‚2P

‚P ‚ ‚U

= P +β 2 (2.100)

‚t ‚x ‚x ‚x

© 2001 by Chapman & Hall/CRC

Equation (2.100) can be converted to the Schroedinger form by en-

forcing the transformation

P = Peq ψ

= e’U/2β ψ (2.101)

where Peq = P0 e’U/β and the normalization condition Peq (x)dx =

1 ¬xes P0 = 1. Note that by setting ‚P = 0 we get the equilibrium

‚t

’»t φ(x) we ¬nd that (2.100) transforms

distribution. Setting ψ = e

to

U2 U

’ βφ + ’ φ = »φ (2.102)

4β 2

where we have used (2.101). Equation (2.102) can be at once recog-

nized to be of supersymmetric nature since we can write it as

βA+ Aφ = »φ (2.103)

where

‚

A= + W (x)

‚x

‚

A+ =’ + W (x) (2.104)

‚x

and the superpotential W (x) is related to U (x) as W (x) = ‚U /2β.

‚x

The zero-eigenvalue of (2.a03) coresponds to Aφ0 = 0 yielding φ0 =

x

1

’ 2β W (y)dy

Be , B a constant. The supersymmetric partner to H+

(the quantity β can be scaled appropriately) is given by H’ ≡ βAA+ .

The eigenvalue that controls the rate at which equilibrium is

approached is the ¬rst excited eigenvalue E1 of H+ component. (Note

the energy eigenvlaues of H+ in increasing order are 0, E1 , E2 , . . .

while those of H’ are E1 , E2 , . . .). E1 is expected to be exponentially

small since from qualitative considerations the potential depicts three

minima and the probability of tunneling-transitions between di¬erent

minima narrows the gap between the lowest and the ¬rst excited

states exponentially. To evaluate E1 it is to be noted that E1 is the

ground-state energy of H’ . Using suitable trial wave functions, E1

can be determined variationally. Such a calculation also gives E1

to be exponentially small as β ’ ∞. For the derivation of Fokker-

Planck equation and explanation of the variational estimate of E1

see [88].

© 2001 by Chapman & Hall/CRC

2.6 Superspace Formalism

An elegant description of SUSY can be made by going over [7,89,90]

to the superspace formalism involving Grassmannian variables and

then constructing theories based on super¬elds of such anticommut-

ing variables. The simplest superspace contains a single Grassmann

variable θ and constitutes what is known as N = 1 supersymmetric

mechanics. The rule for the di¬erentiation and integration of the

Grassmann numbers is given as follows [91]

d

1 = 0,

dθ

d

θj = δij ,

dθi

d

(θk θl ) = δij θl ’ δil θk (2.105)

dθi

dθi θj = δij ,

dθi = 0 (2.106)

The above relations are su¬cient to set up the underlying supersym-

metric Lagrangian.

In the superspace spanned by the ordinary time variable t and

anticommuting θ, we seea invariance of a di¬erential line element un-

der supersymmetric transformations parametrized by the Grassmann

variable . It is easy to realize that under the combined transforma-

tions

θ’θ = θ+

t’t = t+i θ (2.107)

the quantity dt ’ iθdθ goes into itself.

dt ’ iθ dθ ’ dt ’ iθdθ (2.108)

Note that the factor i is inserted in (2.108) to keep the line element

real.

We next de¬ne a real, scalar super¬eld ¦(t, θ) having a general

form

¦(t, θ) = q(t) + iθψ(t) (2.109)

© 2001 by Chapman & Hall/CRC

where ψ(t) is fermionic. Since we are dealing with a single Grassmann

variable θ, the above is the most general representation of ¦(t, θ).

Writing

δ¦ = ¦(t , θ ) ’ ¦(t, θ) (2.110)

we can determine δ¦ to be

δ¦ = i ψ ’ i q

™ (2.111)

Comparison with (2.109) reveals the following transformations for q

and ψ

δq = i ψ, δψ = q ™ (2.112)

These point to a mixing of fermionic (bosonic) variables into the

bosonic (fermionic) counterparts. Further δ¦ can also be expressed

as

δ¦ = Q¦ (2.113)

where,

‚ ‚

Q= + iθ (2.114)

‚θ ‚t

™ ‚

Notice that QQ¦ = iq ’ θψ = i ‚t ¦ implying that Q2 on ¦ gives

™

the time-derivative. Since the generator of time-translation is the

Hamiltonian we can express this result as

{Q, Q} = H (2.115)

Identifying Q as the supersymmetric generator we see that (2.115)

is in a fully supersymmetric form. Also replacing i by ’i in Q we

can de¬ne another operator

‚ ‚

D= ’ iθ (2.116)

‚θ ‚t

which apart from being invariant under (2.107) gives {Q, D} = 0.

The Hamiltonian in (2.115) corresponds to that of a supersym-

metric oscillator. To ¬nd the corresponding Lagrangian we notice

™

™

that D¦ = iψ ’ iθq, and as a result D¦¦ = iψ q + θ(ψ ψ ’ iq 2 ).

™ ™ ™

We can therefore propose the following Lagrangian for N = 1 SUSY

mechanics

i ™

L= dθD¦¦ (2.117)

2

© 2001 by Chapman & Hall/CRC

™

Using (2.106), L can be reduced to L = 1 q 2 + 2 ψ ψ which describes

i

2™

a free particle. It may be checked that δL yields a total derivative.

In fact using (2.109) we ¬nd δL = i2 dt (ψ q).

d

™

In place of the scalar super¬eld ¦(t, θ) if we had considered a

3-vector ¦(t, θ) we would have been led to the nonrelativistic Pauli

Hamiltonian for a quantized spin 1 particle. If however, a 4-vector

2

super¬eld ¦µ (t, θ) was employed we would have gotten a relativistic

supersymmetric version of the Dirac spin 1 particle.

2

We now move on to a formulation of N = 2 supersymmetric

mechanics which involves 2 Grassmannian variables. Let us call them

θ1 and θ2 . The relevant transformations for N = 2 case are

θ1 ’ θ1 = θ1 + 1,

θ2 ’ θ2 = θ2 + 2,

t’t = t + i 1 θ1 + i 2 θ2 (2.118)

with 1 and 2 denoting the parameters of transformations. Ob-

viously, under (2.118) the di¬erential element dt ’ iθ1 dθ1 ’ iθ2 dθ2

remains invariant. For convenience let us adopt a set of√complex

√

representations θ, θ = (θ1 “ iθ2 )/ 2 and , = ( 1 “ i 2 )/ 2. Note

2

that θ2 = 0 = θ and {θ, θ} = 0. Similarly for and . Further, θ

and θ can be considered as complex conjugate to each other.

In the presence of 2 anticommuting variables θ and θ, the N = 2

super¬eld ¦(t, θ, θ) can be written as

¦(t, θ, θ) = q(t) + iθψ(t) + iθψ(t) + θθA(t) (2.119)

where q(t) and A(t) are real variables being bosonic in nature and

(ψ, ψ) are fermionic.

We can determine δ¦ to be

δ¦ = Q+ Q ¦ (2.120)

where Q and Q are the generators of supersymmetric transformations

‚ ‚

Q= + iθ

‚t

‚θ

‚ ‚

Q= + iθ (2.121)

‚θ ‚t

© 2001 by Chapman & Hall/CRC

Note that the supersymmetric transformations (2.118) induce the

following transformations among q(t), ψ(t), ψ(t) and A(t)

δq = i ψ + i ψ

™ ™

δA = ψ ’ ψ

δψ = ’ (q + iA)

™

δψ = ’ (q ’ iA)

™ (2.122)

Further the derivatives

‚ ‚

D= ’ iθ

‚t

‚θ

‚ ‚

D= ’ iθ (2.123)

‚θ ‚t

anticommute with Q and Q. These derivatives act upon ¦ to produce

™

D¦ = iψ ’ θA ’ iθq + θθψ

™

™

D¦ = iψ + θA ’ iθq ’ θθ ψ

™ (2.124)

An educated guess for the N = 2 supersymmetric Lagrangian is

1

L= dθdθ D¦D¦ ’ U (¦) (2.125)

2

where U (¦) is some function of ¦. Expanding U (¦) as U (¦) = U (0)

2

+¦U (0)+ |¦| U (0)+. . . where the derivatives are taken for θ = 0 =

2

θ, we ¬nd on using (2.124), U (¦) = θθ(AU + ψψU ) + . . ., where a

prime denotes a derivative with respect to q. Carrying out the θ and

θ integrations we are thus led to

1 1 ™

L = q 2 + A2 ’ AU + iψ ψ ’ ψψU

™ (2.126)

2 2

‚L

An immediate consequence of this L is that ‚A = 0 (since L

™

is independent of any A term) yielding A = U . L can now be

rearranged to be expressed as

1 12 ™

L = q 2 ’ U + iπ ψ ’ ψψU

™ (2.127)

2 2

© 2001 by Chapman & Hall/CRC

It is not di¬cult to see that if one uses the equations of motion

[66]

™

ψ = iU ψ,

™

ψ = iU ψ (2.128)

L is found to possesi a conserved charge ψψ which is the fermion