Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch) TITLE: Path integral action of a particle in the noncommutative plane and the Aharonov-Bohm effect

Path integral action, Aharonov-Bohm effect in
the noncommutative plane and dualities from
exact renormalization group
[Phys.Rev.Letters 102 (2009) 241602][J.Phys.A 47 (2014) 075301][J.Phys.A 47 (2014) 235301]
Sunandan GangopadhyayNITheP, Stellenbosch
Department of Physics,
West Bengal State University, Kolkata, India

Motivation : Formulate the path integral representation of
noncommutative quantum mechanics.
Noncommutative geometry implies the absence of
common position eigenkets.
The problem can be circumvented by taking coherent
states to dene the propagation kernel.
The coherent states, being the eigenstates of complex
combinations of the position operators, act as a meaningful
replacement for the position eigenstates admissible only in
the commutative theory.

Formalism
In two dimensions, the coordinates of noncommutative
conguration space satisfy
[^x;^y] =i
The annihilation and creation operators dened by
b=
1
p
2
(^x+i^y),b
y
=
1
p
2
(^xi^y)satisfy the Fock algebra
[b;b
y
] =1.
The noncommutative conguration space is then
isomorphic to the boson Fock space
Hc=spanfjni=
1
p
n!
(b
y
)
n
j0ig
n=1
n=0
where the span is take over the eld of complex numbers.

The next step is to introduce the Hilbert space of the
noncommutative quantum system.
We consider the set of Hilbert-Schmidt operators acting on
noncommutative conguration space
Hq=
n
(^x;^y) : (^x;^y)2 B(Hc);trc(
y
(^x;^y) (^x;^y))<1
o
:
Heretrcdenotes the trace over noncommutative
conguration space andB(Hc)the set of bounded
operators onHc.Hqis the Hilbert space of the
noncommutative quantum system.
Notation : We denote states in the noncommutative
conguration space byjiand states in the quantum
Hilbert space by (^x;^y) j ).
Assuming commutative momenta, a unitary representation
of the noncommutative Heisenberg algebra in terms of
operators
^
X,
^
Y,
^
Pxand
^
Pyis easily found to be
^
X (^x;^y) =^x (^x;^y);
^
Y (^x;^y) =^y (^x;^y)

^
Px (^x;^y) =
~

[^y; (^x;^y)];
^
Py (^x;^y) =
~

[^x; (^x;^y)]: Notation : It is also useful to introduce the following
quantum operatorsB=
1
p
2

^
X+i
^
Y

,
B
z
=
1
p
2

^
Xi
^
Y

,
^
P=
^
Px+i
^
Pyand
^
P
z
=
^
Pxi
^
Py
which act in the following way
B (^x;^y) =b (^x;^y);B
z
(^x;^y) =b
y
(^x;^y)
P (^x;^y) =i~
r
2
[b; (^x;^y)]
P
z
(^x;^y) =i~
r
2
[b
y
; (^x;^y)]:

The minimal uncertainty states on noncommutative
conguration space are well known to be the normalized
coherent states
jzi=e
zz=2
e
zb
y
j0i;z=
1
p
2
(x+iy) Corresponding to these states we can construct a state
(operator) in quantum Hilbert space as follows
jz;z) =
1
p
2
jzihzj
which have the property
Bjz;z) =zjz;z):
To construct the path integral, we also need to introduce
momentum eigenstates normalised such that
(p
0
jp) =(pp
0
)
jp) =
r
2~
2
e
i
q
2~
2
(pb+pb
y
)
;
^
P
ijp) =p
ijp):

Crucial to the path integral construction are the following
completeness relations
Z
d
2
pjp)(pj=1
Q;
Z
2dzdzjz;z)?(z;zj=1
Q
where the star product between two functionsf(z;z)and
g(z;z)is dened as
f(z;z)?g(z;z) =f(z;z)e

@z
!
@z
g(z;z):
The following overlap also plays an impotant role in the
derivation
(z;zjp) =
1
p
2~
2
e

4~
2
pp
e
i
q
2~
2
(pz+pz)
:

With these results in hand, we now proceed to write down
the path integral for the propagation kernel on the two
dimensional noncommutative space. This reads
(z
f;t
fjz0;t0) =lim
n!1
Z
(2)
n
(
n
Y
j=1
dz
jdz
j) (z
f;t
fjzn;tn)?n
(zn;tnj::::jz1;t1)?1(z1;t1jz0;t0):
The next step is to compute the propagator over a small
segment in the above path integral :
(z
j+1;t
j+1jz
j;t
j) =
Z
+1
1
d
2
p
je

2~
2
p
2
je
i
q
2~
2[p
j(z
j+1z
j)+p
j(z
j+1z
j)]
e

i
~
[
p
2
j
2m
+V(z
j+1;z
j)]
+O(
2
)
where,H=
~
P
i
2
2m
+ :V(B
y
;B) :is the Hamiltonian acting on
the quantum Hilbert space andV(
^
X;
^
Y)is the normal
ordered potential expressed in terms of the annihilation
and creation operators (B,B
y
).

Substituting the above expression in the path integral and
computing the star products explicitly, we obtain
(z
f;t
fjz0;t0) =
Zn
Y
j=1
(dx
jdy
j)
n
Y
j=0
d
2
p
j
exp
n
X
j=0
"
i
r
2~
2

p
j(z
j+1z
j) +p
j(z
j+1z
j)

+p
j
p
j
i
~
V(z
j+1;z
j)

exp
0
@

2~
2
n1
X
j=0
p
j+1
p
j
1
A
where=(
i
2m~
+

2~
2). An important difference between the commutative and
noncommutative cases is that in the latter the momentum
integral involves off diagonal terms that couplep
jandp
j+1.

To perform the momentum integral, it is convenient to
make the following identicationpn+1=p0and to recast
the integrand of the above integral into the following form:
exp

~
@z
f
~
@z0

exp
n
X
j=0
"
i
r
2~
2
[p
j(z
j+1z
j) +p
j(z
j+1z
j)]
+p
j
p
j
i
~
V(z
j+1;z
j) +

2~
2
p
j+1
p
j

:
The purpose of the boundary operator in the above
expression is to cancel an additional coupling that has
been introduced betweenp0andpn.
Performing thep-integral and taking the limit!0, we
nally obtain the path integral representation of the
propagator
(z
f;t
fjz0;t0) =N
Z
DzDzexp

~
@z
f
~
@z0

exp(
i
~
S)

whereSis the action given by
S=
Z
t
f
t0
dt

1
2
_z(t)(
1
2m
+
i
2~
@t)
1
_z(t)V(z(t);z(t))

: Free particle propagator :
The classical equation of motion obtained from the above
action is of the following form
Kzc(t) =0;K= (
1
2m
+
i
2~
@t)
1
:
Solving the above equation subjected to the boundary
conditions that att=t0,zc=z0,t=t
f,zc=z
f, we obtain
zc(t) =z0+
z
fz0
T
(tt0):
Substituting the above solution in the path integral (with
V=0), we obtain
(z
f;t
fjz0;t0) =Nexp

m
2(i~T+m)
(~x
f~x0)
2

; (n+1)=T
where we have dumped the contribution coming from the
uctuations in the normalisation constantN.

To determine this constant, we use the following identity
(z
f;t
fjp) =2
Z
dz0dz0(z
f;t
fjz0;t0)?0(z0;t0jp)
which xes the constantN=
m
2(m+i~T)
. Harmonic oscillator :
We now include the harmonic oscillator potential
V=
1
2
m!
2
(
^
X
2
+
^
Y
2
)in the Hamiltonian.
Normal ordered form of this potential in terms of the
creation and annihilation operators is:V:=kB
y
B,
(k=m!
2
).
The action for the harmonic oscillator therefore reads
S=
Z
t
f
t0
dt

1
2
_z(t)(
1
2m
+
i
2~
@t)
1
_z(t)kz(t)z(t)

:
The equation of motion following from the above action is
of the following form
Kzc(t) +2kzc(t) =0:

Making an ansatz of the solution of the above equation in
the form
zc(t) =a1e
i1t
+a2e
i2t
leads to the following energy eigen-values for the harmonic
oscillator
1=
1
2~
(m!
2
+!
p
m
2
!
2

2
+4~
2
)
2=
1
2~
(m!
2
+!
p
m
2
!
2

2
+4~
2
):
Remarkably, the energy spectrum computed from the path
integral matches with the existing results in the literature
obtained by Bopp-shift or by working with noncommutative
variables only.
Central result : We obtain the action of a particle in an
arbitrary potential moving in the noncommutative plane
from the path integral.

Path integral in the presence of a magnetic eld
The Hamiltonian (acting on the quantum Hilbert space) for
a particle in a magnetic eld in the presence of a potential
on the noncommutative plane reads
^
H=
(
^
~
Pe
^
~
A)
2
2m
+ :V(
^
B
y
;
^
B) : In the symmetric gauge
^
~
A=

B
2
^
Y;
B
2
^
X

the above Hamiltonian takes the form
^
H=
^
~
P
2
2m
+
e
2
B
2
8m
(
^
X
2
+
^
Y
2
)
eB
2m
(
^
X
^
Py
^
Y
^
Px)+ :V(
^
B
y
;
^
B) ::

Rerunning the earlier method with this Hamiltonian, we
arrive at the path integral representation of the propagator
with the action being given by
S=
Z
t
f
t0
dt
"

2

_z(t)
ieB
2m
z(t)

1
2m
+
i
2~
@t

1

_z(t) +
ieB
2m
z(t)

e
2
B
2

4m
z(t)z(t)V(z(t);z(t))

: In theV!0 limit, the above expression yields the two
frequencies for the particle in a magnetic eld on the
noncommutative plane to be
=
eB
m

1+
eB
4~

;0:

Aharonov-Bohm effect
To proceed, we rst observe that the action can be recast
in the following form
S=
Z
t
f
t0
dt
"
m

1+
eB
2~

2
_z(t)

1+
im
~
@t

1
_z(t)
+ieB

1+
eB
4~

_z(t)z(t)

: This can be mapped to a particle of zero mass moving in
the commutative plane and in a magnetic eld given by
B=
2~
e
:

Indeed, with this choice of the magnetic eld:
S=
ieB
2
Z
t
f
t0
dt_z(t)z(t)
=
eB
4
Z
t
f
t0
dt[_x(t)y(t)_y(t)x(t)]
=
e
2
Z
~x
f
~x0
~
A:d~x It is evident from the rst line that this is a constrained
system with the following second class constraints
1=px+
eB
4
y0
2=py
eB
4
x0:

Introducing the Dirac bracket and replacing
f:; :g
DB!
1
i~
[:; :]yield the following noncommutative
algebra
[x
i;x
j] =i
2~
eB

ij=i
ij; [x
i;p
j] =
i~
2

ij;
[p
i;p
j] =i~
eB
8

ij=
i~
2
4

ij; (i;j=1;2) Note that this noncommutativity was observed earlier by
observing that in the limitm!0, they-coordinate is
effectively constrained to the momentum canonical
conjugate to thex-coordinate. However, in the path
integral approach, the mass zero limit arises naturally.

With the usual Aharonov-Bohm experimental set up, one
can now easily read off the Aharonov-Bohm
phase-differencefrom the action to be=
eBA
2~
=
e
2~
whereAis the area enclosed by the loop around which the
particle is transported andis the total magnetic ux
enclosed by this loop.
The general result for the AB phase is
=
eB
~

1+
eB
4~

A:

An elegant way of obtaining the Aharonov-Bohm-phase is
by transporting a particle in a closed loop.
This can be done by the action of a chain of translation
operators on the wave-function as follows
e

i
~
^yy
e

i
~
^xx
e
i
~
^yy
e
i
~
^xx
:
Now using the identity
^
S
1
e
^
A^
S=e
^
S
1^
A
^
S
and the
Baker-Campbell-Hausdorff formula, the above expression
can be simplied toe
i
~
xyeB(1+
eB
4~)

which yield the AB phase.

Path integral in phase-space representation
The phase-space representation of the path integral reads
(z
f;t
fjz0;t0) =lim
n!1
Zn
Y
j=1
(dz
jdz
j)
n
Y
j=0
d
2
p
je

~
@z
f
~
@z
0
exp
n
X
j=0
"
i
~
r
2

p
j

z
j+1z
j

+c:c:

+p
jp
j
+

2~
2
p
j+1
p
j+
eB
2m~
r
2
(p
j
z
j+1p
jz
j)
#
: Now using
p
jp
j+

2~
2
p
j+1
p
j=
i
2m~
p
jp
j+

2~
2
p
j(p
j+1p
j)

and the fact thatz
j=z(j)andz
j+1=z
j+_z(j) +O(
2
)
followed by the!0 limit leads to the phase-space form of the
path integral with the following form of the action
S=
Z
t
f
t0
dt
"r
2
(p_z+p_z)
pp
2m

i
2~
p_p
ieB
2m
r
2
(pzpz)
#
=
Z
t
f
t0
dt
"
(p
i+eA
i)_x
i+

2~

ijp
i
_p
j
p
2
i
2m
#
:whereA
i=
B
2

ijx
j;(i;j=1;2). Thep_pterm is the Chern-Simons term (in momentum)
noncommutative in origin.

Key observations
We have obtained the action for a particle in a
noncommutative plane from a path integral formulation.
The action indicates that noncommutative theories are
higher order time derivative theories.
The Aharonov-Bohm phase involves a correction in the
noncommutative parameter.
The result is exact upto all orders in. Noncommutative quantum mechanics is basically a
quantum mechanical theory with a Chern-Simons term in
momentum.

Exact renormalization group and noncommutativity
The central idea involves the introduction of an UV cutoff
function
K(p
2
=`
2
), which has the property that it vanishes
when
p> `. The ERGE is then obtained by requiring that the process of
reducing the number of degrees of freedom leaves the
generating functional
Z[J]invariant. Condition imposed :J(p) =0 forp> `,@`K
1
(p
2
=`
2
) =0
for smallp.
This implies that the effective theory can only yield
information on correlation functions of the original theory in
as far as they are computed below the momentum cutoff.
Question : Whether the approach can be extended by
relaxing the conditions.

This would allow the computation of all the correlation
functions of the original theory in terms of the correlation
functions of the effective theory, thereby establishing a
complete duality between them.
This necessitates the ow of the sources together with the
interacting part of the action.
We start with the following action :
S[;

] =
Z
d!

(!)K(!; `)(!) +S
I[;

] +J`[;

]
K(!; `)takes the standard form!
2
in the`!0 limit and
J`[;

]is a generalised source term determined by the
requirement of invariance of the generating functional.

We consider actions which are quadratic in the elds for
which it is sufcient to limit the form ofJ`[;

]to be linear,
i.e., we take
J`[;

] =
Z
d![J0(l) +J

0
(l) +J1(l)

(!) +J

1
(l)(!)]
Impose the initial conditions
J0(`)j`=0=J

0
(`)j`=0=0
J1(`)j`=0=J1(0);J

1
(`)j`=0=J

1
(0):
The normalised generating functional is given by
Z[J`] =
R
[dd

]e
(S0[;

]+S
I[;

]+J`[;

])
R
[dd

]e
(S0[;

]+S
I[;

])
: Invariance of the generating functional
@`Z[J`] =0:

Equations for the interacting part and source terms :
@`S
I=
Z
d! @`K
1

S
I

(!)
S
I
(!)

2
S
I

(!)(!)

@`J`=
Z
d! @`K
1

S
I
(!)
J`

(!)
+
S
I

(!)
J`
(!)
+
J`

(!)
J`
(!)

2
J`

(!)(!)

: These equations can easily be solved when the interaction
term is quadratic in the elds, i.e.,
S
I[;

] =
Z
d!g(!; `)

(!)(!):
Source terms :
@`J1(`) =@`K
1
(!; `)g(!; `)J1(`)
@`[J0(`) +J

0
(`)] =@`K
1
(!; `)jJ1(`)j
2
:

Solutions :
J1(`) =J1(0)exp
Z
`
0
d`
0
g(!; `
0
)@`
0K
1
(!; `
0
)

J0(`) +J

0
(`) =jJ1(0)j
2
Z
`
0
d`
0
@`
0K
1
(!; `
0
)
exp

2
Z
`
0
0
d`
00
g(!; `
00
)@`
00K
1
(!; `
00
)
!
:
Landau problem :
S=
Z
d![mz(!)!
2
z(!) +eB!z(!)z(!)
+J(!)z(!) +J

(!)z(!)]:
Choice ofK(!; `):
K(!; `) =
m!
2
(1
m!`~
)
:

Initial condition
g(!; `)j`=0=eB!:
ow equation for the coefcientg(!; `):
@g(!; `)
@`
=
1
~!
g
2
(!; `): Integration of this equation subject to the initial condition
gives
g(!; `) =e
~
B(`)!;
~
B(`) =
B
(1+
eB`~
)
:
Solution for the sources :
J1(`) =
J1(0)
(1+
eB`~
)
J0(`) +J

0
(`) =
jJ1(0)j
2
`
~!(1+
eB`~
)
:

Rescaling the coordinates as
~z(!) =
s
1
e
~
B(`)`
~
z(!)we obtain the nal form of the action :
S=
Z
d!
"
~z(!)K(!; `)~z(!) +
eB!
(1
m!`~
)
~z(!)~z(!)

jJ1(0)j
2
`
~!(1+
eB`~
)
+
0
@
J1(0)
q
1+
eB`
~
~z(!) +c:c:
1
A
3
5:
Action of a particle moving in a magnetic eldB

in a
noncommutative plane with noncommutative parameter`
and source terms:S
NC=
Z
d!
"
~z(!)K(!; `)~z(!) +
eB

(1+
eB

`
4~
)!
(1
m!`~
)

~z(!)~z(!) +J
~z(!) +

J~z(!)
i
:

Blocking procedure
This result admits a different interpretation in terms of a
blocking procedure, where the blocking is performed over
time.
It is possible to interpret the renormalization ow as a
change of variables in the path integral.
In this case a simple change of variables involving a!
dependent scaling,
~z(!) =
p
1m!`=~z(!)
, transforms
the commutative action into the noncommutative action.
This corresponds to the following blocking relation
~z(t) =
Z
+1
1
dt
0
e
i!0(t
0
t)
p
!0(t
0
t)
3=2
z(t
0
)where!0=
~
m`
.

This relation clearly reveals the fact that spatial
noncommutativity can be seen as a blocking procedure
over time.
It is also reassuring to check, using the commutator ofz(t)
andz(t
0
)(t6=t
0
)
[z(t);z(t
0
)] =2~(e
ieB(tt
0
)=m
1)=(eB)
that the equal time commutator of~z(t)and
~z(t)is given by[~z(t);
~z(t)] =c`. This is consistent with our rationale of mapping the original
theory onto a noncommutative theory with
noncommutative parameter`.
It is important to note that this only happens if the coarse
graining relation above is used.

Observations
We have constructed one parameter families of dual
theories using the ERGE for the Landau problem.
The precise choice made here for the kinetic energy term
established these dual families to be noncommutative
theories.
We also observe a subtle link between noncommutative
theories and the blocking procedure in the ERG approach.
The latter may open up an avenue to explore these
relations in higher dimensional interacting eld theories.

Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch) TITLE: Path integral action of a particle in the noncommutative plane and the Aharonov-Bohm effect

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Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch) TITLE: Path integral action of a particle in the noncommutative plane and the Aharonov-Bohm effect

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