···

These terms in the effective field theory are non-local ! It involves integrating

over many different points in

S

1

. In fact, we should have expected this non-

locality from the corresponding Feynman diagrams. The first term corresponds

to

x(t)

x(t)

Here G(t, t) corresponds to the y field propagator, and the −

λ

2

comes from the

vertex.

The second diagram we have looks like this:

x(t)

x(t)

x(t

0

)

x(t

0

)

We see that the first diagram is local, as there is just one vertex at time

t

. But

in the second diagram, we use the propagators to allow the

x

at time

t

to talk

to x at time t

0

. This is non-local!

Non-locality is generic. Whenever we integrate out our fields, we get non-local

terms. But non-locality is terrible in physics. It means that the equations of

motion we get, even in the classical limit, are going to be integral differential

equations, not just normal differential equations. For a particle to figure out

what it should do here, it needs to know what is happening in the far side of the

universe!

To make progress, we note that if

M

is very large, then we would expect

G

(

t, t

0

) could be highly suppressed for

t 6

=

t

0

. So we can try to expand around

t = t

0

. Recall that the second term is given by

Z

dt dt

0

G(t, t

0

)

2

x

2

(t)x

2

(t

0

)

We can write out x

0

(t

2

) as

x

0

(t

2

) = x

2

(t) + 2x(t) ˙x(t)(t

0

− t) +

˙x

2

(t) +

1

2

x(t) ˙x(t)

(t − t

0

)

2

+ ··· .

Using the fact that

G

(

t, t

0

) depends on

t

0

only through

M

(

t

0

−t

), by dimensional

analysis, we get an expansion that looks like

1

M

2

Z

dt

α

M

x

4

(t) +

β

M

3

x

2

˙x

2

+

1

2

x

2

¨x

+

γ

M

5

(4-derivative terms) + ···

Here α, β, γ are dimensionless quantities.

Thus, we know that every extra derivative is accompanied by a further power

of

1

M

. Thus, provided

x

(

t

) is slowly varying on scales of order

1

M

, we may hope

to truncate the series.

Thus, at energies

E M

, our theory looks approximately local. So as long

as we only use our low-energy approximation to answer low-energy questions, we

are fine. However, if we try to take our low-energy theory and try to extrapolate

it to higher and higher energies, up to

E ∼ M

, it is going to be nonsense. In

particular, it becomes non-unitary, and probability is not preserved.

This makes sense. By truncating the series at the first term, we are ignoring

all the higher interactions governed by the

y

fields. By ignoring them, we are

ignoring some events that have non-zero probability of happening, and thus we

would expect probability not to be conserved.

There are two famous examples of this. The first is weak interactions. At

very low energies, weak interactions are responsible for

β

-decay. The effective

action contains a quartic interaction

Z

d

4

x

¯

ψ

e

nν

e

p G

weak

.

This coupling constant

G

weak

has mass dimensional

−

1. At low energies, this is

a perfectly well description of beta decay. However, this is suspicious. The fact

that we have a coupling constant with negative mass dimension suggests this

came from integrating some fields out.

At high energies, we find that this 4-Fermi theory becomes non-unitary, and

G

weak

is revealed as an approximation to a

W

-boson propagator. Instead of an

interaction that looks like this:

what we really have is

W

There are many other theories we can write down that has negative mass

dimension, the most famous one being general relativity.