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given by [87]
‚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,

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)


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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

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( 42 .)



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