Measure factors, tension, and correlations of fluid membranes

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membranes

W. Cai, T. Lubensky, P. Nelson, T. Powers

To cite this version:

W. Cai, T. Lubensky, P. Nelson, T. Powers. Measure factors, tension, and correlations of fluid membranes. Journal de Physique II, EDP Sciences, 1994, 4 (6), pp.931-949. �10.1051/jp2:1994175�.

�jpa-00248017�

Classification Physics Abstracts

87.10 II-Ii 68.15 87.20

Measure factors, tension, and correlations of fluid membranes

W.

Cai,

T-C-

Lubensky,

P. Nelson and T. Powers

Physics Department, University of Pennsylvania, Philadelphia, PA 19104 U-S-A-

(Received

18 January1994, accepted 10 March

1994)

Abstract We study two geometrical factors needed for the correct construction of statistical ensembles of surfaces. Such ensembles _appear in the study of fluid bilayer membranes, though

our results _{are more} generally applicable. The naive functional _{measure over} height fluctuations must be corrected by these factors in order _to give correct, self-consistent formulas for the free energy and correlation functions of the height. While _one of these corrections the Faddeev,

Popov determinant has been studied extensively, _our derivation proceeds from very simple geometrical ideas, which _we hope _{removes some} of its mystery. The other factor is similar to the Liouville correction in string theory. Since _our formulas differ from those of previous authors,

we include _some explicit calculations of the effective frame tension and two-point function to show that _our version indeed _secures coordinate-invariance and consistency to lowest nontrivial order in _a temperature expansion.

1. Introduction and summary.

Amphiphilic

molecules in _water tend to aggregate into thin flexible

bilayers,

^{which in} turn form

a wide

variety

of

nearly

two-dimensional structures with characteristic size of order microns

[I].

The

study

of these _{structures may} become

significant

for

biology; they

_are

certainly

important

technologically

_as the

underlying

elements of microemulsions and other

complex

fluids.

To understand the static and

dynamic properties

of

bilayer membranes,

and the transitions between different

morphologjes,

_we must first understand the role of thermal fluctuations in

two-dimensional

surfaces,

_a subtle

problem

in statistical mechanics. Thermal fluctuations _are important because

regardless

^how still _a membrane _may

be,

_very

long-wavelength

undulations in its

shape

_are

always allowed;

_on

long enough length

scales any membrane will appear flexible

and will undulate

significantly,

_a

phenomenon

first observed in the cell walls of red blood cells in the 19th century

[2](~).

(~) Strictly speaking _{we are} here talking about "fluid membranes", those with _no internal structure

or order within the surface. In this _paper

we discuss only fluid membranes, but

our measure factors

can be used in _more complicated situations too [3].

Since the

length

scales _we wish _to

study

are much

larger

than the size of the constituent

molecules,

_we expect ^that a membrane will be characterized

by just

_a few effective parameters,

summarizing

the effects of the

complicated

molecular forces.

Indeed,

the famous Canham- Helfrich model _[4] describes the

equilibrium

statistical mechanics of membranes in terms of

just

two parameters: a chemical

potential

for the addition of surfactant

molecules,

which _we will call _po, and _a

bending

_energy

coefficient,

the stiffness _~o.

(Another

_parameter, the Gaussian

stiffness Ku,

only

enters when _we consider

topology change.)

The bare coefficients _{po, ~o are} not

directly observable,

but from them _{we can} derive _{some more} relevant

phenomenologjcal

parameters. We will

mainly study

the "frame tension" _T, which is the free _{energy per} unit frame _area of the

fluctuating membrane,

and the

two-point

correlation

(h(u)h(0))

of the

height h(u)

of the membrane from its average

plane,

which _we fix

by fixing

the

edges

of the membrane to lie _{on a square} frame. The

leading

behavior of this

two-point

function at small wavenumber

defines another

phenomenological

parameter, the "q~ coefficient" _r via

(hlq)hl-q))

₌

~~~

()~~~

,

(l.1)

where AB is the _area of the

frame,

T is the temperature

(we

set Boltzmann's _constant kB

equal

to

one),

and _q is _a wavenumber. Another _reason to introduce _r is that in

dynamics problems

we can still calculate correlation functions like

(I.I _),

while there is _no

analog

of the free _energy from which to compute T. Two of _us

study

the

dynamical problem

in [5].

in this _{paper we} will

explore

^the relation between the various coefficients _{po, ~o, T,} and _r.

More

generally,

_on

long length

scales the

precise

size of the constituent molecules _is immaterial:

we can get ^the same answers

using

^{molecules of} size _a with coefficients _{po, ~o or} with rescaled

molecules of size

b~~a,

b < I, and effective coefficients

pe~(b), ~e~(b).

^The scale

dependence

of _~e~ is well-known

[6-11],

^but various _answers for _pe~ have been

given (for example,

_see [8, 9,

11-15]).

^In part the differences reflect

convention,

but there is _a

key physics point

^which

we will address: to get correct _{answers we} must define _our functional _measure

properly.

Two of _us have

already

discussed such matters in [3], ^{but here} we will add _a number of

points,

_as

follows.

We first introduce the effective action

r[h]

of the

fluctuating

membrane and recall the argument of [3] that ^{its form is} constrained

by

the coordinate invariance of the

underlying theory.

In

particular

_we show in section 2 that the coefficients _r and _T defined above must be

equal.

We then set up a naive,

uncorrected,

calculational scheme which

yields (consistent

with

J51)

~ ~° ~

/~ ^~~(2

~°~

~~°~~

~ ~°~~~

~~~

' ~~'~~

where A

=

27rla

and _a is the linear size of the constituent molecules.

Following

Morse and MiIner _we have introduced the thermal de

Broglie wavelength

I ₌

h/$4

,

(1.3)

where _m is the _mass of _a surfactant molecule [16]. ^The same calculational scheme however

yields

_{a very} different formula for _r. In fact

(1.2)

is _our final

result,

but

clearly

_we need _to

work _a bit harder to understand _r.

The first correction factor, the

"Faddeev-Popov" factor,

is well known. In [3] ^we gave a

formula for this factor which differs from the _one

given by

David [14]; ^{here in section 3} we

give

a

simple geometrical

motivation followed

by

_a derivation of _our formula which is

simpler

than the _one in [3]. ^While

unimportant

in the calculation of _T, this factor does

modify

the naive

calculation of _r

(and higher

correlation functions

too), yielding

_{an rfp} which still

disagrees

With _T.

Next in section 4, _{we argue} that another correction factor is needed.

Physically

_{our mem-}

brane consists of constituent molecules each

occupying

a fixed _area a2 in

physical 3,space.

As the surface bows outward _away from its flat

equilibrium configuration

its _area increases and additional molecules must be

brought

in from

solution, effectively increasing

the number of

degrees

of freedom in the membrane

problem.

A statistical _measure with variable number of

degrees

of

freedom, depending

_on the

configuration itself,

is rather

complicated.

We would

prefer

_{a measure} with fixed number of

degrees

of

freedom,

_as indeed _we used

implicitly

in the naive derivation of _rFP. Since

changing

the number of

degrees

of

freedom,

_or

equivalently

the ultraviolet

cutoff,

_can be

compensated by renormalizing

the bare _{energy, we} expect that _a countertenn correction will need _to be added _to the naive derivation.

We fix this counterterm

by requiring consistency

between two different

Monge-gauge

cal- culations of the free _energy

(in

the

Appendix

_we

give

_a different

strategy).

In this _{way we} discover that

~ ~ ~~~ ~

d

~~Al

~~,~

' ~~'~~

where _we introduced _an anisotropic wavenumber cutoff

(Al, _A2).

In

fact, substituting

_our

formulas for _T and _rFP into lA shows that

they

indeed

obey (lA).

The

required

counterterm

turns out to make _a contribution

precisely changing

the q~ coefficient from _rFP to r = T.

The introduction of _our second correction factor _{may seem} like ad hoc

wish-fulfillment,

but in the

Appendix

_we

give

some detailed calculations

shoring

how it and the Faddeev-

Popov

factor

together

_are crucial to obtain consistent _answers; in

particular

_we show that the

specific

coefficient of the counterterm

implied by consistency

is the

right

_one to get r = T.

Furthermore,

_an

analogous

correction is well-known in the

string theory literature,

where it is called the "Liouville factor"

[li,

_18]. This factor sometimes appears in the

previous

membrane

literature

(e.g. [19]),

but _one gets the

impression

that it matters

only

in the low-stiffness

regime beyond

the

persistence length; again,

_we will _see that it is needed to get the correct correlation functions _even in the _more

physical

stiff

regime.

Our correction differs in detail from the Liouville factor because the latter is

appropriate

for conformal _gauge, but in each _case the motivation is the _same. We will also argue that the usual

interpretation

of the Liouville

factor _{as a} Jacobian is

misleading; really

_as

argued

^above it reflects renormalization.

Again

_our counterterm is

important

^{because without}

it, ordinary diagrammatic perturbation theory gives

incorrect results due to the

subtlety

in the statistical _measure discussed above.

Finally

_we conclude in section 5.

2. General

arguments.

We will consider fluid

membranes,

those with _no internal order _or structure. Real membranes

are

typically

fluid _at temperatures

high enough

to

destroy in-plane

order, but in any case such order _can

easily

be

incorporated

into the argument ^below [3]. ^The constituents of _a membrane have _a

preferred spacing,

with _a

high

_energy cost for local deviations from that

density. Thus,

in this

regime

the free _{energy cost} of _a membrane

configuration depends only

_on its

shape.

As mentioned in the

Introduction,

_{we can} thus capture ^the

physics

^of

length

scales much

longer

than the constituent molecule size

by regarding

the surface _as continuous and

using

the Canham-Helfrich free _{energy [4]}

7i=po /dS+) _/dS(

+

~ (2.I)

1 2

Here dS is the element of surface _area,

f

dS is the total _area of the

surface,

and

Ri,

R2 _are the

principal

radii of _curvature of the surface. The two coefficients _{po, ~o are}

respectively

the

area cost and

bending rigidity.

The form of

(2.I)

is

severely

constrained

by

the

requirement

that the free _energy

depend only

on the membrane's

shape,

and _{not on} how _one chooses _to label

points

_on the surface. While this property ^{is manifest in}

(2.I),

to do calculations _{a more}

explicit

version of this

equation

proves necessary, in which _we do choose coordinates _u

=

(u~, u~)

to label

points R(u)

of the surface. As

usual,

_we then get tangent vectors e~ =

d~R

_e

dR/du~

at u and _a metric tensor

g~b " ea eb at each

point.

The unit normal is then _n

=

(ei

_x

e2)/(ei

_x

e2(,

and the curvature tensor is

Kab

_{" n}

V~dbR,

where V~ is the covariant derivative associated _to the metric g~b.

Letting

_{g +} det [g~b), [g~~) = [gab)

~~,

and

K]

_%

g~~Kab, equation (2.I)

becomes

7i = po

/ _d~u/j

+

) / _{d~u/j (K])~ (2.2)}

If _we

change

the coordinates from u~ to fi~

=

fi~(u)

_we

just

relabel

points

in the

plane.

For

an infinitesimal

change

fi~

= u~ ₊ e~

iv)

_{we get a new}

k(fi)

=

R(u) describing

the _same surface:

kin)

=

Rju) e~ju)ea(u) (2.3)

The

change

of R is _a

tangential motion; conversely, tangential changes

^{of R} _are not

physical cllanges

to the

shape

of the surface and

ought

not to be counted

separately

in the statistical

sum. We thus need to

supplement (2.2) by choosing

_a

protocol

for

assigning

_a

single

coordinate system to each

shape,

and _a procedure ^for

discarding

_every surface

R(u)

not

parameterized

in this _way.

Any

such choice is called _a

"gauge,fixing" procedure.

A

popular

class of

gauge-fixing procedures

_are the so-called "normal

gauges." Suppose

we have

arranged

that

configurations

_very close _{to one} reference surface will dominate the statistical _sum

(by stretching

the membrane _{across a} fixed

frame,

for

instance).

We choose

once and for all _a

parameterization

Ito

iv)

for this reference surface and compute its unit normal

no(u).

Then other

nearby

surfaces _can be written _as

R(u)

₌

Ruin)

+

hju)noiu) j2.4)

where the

height

^field h at any

point

is

uniquely

defined _as the normal distance from that

point

to the reference surface. We thus describe each distinct surface _once when _{we sum over} all functions

h(u).

Of _course the

point

of this _paper is that the correct definition of this _sum is _a

tricky

business, but let _us

proceed formally

to obtain _some

general properties constraining

the correct

prescription;

Let _us consider _a membrane confined to span a square, flat frame of _area

AB.

We suppose surfactant molecules _can

join

or leave the membrane with _a fixed cost in free _energy of _po per added _area

(one

_can

easily

_pass to _an ensemble with _a fixed number of molecules

by

_a

Legendre transformation).

The full partition ^function of _our system

Z(AB)

then

depends

_on the base _area, and

implicitly

_{on po,} the stiffness _~o

appearing

in

(2.I)

the size _{a a} 27r

IA

of the

constituent

molecules,

and the temperature ^T.

Since _our frame is

flat,

the

equilibrium (zero-temperature)

surface will be flat too, and _so it is convenient to work in

Monge

_{gauge, a} normal _gauge with Ro

Iv)

the flat surface

spanned by

the

frame,

with Cartesian coordinates u~

running

^from _zero to

/G.

In addition _to

Z(AB)

_we

will find it convenient _to introduce the "effective action"

r[h].

To define

r[h]

_we introduce _a

forcing

term into

7i, 7ij

₌ 7i

f hj,

and

adjust j

to ensure that

(h(u))

is the desired function

h(~). Computing

the

partition

function

Z[AB, ii

₌

/[Dh]e~~/T~(i/T) ^f

^hj '

(2.5)

we then let

r[h]

₊ -T

log Z[AB, j]

+

hj _(2.6)

The frame tension is the free energy per area of the unforced

membrane,

_{so we} have

T =

-) _{log Z(AB)}

"

~ _r[h

= o] «

r~°) (2.7)

B B

AB

Physically

there is

nothing special

about

Monge

_gauge. Even if _{we agree to} work in

Monge

gauge there is

nothing special

about _a reference surface

Ito(u) coinciding

with the

plane

of the

frame;

_a tilted reference surface _must

give

the _same free _energy. All that matters is the

physical

_area of the frame. Thus the _terms of

r[h] involving

at most first derivatives of

h (higher

derivatives _are insensitive to

tilting)

_{must enter} in the combination

f

^d~u

/J,

where

jab " bab +

dahdbh

is the induced metric and g = det g~b as usual. In other

words,

rjhj

= T

/ _d2u

(1 ⁺

)(oh)2

^{+ +} _,

^(2.8)

where the first

ellipsis

involves

O(h~)

while the second involves _more derivatives than

l's.

_We

gave a more detailed argument for this conclusion in [3].

Whatever the

precise

form of the _measure

[Dh] appearing

in

(2.5),

it does not

depend

_on

j,

and _so the first _moment

.6)

nding

(h)j

=

h,

we

&r

^~

ⁱ

^&j(u')

6z j

^j(u')

-

Differentiating again

and

using (h)j=o

_" 0, _{we see} that

jh(u)hju'))~=o

=

z-iT2

~~

/))~~,~

~

=

z-iT ~

~

(zh(u'))

&h(u')

_&j

(vi

^~~

~

&2r ^~~ ~~'~~~

~

bi(Ul

o

~

bh(U')

o

bh(U)bh(U')

o

(~) This is

a formal trick to obtain correlations of height; we do not imagine j _as coming from _any physical force. Mathematically j ₌ j12d~u _is

a density _on parameter space, so that 616

j(u)

is _a scalar.

For _more details _see [3].

Thus the inverse of the

two-point

function is

just

the

quadratic

part ^{of the effective action.}

We have

repeated

the above well-known argument to make _a point: it in _no way

depended

upon the niceties of the _measure

[Dh]. Any

exotic

gauge-fixing

_or

cutoff-stretching

factor which

we may

eventually

fold into

[Dh]

will not alter the above

conclusion, though they

_can and will affect the relation of the

quadratic

part to the _constant part

r(°) Combining (2.ll)

with

(2.8), (1.I)

at _once shows that the q~ coefficient _r

equals

the frame tension _T.

Let _us make _a first attempt at

calculating

_r and _T. If the temperature T is small _{we can}

expand everything

around the

equilibrium configuration

h ₌ 0. The _energy functional

(2.2)

then becomes

ill]

7i = po

/ _{d~u (I}

+

jfidh)~ ((dh)~)

+

i / d~u [(d~h)~ (dh)~(d~h)~ 2(d~h)d~hdbhd~dbh)

+

O(h6)

~~ ~~~

2 2

In this formula d~

=

d/du~,

^d~

= b~~d~db is the flat

Laplacian,

and all indices _are contracted with the Kronecker

symbol.

^{Thus all}

h-dependence

is

explicitly

in view in

(2.12).

To calculate

Z(AB)

_we need _a

proposal

for

[Dh].

The

simplest

choice is _to choose _a mesh of

grid

points

(u~j)

_{on u space,}

spaced by

_a, and let

[Dh]~a;ve

=

fl(dh(ujj)/1) (2.13)

iJ

We introduced the thermal

wavelength

I here to render the _measure

properly dimensionless(3 ).

We will argue later that

[Dh]~a;ve

^is _wrong, ^{but for} _now we take it _{as our}

starting point.

For low temperature _we

approximate Z(AB)

_as

^e~~°~B/~

times _a Gaussian

integral.

Col-

lecting

the

quadratic

terms of

(2.12)

and

writing f

d~u

= a~

£~~

we find

T

~

_~~~

^~~~~ _(p~q~

₊

^oQ~)j

r(o)

~ -T

log Z(AB)

_"

P°~B

~

i

_2~~

Now the _sum is _over the

points

_q of

reciprocal

_space. The _upper limit of this _sum is related to the constituent size _a; for

example,

if in

(2.13)

_we choose points on a square

grid,

then _our

Brillouin _zone is a square of

length

27r

la.

Thus _{we recover} the announced formula

(1.2).

Next _we turn to the

two-point function,

which _we

provisionally

define _as

jhju)h(0))na;ve

₌

^Z~~ /jDhjna;vee~~/~hju)hj0) ^j2.14)

This time however the constant term of

(2.12) drops

out, the

quadratic

term

gives

the

leading

contribution

(h(q)h(-q))o

₌

j~~~

~, and the

quartic

terms

yield

the desired thermal

POQ + ~oQ

correction to

(hh)o

via Wick's theorem. To shorten the

formulas,

in the rest of this

paragraph

(~) In [15] ^and [3] ^{this scale} was taken to be the molecule size _{a or a} constant multiple of it. This is legitimate when _we do not care about the bare chemical potential, for example when _we just want

to dial ~o to zero physical tension. But _~0 does have physical meaning, ^and MiIner and Morse have argued that the factor in

(2.13)

is the correct choice [16].

we will

drop

all factors of

AB,

or set

AB

₌ I.

Retaining only

those _terms

contributing

_{to r we} get

(~iU)~(°))na>ve

_"

i~(U)h(0))0

+

~4 / d~U'(h(U)dahlU')lolh10)dahlU')loldbh(U')0bhlU'))o

+ 8

/ _d~U' ~(U)°ahiU'))01~i°)°bh(U'))0(°ahiU')°bh(U'))01

+

j

₂

/ _d~U' (h(U)0ah(U'))0(h(0)bah(U'))0(b~hjU')b~h(U'))0

+

j

₂

/ d~U'(hjU)bahjU'))01hj0)bbhjU'))0i0a0bhjU')b~hjU'))0

,

or

into) hi -Q)ma>ve

₌

~~q4

~~q~

+ ~~q4

~~q~ / I ll))l~ill))l~

= T

P°Q~

~i~ /1 ³

~~iq,iil'liq,~~

⁺

°iQ~)l~

~~

T

/~

^d~Q

³

^P°Q~

(2.15)

~"~'~~ " ~°

§ @

_{~uq4 +}

_poq2

Equation (2.15)

is

essentially

the result of Meunier [13]. ^It

certainly

does not

equal (1.2), notwithstanding

_our formal

argument

^that r = T.

3.

Gauge fixing.

Apparently

_our troubles stern from

(2.13).

How should _{we sum over} all

configurations?

We will return in the next section to the

question

of the

spacing

of

grid

points, ^but even with this choice made

correctly, equation (2.13)

is not correct. Consider the two surfaces

R,

R + dR in

figure

I.

According

to

(2.4),

_we describe the

displacement

from R to R ₊ dR

by

_an increment

bh(u)

in

h(u).

But this

displacement

is

only partly physical,

I-e-,

only partly

normal to

R,

since it is

along no(u),

which is _not

equal

to the normal

n(u)

at

R(u),

The actual normal

displacement

from R _to R ₊ dR is

simply n(u)

_no

(u)dh(u)

_[22], and _we need to

replace [Dh]na;ve

in

(2.13) by it

_[hi

_[Dh]na;ve

where

Llhl

₊

fl(n(u,j) no(u,j)) (3.i)

,j

The Jacobian

it

_{[hi is the first correction factor}

(the "Faddeev-Popov

determinant" mentioned in the Introduction.

Our derivation of

(3.I)

_was heuristic. Nevertheless the result _agrees with _{a more} formal derivation which _we will give in _a moment, and the heuristic discussion

fives it[h]

_{a very}

simple geometric meaning.

The formula

(3.I)

also _agrees with the

approach

based _on metrics

on field _space [3]. ^For

example, taking Ito(u)

to be

flat

_we have _{n no}

=

g~~/~,

and

it

_{is the}

R+dR

-, ^{' '}

''

'~h

n ^{'' ''}

'

.,oooo,o2~o.o..

'

.,o..,

o ^o"'' ', _'"..

, ,o.""" _". ~ ,." """". _',

, ."' °. ." ~ ^{'~ /"°o, '}

, ~o"' "o:.."' '".~ ,

'.°' '"°...~

h

no

R~

Fig. I. Two surfaces R, R + dR described in the normal gauge associated to Ro.

Monge

_gauge ^{factor found} in [3]. ^{It is not} unity, contrary to reference [14]; ^{while it does} not affect the

one-loop

renormalization of _~o, it will be crucial for _{r, as} shown below and in [3].

Our second derivation of

(3.I)

is couched in

language

familiar from _gauge theories [20]. ^{It is}

more

precise

than the motivation

just given

for equation

(3.I),

and

again reproduces

the _answer

given

in [3]. ^{We think that each derivation sheds}

light

on

(and

increases _our confidence

in)

the final _answer. In order to form _a statistical _{sum over} distinct surfaces _we

imagine integrating

over all

parameterized surfaces,

with _a correction factor to eliminate

overcounting. Thus,

_we

write the

partition

function _as

Z =

/[DR] lj) _{e~~/~ (3.2)}

V°

Here

[DR] just

^describes the

possible displacement

of the individual

mass-points

_on the surface and

vol[R]

is _a factor _to be discussed below. All _we will need _to know for the _moment is that

the _measure

[DR]

must be

coordinate-invariant,

since _no

preferred

coordinate system is

given

on a fluid membrane;

[DR]

must also be invariant under

spatial

translations and rotations.

To finish

specifying (3.2)

_we need to

specify

the extra volume factor

vol[R] appearing

there.

Starting

^from _any ^surface

R(u)

and

applying

all

possible reparameterizations

we sweep ^out a

subspace

of all

parameterized

surfaces. The volume

vol[R]

of this "orbit" may well

depend

_on which surface _we start

with,

but it cannot

depend

_on how

R(u)

is

parameterized.

Since it _proves inconvenient to include

vol[R] explicitly

in

(3.2),

_{we now} fix _a coordinate choice

by introducing

two constraints _on

R(u)

at every

point

u:

ia(Rju))

= 0 a = 1,2

For

example,

in normal gauge we

require

falRlU))

₊

l~~ea,olU) (R(U) Ro(u))

= o

(3.3)

We have introduced _a factor of l~~

(see 11.3))

to make

f~

dimensionless.

Any length

scale will do here since _{our answers} will be

altogether independent

of _our choice of

f~.

We _now define the

Faddeev-Popov

determinant

Ai[R]

via

e

a~jRj l'jDkj &ji~jk)j j3.4)

The

integral

is

only

over the space of

k's

_which

are

repararneterizations

of

R;

hence _we also have

,

vol[R]

=

[DR] (3.5)

To

keep vol[R],

^and hence

(3.2),

dimensionless _we include _two factors of l~~ _{at every}

point

in

(3.5).

Since _every

kin)

=

R(fl(u))

is related to R

by

_a reparameterization, u~ _-+

fl~(u),

we can also write

f'[Dk]

as

f[Dfl]D[fl,R],

where

formally D[fl, R]

₌ det

d(R(fl))/dfl

is _a Jacobian factor related to the

change

of variables. Let _{us now} insert

(3A)

into

(3.I)

_{to get}

Z

"

/lDRl ^~~~~j e~~~~~~AflRl /lDflllllfl, RldlfalRlfl))I

Since _a

reparameterization

fl has

meaning independent

of the surface R _we

apply

it to, _we can

do the fl

integral

last.

Letting k

=

R(fl),

_{we use} the fact that

[DR], 7i, vol[R]

and

Ai

_are

reparameterization

invariant to rewrite Z _as

Z ₌

/[Dfl] /[Dk] lfi) e~~l~l/~'Ai[k]d[f~(k)]D[fl, R]

vol[R]

Also,

the fact that

[DR]

is coordinate-invariant _means that small

displacements

of fl _sweep out

equal

volumes _{no matter} where _on-a

given

orbit

they

act, ^I-e-

D[fl,R]

=

D[fl, R]. Renaming k

as R and

using (3.5)

_we ^thus get

Z =

/[DR]e~~l~l/~'Ai ^{(Rid( f~(R)] (3.6)}

The

only configurations entering (3.6)

_are those

obeying

the _gauge

condition,

_as

desired,

but

now we must compute

hi [R]

^{from the definition}

(3.4).

While

(3.6)

is

quite general,

_we will _now

specialize

to the normal _gauge

(2.4)

associated to a

surface

Ito.

The functional delta-function in

(3.6)

tells _{us we} need

only

to examine

hi [R]

^for R

obeying

the

normal-gauge condition;

the delta-function in

(3.4)

then tells _{us we} need

only study k infinitesimally

close to R. Thus _we write

k

_{in the form}

(2.3), kin)

=

R(u) e~(u)e~(u).

A reasonable

choice(~)

for

[Dk]

_{near a}

given

R is then

lDkl

_%

flld~klU,j)/>~) 13.7)

In

(3.7)

_we

imagine

that whatever coordinate system we

choose,

_a

grid

has been laid down with each

point carrying

constant _mass. Since each constituent molecule

occupies

a fixed

physical

area, we can

equivalently specify

that each

grid

cell of size

(Au~, Au~)

_occupy

equal

_areas in 3-space.

(This,

in turn,

typically

_means that the Au~ themselves _are not all the _same. With this

understanding [Dk]

is indeed coordinate-invariant-

Submitting

R to _an infinitesimal coordinate transformation u~ _-+ u~ +

e~(u),

_{we see} that at each u,j

R(u,j)

_moves in the tangent

plane;

the _measure

d3k(uq)/13

_in

(3.7),

restricted _to this

plane,

becomes

l~~jei

x

e2(d~e(uq)

=

l~~fid~e(u,j) _(3.8)

We _{can now} evaluate

(3.4):

AflRl~~

_"

/fll>~~vld~flU~j))d~~~lfalRlUlj

⁺

^f~lUij))))

~

=

/ fl(l~~/jd~e(uq))det ~~[J~b(uq)]d(~)(e~(u~))

,

~

(~) The nonlinear _measure correction to be discussed in the next section will _not affect _{our answer;}

see the Appendix.

where

J~b(u)

=

df~(R(u

+

e))/de~

=

l~~e~,o

_eb in normal _gauge

using (3.3).

Thus

hi [R]

₌

fl

^{~~~~~ ~~}

_(3.9)

~~

To finish _our

job

of

getting

_an

explicit

expression ^{for the}

partition function,

_{we now express}

[DR]d[f~(R)] appearing

in

(3.6)

in terms of the nonredundant

height

^variable

h(u) _appearing

in

(2.4).

We _can

conveniently

_{express an}

arbitrary

R in terms of _a

height change plus

the rest:

Rju)

₌

Roju)

₊

h(u)noiu)

₊

ea,oiu)v~ju) j3.10)

The calculation is not

quite

the _{same as} the _one

leading

to

(3.8);

instead of infinitesimal

reparameterizations

e~ from _an

arbitrary normal-gauge

surface

R,

_{now we} have

displacements

from the

fixed

reference surface Ito.

Accordingly,

instead of

(3.8)

_we find

d~R(u)

=

fi$d~vdh _(3.ll)

Next, (2A)

with

(3.lo) gives f~(R)

₌

l~~e~,o.eb,ov~,

_or ^with

(3.lo) d(~)(f~(R))

₌ ^l~det

~~(e~,o.

eb,o) d(~l(v~).

We _{can now} do the v~

integrals

to get ^from

(3.6)

to

Z =

/ _{lfl l~~dh(u~ iie~~/~'}

,

(3.12)

iJ

Wlle~~

~

^det

^{(ea,o eb)}

~

fl

~ ~~(u~~

13.13)

~

i~

vlW

iJ

It is understood that 7i and

it

are evaluated _on the surface

given

in terms of

h(u) by (2.4).

The last step ^{follows because det}

(e~,o eb)

_"

(ei,o

_{x e2,o}

(ei

_x

e2)

₌

/fino vln.

Note that each factor in

(3.12)

is

explicitly coordinate-invariant,

in contrast with the

analogous

formula in

[14j.

We _see that

(3.13)

indeed does _agree with _our heuristic formula

(3.I). Specializing

to

Monge

gauge, no " I, _{we recover} the result of [3],

it

= ~j~~~

g~l/2

An alternate form for

it

can be obtained

using

eb " dbR

= eb,o ⁺

db(hno

" eb,o

he(Kbc,0

+ dbhno

,

(3.14)

where Kbc,0 ^{is the} curvature tensor of the

background

surface. From this _we find

~

~~ ~~)

~(_ _dahdah

₊

_{O(h ))}

' ~~ ~~~

£

=

§

^~~~

_~~~$

jf

^{~ ~~~}

~j

~

which

again

differs

slightly

from the formula in [14]. ^We see that

getting

the correct

/j

factors in the denominator is crucial to avoid _a

spurious

term linear in h.

Returning

to the

unhappy ending

of section 3, we now see that the

Monge-gauge

Faddeev-

Popov

^factor

it

= exp

~j

^~

_log(1

₊

_(0h)~)

= exp

~j

^~

_(dh)~

₊

(h~)1(3.16)

~

"

~ ~

"

~

effectively

contributes _{a new term to}

(2.12),

_one which is

already

of order T.

Following

the steps

leading

to

(2.15),

_{we get} the corrected version

~~~ ~°

/~ ^{~~(2 ~ oo~i~ioo~}

_{~~ ~~~}

On the other hand

it

_{cannot contribute to}

our calculation of _{T since}

being already

of order T

we

simply

evaluate it at h

= o, ^where it

equals

_one. We still have the

problem

that _rFP

#

_T.

4. Nonlinear _measure.

To _see where _we have erred, let _us first

give

another derivation that _r

= r. Note that there

are many different

Monge

_gauges

corresponding

to

projections

to various reference

planes (~).

So far _we have used _a reference

plane coinciding

with the

plane

of the frame

holding

the

membrane,

_so the

equilibrium configuration

is h

= o, ^but we could instead choose

It~

tilted

by

a small

angle

_~b. We will

always

work to lowest nontrivial order in _~b. In the _new coordinate system the _same

equilibrium

surface is _now described

by h'iv)

= ~bu~

(Fig. 2);

furthermore the

projected

_area has been foreshortened in _one

direction, giving A)

= AB _{cos ~b =}

AB(I

~~b~).

2 Since the two calculations compute the _same

quantity

_we must have that

rjA~;

^h

= oj a TA~

"

~[~B ji~~~Bl ^h'(U)

^{" i~U~)}

(4.1)

~~~

~ ~~~~

_~

~ ~~~~

In the last step we wrote the

O(h'~)

term of r in _terms of

r(0h')~

= r~b~. Thus _we

again

conclude

r = T, contrary to _our still

incomplete

calculation.

v _~

A'~

Fig. 2. Tilting the reference surface foreshortens both the base

area and the coordinate cutoff.

What have _we done

wrong? Throughout

_our discussion of the

previous

section _we assumed

that the _measure

[DR],

and its reduction

[Dh],

were invariant under

changes

^of parameteriza-

tion of the surface. But the choice made in

(2.13)

does not have this property. We chose the

(~) Recall that _a Monge _gauge is _a normal _gauge where the reference surface Ro in figure I is flat.

Alternatively _we could imagine two different physical surfaces related by _a rotation, both viewed in the _same Monge _gauge.

grid points (u~ ) evenly spaced

in the coordinates u~, ^{which has different}

meaning

for different coordinate systems. In

particular

for _our second

description

_we have _a total of

A[ la~ degrees

of

freedom,

which is different from the true number

ABla~

of _mass

points!

More

formally, equation (3.7)

tells _us to use

grid

cells of constant metric area Au~Au~

=

a~llj,

while

(2.13) specifies

constant coordinate _area Au~Au~

= a~. _{We can}

easily

show that this _error accounts for the

remaining discrepancy

between _T and _rFP,

equations (1.2)

and

(3.17).

Let _us denote

by

_{rfp [AB,} A~, A~;

hi

the effective action calculated with the correct Faddeev-

Popov

factor

it

_{but still}

_using

_{the naive}

(coordinate-dependent)

_measure

(2.13).

We momentar-

ily

allow _an

anisotropic

cutoff

(A~, A~),

^which we write

explicitly;

at the end of the discussion

we will set A~

= A~

=

27rla.

The

foregoing

discussion

implies

that _we may use rFP if _we remember to foreshorten the spacing of constituents

(increase

^Al _as well _as the base

plane (Fig. 2),

_so

(4.I)

becomes

rfp

[AB, ~~, ~~,

h

=

0j

⁼

^TAB

~ ~

~ ~

(4.2)

= rfp

[AB -ifi~AB, (I

+

-il~)~, ~; _h'(u)

=

ilu~j

or

since _rFP is the q~ coefficient _we calculated with the naive _measure. Thus _we find

~ ~~~ ~

l~)A~

' ~~'~~

which

really

is

obeyed by

_our calculated

expressions (1.2, 3.17).

We have thus identified the

problem:

the naive _measure

spreads

_mass

points evenly

in coordinate _space, not in real space. For flat

equilibrium

surfaces

(h

linear in

u~),

the

right prescription

_{was easy} to find: _we

keep

^the

grid points

uniform but

change

their

density

to

get ^{the RHS of}

(4.2). Clearly

for nonflat

h(u)

_we will have to be _more

clever,

since here the cutoff is nonuniform and indeed

field-dependent:

the desired _measure

[Dh]

^{is nonlinear in h.}

Incidentally,

the _same

problem

_we face here arises

again

in the

dynamics

context [5], ^where a

naive calculation

fives

the

two-point

function controlled

by

_{our rFP.}

We hinted at the _answer in the Introduction: _a

change

in the

cutoff,

_{even a}

spatially-varying cllange,

amounts to _a local

change

in the number of

degrees

of freedom. We _can

imagine passing

from the desired nonlinear

[Dh]

to the convenient

[Dh]na;ve by

a sort of decimation

procedure,

where _we

integrate e~~/~

over the

discrepant degrees

^{of freedom} to obtain

e~~eff/~

_The

difference d7i

= 7ie~ 7i is

simply

_a counterterm; ^while in

general

it is _a nonlocal functional of

h,

still when _we ask

long-scale

questions it may be

replaced by

_a renormalizable truncation,

just

as we did for 7i itself.

We therefore seek _a correction factor d7i with the property that

"

[Dh]

=

e~~~l~l/~[Dh]na;ve

^"

(4.5)

This

equation

is in quotes ^because as it stands it makes _{no sense: we} cannot equate two

measures with different numbers of

degrees

of freedom. In

particular

d7i is not a Jacobian