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 inthe 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 Gangopadhyay

NITheP, Stellenbosch

Department of Physics,

West Bengal State University, Kolkata, India

Motivation : Formulate the path integral representation ofnoncommutative quantum mechanics.

Noncommutative geometry implies the absence ofcommon position eigenkets.

The problem can be circumvented by taking coherentstates to define the propagation kernel.

The coherent states, being the eigenstates of complexcombinations of the position operators, act as a meaningfulreplacement for the position eigenstates admissible only inthe commutative theory.













FormalismIn two dimensions, the coordinates of noncommutativeconfiguration space satisfy

[x , y ] = iθ

The annihilation and creation operators defined byb = 1√

2θ(x + i y), b† = 1√

2θ(x − i y) satisfy the Fock algebra

[b,b†] = 1.

The noncommutative configuration space is thenisomorphic to the boson Fock space

Hc = span|n〉 =1√n!

(b†)n|0〉n=∞n=0

where the span is take over the field of complex numbers.


[x , y ] = iθ


2θ(x + i y), b† = 1√


[b,b†] = 1.



(b†)n|0〉n=∞n=0



[x , y ] = iθ


2θ(x + i y), b† = 1√


[b,b†] = 1.



(b†)n|0〉n=∞n=0


The next step is to introduce the Hilbert space of thenoncommutative quantum system.

We consider the set of Hilbert-Schmidt operators acting onnoncommutative configuration space

Hq =ψ(x , y) : ψ(x , y) ∈ B (Hc) , trc(ψ

†(x , y)ψ(x , y)) <∞.

Here trc denotes the trace over noncommutativeconfiguration space and B (Hc) the set of boundedoperators on Hc . Hq is the Hilbert space of thenoncommutative quantum system.

Notation : We denote states in the noncommutativeconfiguration space by |·〉 and states in the quantumHilbert space by ψ(x , y) ≡ |ψ).

Assuming commutative momenta, a unitary representationof the noncommutative Heisenberg algebra in terms ofoperators X , Y , Px and Py is easily found to be

Xψ(x , y) = xψ(x , y) , Yψ(x , y) = yψ(x , y)

The next step is to introduce the Hilbert space of thenoncommutative quantum system.We consider the set of Hilbert-Schmidt operators acting onnoncommutative configuration space


†(x , y)ψ(x , y)) <∞.







†(x , y)ψ(x , y)) <∞.







†(x , y)ψ(x , y)) <∞.





Pxψ(x , y) =~θ[y , ψ(x , y)] , Pyψ(x , y) = −~

θ[x , ψ(x , y)] .

Notation : It is also useful to introduce the followingquantum operators B = 1√

2θ

(X + i Y

),

B‡ = 1√2θ

(X − i Y

), P = Px + i Py and P‡ = Px − i Py

which act in the following way

Bψ(x , y) = bψ(x , y) , B‡ψ(x , y) = b†ψ(x , y)

Pψ(x , y) = −i~√

2θ[b, ψ(x , y)]

P‡ψ(x , y) = i~√

2θ[b†, ψ(x , y)].

The minimal uncertainty states on noncommutativeconfiguration space are well known to be the normalizedcoherent states

|z〉 = e−zz/2ezb† |0〉 , z =1√2θ

(x + iy)

Corresponding to these states we can construct a state(operator) in quantum Hilbert space as follows

|z, z) =1√2πθ

|z〉〈z|

which have the property

B|z, z) = z|z, z).

To construct the path integral, we also need to introducemomentum eigenstates normalised such that(p′|p) = δ(p − p′)

|p) =

√θ

2π~2 eiq

θ

2~2 (pb+pb†), Pi |p) = pi |p) .


|z〉 = e−zz/2ezb† |0〉 , z =1√2θ

(x + iy)


|z, z) =1√2πθ

|z〉〈z|


B|z, z) = z|z, z).


|p) =

√θ

2π~2 eiq

θ

2~2 (pb+pb†), Pi |p) = pi |p) .


|z〉 = e−zz/2ezb† |0〉 , z =1√2θ

(x + iy)


|z, z) =1√2πθ

|z〉〈z|


B|z, z) = z|z, z).


|p) =

√θ

2π~2 eiq

θ

2~2 (pb+pb†), Pi |p) = pi |p) .

Crucial to the path integral construction are the followingcompleteness relations∫

d2p |p)(p| = 1Q ,

∫2θdzdz |z, z) ? (z, z| = 1Q

where the star product between two functions f (z, z) andg(z, z) is defined 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 thederivation

(z, z|p) =1√

2π~2e−

θ

4~2 ppeiq

θ

2~2 (pz+pz).

Crucial to the path integral construction are the followingcompleteness relations∫

d2p |p)(p| = 1Q ,

∫2θdzdz |z, z) ? (z, z| = 1Q

where the star product between two functions f (z, z) andg(z, z) is defined 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 thederivation

(z, z|p) =1√

2π~2e−

θ

4~2 ppeiq

θ

2~2 (pz+pz).

With these results in hand, we now proceed to write downthe path integral for the propagation kernel on the twodimensional noncommutative space. This reads

(zf , tf |z0, t0) = limn→∞

∫(2θ)n(

n∏j=1

dzjdzj) (zf , tf |zn, tn) ?n

(zn, tn|....|z1, t1) ?1 (z1, t1|z0, t0) .

The next step is to compute the propagator over a smallsegment in the above path integral :

(zj+1, tj+1|zj , tj) =

∫ +∞

−∞d2pj e−

θ

2~2 p2j ei

qθ

2~2 [pj (zj+1−zj )+pj (zj+1−zj )]

×e−i~ τ [

p2j

2m +V (zj+1,zj )] + O(τ2)

where, H =~Pi

2

2m + : V (B†,B) : is the Hamiltonian acting onthe quantum Hilbert space and V (X , Y ) is the normalordered potential expressed in terms of the annihilationand creation operators (B, B†).

With these results in hand, we now proceed to write downthe path integral for the propagation kernel on the twodimensional noncommutative space. This reads

(zf , tf |z0, t0) = limn→∞

∫(2θ)n(

n∏j=1

dzjdzj) (zf , tf |zn, tn) ?n

(zn, tn|....|z1, t1) ?1 (z1, t1|z0, t0) .

The next step is to compute the propagator over a smallsegment in the above path integral :

(zj+1, tj+1|zj , tj) =

∫ +∞

−∞d2pj e−

θ

2~2 p2j ei

qθ

2~2 [pj (zj+1−zj )+pj (zj+1−zj )]

×e−i~ τ [

p2j

2m +V (zj+1,zj )] + O(τ2)

where, H =~Pi

2

2m + : V (B†,B) : is the Hamiltonian acting onthe quantum Hilbert space and V (X , Y ) is the normalordered potential expressed in terms of the annihilationand creation operators (B, B†).

Substituting the above expression in the path integral andcomputing the star products explicitly, we obtain

(zf , tf |z0, t0) =

∫ n∏j=1

(dxjdyj)n∏

j=0

d2pj

expn∑

j=0

[i

√θ

2~2

[pj(zj+1 − zj) + pj(zj+1 − zj)

]+αpj pj −

i~τV (zj+1, zj)

]

×exp

θ

2~2

n−1∑j=0

pj+1pj

where α = −( iτ

2m~ + θ2~2 ).

An important difference between the commutative andnoncommutative cases is that in the latter the momentumintegral involves off diagonal terms that couple pj and pj+1.

Substituting the above expression in the path integral andcomputing the star products explicitly, we obtain

(zf , tf |z0, t0) =

∫ n∏j=1

(dxjdyj)n∏

j=0

d2pj

expn∑

j=0

[i

√θ

2~2

[pj(zj+1 − zj) + pj(zj+1 − zj)

]+αpj pj −

i~τV (zj+1, zj)

]

×exp

θ

2~2

n−1∑j=0

pj+1pj

where α = −( iτ

2m~ + θ2~2 ).

An important difference between the commutative andnoncommutative cases is that in the latter the momentumintegral involves off diagonal terms that couple pj and pj+1.

To perform the momentum integral, it is convenient tomake the following identification pn+1 = p0 and to recastthe integrand of the above integral into the following form:

exp(−~∂zf

~∂z0

)exp

n∑j=0

[i

√θ

2~2 [pj(zj+1 − zj) + pj(zj+1 − zj)]

+αpj pj −i~τV (zj+1, zj) +

θ

2~2 pj+1pj

].

The purpose of the boundary operator in the aboveexpression is to cancel an additional coupling that hasbeen introduced between p0 and pn.

Performing the p-integral and taking the limit τ → 0, wefinally obtain the path integral representation of thepropagator

(zf , tf |z0, t0) = N∫DzDz exp

(−~∂zf

~∂z0

)exp(

i~

S)

To perform the momentum integral, it is convenient tomake the following identification pn+1 = p0 and to recastthe integrand of the above integral into the following form:

exp(−~∂zf

~∂z0

)exp

n∑j=0

[i

√θ

2~2 [pj(zj+1 − zj) + pj(zj+1 − zj)]

+αpj pj −i~τV (zj+1, zj) +

θ

2~2 pj+1pj

].

The purpose of the boundary operator in the aboveexpression is to cancel an additional coupling that hasbeen introduced between p0 and pn.

Performing the p-integral and taking the limit τ → 0, wefinally obtain the path integral representation of thepropagator

(zf , tf |z0, t0) = N∫DzDz exp

(−~∂zf

~∂z0

)exp(

i~

S)

where S is the action given by

S =

∫ tf

t0dt θ

[12

˙z(t)(1

2m+

iθ2~∂t)

−1z(t)− V (z(t), z(t))].

Free particle propagator :The classical equation of motion obtained from the aboveaction is of the following form

K zc(t) = 0 ; K = (1

2m+

iθ2~∂t)

−1 .

Solving the above equation subjected to the boundaryconditions that at t = t0, zc = z0, t = tf , zc = zf , we obtain

zc(t) = z0 +zf − z0

T(t − t0) .

Substituting the above solution in the path integral (withV = 0), we obtain

(zf , tf |z0, t0) = N exp[− m

2(i~T + mθ)(~xf − ~x0)

2]; (n + 1)τ = T

where we have dumped the contribution coming from thefluctuations in the normalisation constant N.

To determine this constant, we use the following identity

(zf , tf |p) = 2θ∫

dz0dz0 (zf , tf |z0, t0) ?0 (z0, t0|p)

which fixes the constant N = m2π(θm+i~T ) .

Harmonic oscillator :We now include the harmonic oscillator potentialV = 1

2mω2(X 2 + Y 2) in the Hamiltonian.Normal ordered form of this potential in terms of thecreation and annihilation operators is : V := kθB†B,(k = mω2).The action for the harmonic oscillator therefore reads

S =

∫ tf

t0dt θ

[12

˙z(t)(1

2m+

iθ2~∂t)

−1z(t)− kz(t)z(t)].

The equation of motion following from the above action isof the following form

K zc(t) + 2kzc(t) = 0 .


(zf , tf |p) = 2θ∫

dz0dz0 (zf , tf |z0, t0) ?0 (z0, t0|p)



2mω2(X 2 + Y 2) in the Hamiltonian.Normal ordered form of this potential in terms of thecreation and annihilation operators is : V := kθB†B,(k = mω2).

The action for the harmonic oscillator therefore reads

S =

∫ tf

t0dt θ

[12

˙z(t)(1

2m+

iθ2~∂t)

−1z(t)− kz(t)z(t)].


K zc(t) + 2kzc(t) = 0 .


(zf , tf |p) = 2θ∫

dz0dz0 (zf , tf |z0, t0) ?0 (z0, t0|p)



2mω2(X 2 + Y 2) in the Hamiltonian.Normal ordered form of this potential in terms of thecreation and annihilation operators is : V := kθB†B,(k = mω2).The action for the harmonic oscillator therefore reads

S =

∫ tf

t0dt θ

[12

˙z(t)(1

2m+

iθ2~∂t)

−1z(t)− kz(t)z(t)].


K zc(t) + 2kzc(t) = 0 .

Making an ansatz of the solution of the above equation inthe form

zc(t) = a1eiγ1t + a2eiγ2t

leads to the following energy eigen-values for the harmonicoscillator

γ1 =12~

(mω2θ + ω√

m2ω2θ2 + 4~2)

γ2 =12~

(−mω2θ + ω√

m2ω2θ2 + 4~2) .

Remarkably, the energy spectrum computed from the pathintegral matches with the existing results in the literatureobtained by Bopp-shift or by working with noncommutativevariables only.Central result : We obtain the action of a particle in anarbitrary potential moving in the noncommutative planefrom the path integral.

Path integral in the presence of a magnetic fieldThe Hamiltonian (acting on the quantum Hilbert space) fora particle in a magnetic field in the presence of a potentialon the noncommutative plane reads

H =(~P − e~A)2

2m+ : V (B†, B) :

In the symmetric gauge

~A =

(−B

2Y ,

B2

X)

the above Hamiltonian takes the form

H =~P2

2m+

e2B2

8m(X 2 + Y 2)− eB

2m(X Py − Y Px)+ : V (B†, B) : .

Path integral in the presence of a magnetic fieldThe Hamiltonian (acting on the quantum Hilbert space) fora particle in a magnetic field in the presence of a potentialon the noncommutative plane reads

H =(~P − e~A)2

2m+ : V (B†, B) :

In the symmetric gauge

~A =

(−B

2Y ,

B2

X)

the above Hamiltonian takes the form

H =~P2

2m+

e2B2

8m(X 2 + Y 2)− eB

2m(X Py − Y Px)+ : V (B†, B) : .

Rerunning the earlier method with this Hamiltonian, wearrive at the path integral representation of the propagatorwith the action being given by

S =

∫ tf

t0dt

[θ

2

˙z(t)− ieB

2mz(t)

(1

2m+

iθ2~∂t

)−1

z(t) +

ieB2m

z(t)− e2B2θ

4mz(t)z(t)− V (z(t), z(t))

].

In the V → 0 limit, the above expression yields the twofrequencies for the particle in a magnetic field on thenoncommutative plane to be

γ =eBm

(1 +

eBθ4~

),0 .

Rerunning the earlier method with this Hamiltonian, wearrive at the path integral representation of the propagatorwith the action being given by

S =

∫ tf

t0dt

[θ

2

˙z(t)− ieB

2mz(t)

(1

2m+

iθ2~∂t

)−1

z(t) +

ieB2m

z(t)− e2B2θ

4mz(t)z(t)− V (z(t), z(t))

].

In the V → 0 limit, the above expression yields the twofrequencies for the particle in a magnetic field on thenoncommutative plane to be

γ =eBm

(1 +

eBθ4~

),0 .

Aharonov-Bohm effect

To proceed, we first observe that the action can be recastin the following form

S =

∫ tf

t0dt

[θm(

1 +eBθ2~

)2˙z(t)

(1 +

iθm~∂t

)−1

z(t)

+ieBθ(

1 +eBθ4~

)˙z(t)z(t)

].

This can be mapped to a particle of zero mass moving inthe commutative plane and in a magnetic field given by

B = −2~eθ.

Aharonov-Bohm effect

To proceed, we first observe that the action can be recastin the following form

S =

∫ tf

t0dt

[θm(

1 +eBθ2~

)2˙z(t)

(1 +

iθm~∂t

)−1

z(t)

+ieBθ(

1 +eBθ4~

)˙z(t)z(t)

].

This can be mapped to a particle of zero mass moving inthe commutative plane and in a magnetic field given by

B = −2~eθ.

Indeed, with this choice of the magnetic field:

S =ieBθ

2

∫ tf

t0dt ˙z(t)z(t)

= −eB4

∫ tf

t0dt [x(t)y(t)− y(t)x(t)]

=e2

∫ ~xf

~x0

~A.d~x

It is evident from the first line that this is a constrainedsystem with the following second class constraints

Ω1 = px +eB4

y ≈ 0

Ω2 = py −eB4

x ≈ 0 .

Indeed, with this choice of the magnetic field:

S =ieBθ

2

∫ tf

t0dt ˙z(t)z(t)

= −eB4

∫ tf

t0dt [x(t)y(t)− y(t)x(t)]

=e2

∫ ~xf

~x0

~A.d~x

It is evident from the first line that this is a constrainedsystem with the following second class constraints

Ω1 = px +eB4

y ≈ 0

Ω2 = py −eB4

x ≈ 0 .

Introducing the Dirac bracket and replacing., .DB → 1

i~ [., .] yield the following noncommutativealgebra

[xi , xj ] = −i2~eB

εij = iθεij ; [xi ,pj ] =i~2δij ;

[pi ,pj ] = −i~eB8εij =

i~2

4θεij ; (i , j = 1,2)

Note that this noncommutativity was observed earlier byobserving that in the limit m → 0, the y -coordinate iseffectively constrained to the momentum canonicalconjugate to the x-coordinate. However, in the pathintegral approach, the mass zero limit arises naturally.

Introducing the Dirac bracket and replacing., .DB → 1

i~ [., .] yield the following noncommutativealgebra

[xi , xj ] = −i2~eB

εij = iθεij ; [xi ,pj ] =i~2δij ;

[pi ,pj ] = −i~eB8εij =

i~2

4θεij ; (i , j = 1,2)

Note that this noncommutativity was observed earlier byobserving that in the limit m → 0, the y -coordinate iseffectively constrained to the momentum canonicalconjugate to the x-coordinate. However, in the pathintegral approach, the mass zero limit arises naturally.

With the usual Aharonov-Bohm experimental set up, onecan now easily read off the Aharonov-Bohmphase-difference φ from the action to be

φ =eBA2~

=eΦ

2~

where A is the area enclosed by the loop around which theparticle is transported and Φ is the total magnetic fluxenclosed by this loop.

The general result for the AB phase is

φ =eB~

(1 +

eBθ4~

)× A.

With the usual Aharonov-Bohm experimental set up, onecan now easily read off the Aharonov-Bohmphase-difference φ from the action to be

φ =eBA2~

=eΦ

2~

where A is the area enclosed by the loop around which theparticle is transported and Φ is the total magnetic fluxenclosed by this loop.The general result for the AB phase is

φ =eB~

(1 +

eBθ4~

)× A.

An elegant way of obtaining the Aharonov-Bohm-phase isby transporting a particle in a closed loop.

This can be done by the action of a chain of translationoperators on the wave-function as follows

e−i~ πy∆ye−

i~ πx∆xe

i~ πy∆ye

i~ πx∆xΨ .

Now using the identity S−1eAS = eS−1AS and theBaker-Campbell-Hausdorff formula, the above expressioncan be simplified to

ei~ ∆x∆yeB(1+ eBθ

4~ )Ψ

which yield the AB phase.



e−i~ πy∆ye−

i~ πx∆xe

i~ πy∆ye

i~ πx∆xΨ .



4~ )Ψ




e−i~ πy∆ye−

i~ πx∆xe

i~ πy∆ye

i~ πx∆xΨ .



4~ )Ψ


Path integral in phase-space representation

The phase-space representation of the path integral reads

(zf , tf |z0, t0) = limn→∞

∫ n∏j=1

(dzjdzj)n∏

j=0

d2pje−~∂zf

~∂z0

×expn∑

j=0

[i~

√θ

2[pj

zj+1 − zj

+ c.c.]+ αpjpj

+θ

2~2 pj+1pj +eB

2m~

√θ

2τ(pj zj+1 − pjzj)

].

Now using

αpjpj +θ

2~2 pj+1pj = − iτ2m~

pjpj +θ

2~2 pj(pj+1 − pj)

Path integral in phase-space representation

The phase-space representation of the path integral reads

(zf , tf |z0, t0) = limn→∞

∫ n∏j=1

(dzjdzj)n∏

j=0

d2pje−~∂zf

~∂z0

×expn∑

j=0

[i~

√θ

2[pj

zj+1 − zj

+ c.c.]+ αpjpj

+θ

2~2 pj+1pj +eB

2m~

√θ

2τ(pj zj+1 − pjzj)

].

Now using

αpjpj +θ

2~2 pj+1pj = − iτ2m~

pjpj +θ

2~2 pj(pj+1 − pj)

and the fact that zj = z(jτ) and zj+1 = zj + τ z(jτ) + O(τ2)followed by the τ → 0 limit leads to the phase-space form of thepath integral with the following form of the action

S =

∫ tf

t0dt

[√θ

2(p ˙z + pz)− pp

2m− iθ

2~pp − ieB

2m

√θ

2(pz − pz)

]

=

∫ tf

t0dt

[(pi + eAi)xi +

θ

2~εijpi pj −

p2i

2m

].

where Ai = −B2 εijxj , (i , j = 1,2).

The εpp term is the Chern-Simons term (in momentum)noncommutative in origin.

Key observations

We have obtained the action for a particle in anoncommutative plane from a path integral formulation.

The action indicates that noncommutative theories arehigher order time derivative theories.

The Aharonov-Bohm phase involves a correction in thenoncommutative parameter θ.

The result is exact upto all orders in θ.

Noncommutative quantum mechanics is basically aquantum mechanical theory with a Chern-Simons term inmomentum.

Key observations






Key observations






Key observations






Key observations






Exact renormalization group and noncommutativity

The central idea involves the introduction of an UV cutofffunction K (p2/`2), which has the property that it vanisheswhen p > `.

The ERGE is then obtained by requiring that the process ofreducing the number of degrees of freedom leaves thegenerating functional Z [J] invariant.

Condition imposed : J(p) = 0 for p > `, ∂`K−1(p2/`2) = 0for small p.

This implies that the effective theory can only yieldinformation on correlation functions of the original theory inas far as they are computed below the momentum cutoff.

Question : Whether the approach can be extended byrelaxing the conditions.

























This would allow the computation of all the correlationfunctions of the original theory in terms of the correlationfunctions of the effective theory, thereby establishing acomplete duality between them.

This necessitates the flow of the sources together with theinteracting part of the action.

We start with the following action :

S[φ, φ∗] =

∫dω φ∗(ω)K (ω, `)φ(ω) + SI [φ, φ

∗] + J`[φ, φ∗]

K (ω, `) takes the standard form ω2 in the `→ 0 limit andJ`[φ, φ

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




S[φ, φ∗] =

∫dω φ∗(ω)K (ω, `)φ(ω) + SI [φ, φ

∗] + J`[φ, φ∗]






S[φ, φ∗] =

∫dω φ∗(ω)K (ω, `)φ(ω) + SI [φ, φ

∗] + J`[φ, φ∗]






S[φ, φ∗] =

∫dω φ∗(ω)K (ω, `)φ(ω) + SI [φ, φ

∗] + J`[φ, φ∗]



We consider actions which are quadratic in the fields forwhich it is sufficient to limit the form of J`[φ, φ

∗] to be linear,i.e., we take

J`[φ, φ∗] =

∫dω [J0(l) + J∗0(l) + J1(l)φ∗(ω) + J∗1(l)φ(ω)]

Impose the initial conditions

J0(`)|`=0 = J∗0(`)|`=0 = 0J1(`)|`=0 = J1(0) , J∗1(`)|`=0 = J∗1(0).

The normalised generating functional is given by

Z [J`] =

∫[dφ dφ∗] e−(S0[φ,φ∗]+SI [φ,φ∗]+J`[φ,φ∗])∫

[dφ dφ∗] e−(S0[φ,φ∗]+SI [φ,φ∗]).

Invariance of the generating functional

∂`Z [J`] = 0.



J`[φ, φ∗] =

∫dω [J0(l) + J∗0(l) + J1(l)φ∗(ω) + J∗1(l)φ(ω)]


J0(`)|`=0 = J∗0(`)|`=0 = 0J1(`)|`=0 = J1(0) , J∗1(`)|`=0 = J∗1(0).


Z [J`] =


[dφ dφ∗] e−(S0[φ,φ∗]+SI [φ,φ∗]).


∂`Z [J`] = 0.



J`[φ, φ∗] =

∫dω [J0(l) + J∗0(l) + J1(l)φ∗(ω) + J∗1(l)φ(ω)]


J0(`)|`=0 = J∗0(`)|`=0 = 0J1(`)|`=0 = J1(0) , J∗1(`)|`=0 = J∗1(0).


Z [J`] =


[dφ dφ∗] e−(S0[φ,φ∗]+SI [φ,φ∗]).


∂`Z [J`] = 0.



J`[φ, φ∗] =

∫dω [J0(l) + J∗0(l) + J1(l)φ∗(ω) + J∗1(l)φ(ω)]


J0(`)|`=0 = J∗0(`)|`=0 = 0J1(`)|`=0 = J1(0) , J∗1(`)|`=0 = J∗1(0).


Z [J`] =


[dφ dφ∗] e−(S0[φ,φ∗]+SI [φ,φ∗]).


∂`Z [J`] = 0.

Equations for the interacting part and source terms :

∂`SI =

∫dω ∂`K−1

δSI

δφ∗(ω)

δSI

δφ(ω)− δ2SI

δφ∗(ω)δφ(ω)

∂`J` =

∫dω ∂`K−1

δSI

δφ(ω)

δJ`

δφ∗(ω)+

δSI

δφ∗(ω)

δJ`

δφ(ω)

+δJ`

δφ∗(ω)

δJ`

δφ(ω)− δ2J`

δφ∗(ω)δφ(ω)

.

These equations can easily be solved when the interactionterm is quadratic in the fields, i.e.,

SI [φ, φ∗] =

∫dω g(ω, `)φ∗(ω)φ(ω).

Source terms :

∂`J1(`) = ∂`K−1(ω, `)g(ω, `)J1(`)

∂`[J0(`) + J∗0(`)] = ∂`K−1(ω, `)|J1(`)|2.


∂`SI =

∫dω ∂`K−1

δSI

δφ∗(ω)

δSI

δφ(ω)− δ2SI

δφ∗(ω)δφ(ω)

∂`J` =

∫dω ∂`K−1

δSI

δφ(ω)

δJ`

δφ∗(ω)+

δSI

δφ∗(ω)

δJ`

δφ(ω)

+δJ`

δφ∗(ω)

δJ`

δφ(ω)− δ2J`

δφ∗(ω)δφ(ω)

.


SI [φ, φ∗] =

∫dω g(ω, `)φ∗(ω)φ(ω).

Source terms :

∂`J1(`) = ∂`K−1(ω, `)g(ω, `)J1(`)

∂`[J0(`) + J∗0(`)] = ∂`K−1(ω, `)|J1(`)|2.


∂`SI =

∫dω ∂`K−1

δSI

δφ∗(ω)

δSI

δφ(ω)− δ2SI

δφ∗(ω)δφ(ω)

∂`J` =

∫dω ∂`K−1

δSI

δφ(ω)

δJ`

δφ∗(ω)+

δSI

δφ∗(ω)

δJ`

δφ(ω)

+δJ`

δφ∗(ω)

δJ`

δφ(ω)− δ2J`

δφ∗(ω)δφ(ω)

.


SI [φ, φ∗] =

∫dω g(ω, `)φ∗(ω)φ(ω).

Source terms :

∂`J1(`) = ∂`K−1(ω, `)g(ω, `)J1(`)

∂`[J0(`) + J∗0(`)] = ∂`K−1(ω, `)|J1(`)|2.

Solutions :

J1(`) = J1(0) exp(∫ `

0d`′ g(ω, `′)∂`′K−1(ω, `′)

)J0(`) + J∗0(`) = |J1(0)|2

∫ `

0d`′ ∂`′K−1(ω, `′)

×exp

(2∫ `′

0d`′′ g(ω, `′′)∂`′′K−1(ω, `′′)

).

Landau problem :

S =

∫dω [mz(ω)ω2z(ω) + eBωz(ω)z(ω)

+J(ω)z(ω) + J∗(ω)z(ω)].

Choice of K (ω, `) :

K (ω, `) =mω2

(1− mω`~ )

.

Solutions :

J1(`) = J1(0) exp(∫ `

0d`′ g(ω, `′)∂`′K−1(ω, `′)

)J0(`) + J∗0(`) = |J1(0)|2

∫ `

0d`′ ∂`′K−1(ω, `′)

×exp

(2∫ `′

0d`′′ g(ω, `′′)∂`′′K−1(ω, `′′)

).

Landau problem :

S =


+J(ω)z(ω) + J∗(ω)z(ω)].


K (ω, `) =mω2

(1− mω`~ )

.

Solutions :

J1(`) = J1(0) exp(∫ `

0d`′ g(ω, `′)∂`′K−1(ω, `′)

)J0(`) + J∗0(`) = |J1(0)|2

∫ `

0d`′ ∂`′K−1(ω, `′)

×exp

(2∫ `′

0d`′′ g(ω, `′′)∂`′′K−1(ω, `′′)

).

Landau problem :

S =


+J(ω)z(ω) + J∗(ω)z(ω)].


K (ω, `) =mω2

(1− mω`~ )

.

Initial condition

g(ω, `)|`=0 = eBω.

flow equation for the coefficient g(ω, `) :

∂g(ω, `)

∂`= − 1

~ωg2(ω, `).

Integration of this equation subject to the initial conditiongives

g(ω, `) = eB(`)ω ; B(`) =B

(1 + eB`~ )

.

Solution for the sources :

J1(`) =J1(0)

(1 + eB`~ )

J0(`) + J∗0(`) = − |J1(0)|2`~ω(1 + eB`

~ ).

Initial condition

g(ω, `)|`=0 = eBω.


∂g(ω, `)

∂`= − 1

~ωg2(ω, `).


g(ω, `) = eB(`)ω ; B(`) =B

(1 + eB`~ )

.


J1(`) =J1(0)

(1 + eB`~ )

J0(`) + J∗0(`) = − |J1(0)|2`~ω(1 + eB`

~ ).

Initial condition

g(ω, `)|`=0 = eBω.


∂g(ω, `)

∂`= − 1

~ωg2(ω, `).


g(ω, `) = eB(`)ω ; B(`) =B

(1 + eB`~ )

.


J1(`) =J1(0)

(1 + eB`~ )

J0(`) + J∗0(`) = − |J1(0)|2`~ω(1 + eB`

~ ).

Initial condition

g(ω, `)|`=0 = eBω.


∂g(ω, `)

∂`= − 1

~ωg2(ω, `).


g(ω, `) = eB(`)ω ; B(`) =B

(1 + eB`~ )

.


J1(`) =J1(0)

(1 + eB`~ )

J0(`) + J∗0(`) = − |J1(0)|2`~ω(1 + eB`

~ ).

Rescaling the coordinates as

z(ω) =

√1− eB(`)`

~z(ω)

we obtain the final form of the action :

S =

∫dω

[¯z(ω)K (ω, `)z(ω) +

eBω(1− mω`

~ )¯z(ω)z(ω)

− |J1(0)|2`~ω(1 + eB`

~ )+

J1(0)√1 + eB`

~

¯z(ω) + c.c.

.

Action of a particle moving in a magnetic field B∗ in anoncommutative plane with noncommutative parameter `and source terms:

SNC =

∫dω

[¯z(ω)K (ω, `)z(ω) +

eB∗(1 + eB∗`4~ )ω

(1− mω`~ )

×

¯z(ω)z(ω) + J ¯z(ω) + J z(ω)].

Rescaling the coordinates as

z(ω) =

√1− eB(`)`

~z(ω)

we obtain the final form of the action :

S =

∫dω

[¯z(ω)K (ω, `)z(ω) +

eBω(1− mω`

~ )¯z(ω)z(ω)

− |J1(0)|2`~ω(1 + eB`

~ )+

J1(0)√1 + eB`

~

¯z(ω) + c.c.

.Action of a particle moving in a magnetic field B∗ in anoncommutative plane with noncommutative parameter `and source terms:

SNC =

∫dω

[¯z(ω)K (ω, `)z(ω) +

eB∗(1 + eB∗`4~ )ω

(1− mω`~ )

×

¯z(ω)z(ω) + J ¯z(ω) + J z(ω)].

Blocking procedure

This result admits a different interpretation in terms of ablocking procedure, where the blocking is performed overtime.

It is possible to interpret the renormalization flow as achange of variables in the path integral.In this case a simple change of variables involving a ωdependent scaling, z(ω) =

√1−mω`/~ z(ω), transforms

the commutative action into the noncommutative action.This corresponds to the following blocking relation

z(t) =

∫ +∞

−∞dt ′

e−iω0(t ′−t)

√ω0(t ′ − t)3/2 z(t ′)

where ω0 = ~m` .

Blocking procedure

This result admits a different interpretation in terms of ablocking procedure, where the blocking is performed overtime.It is possible to interpret the renormalization flow as achange of variables in the path integral.

In this case a simple change of variables involving a ωdependent scaling, z(ω) =



z(t) =

∫ +∞

−∞dt ′

e−iω0(t ′−t)

√ω0(t ′ − t)3/2 z(t ′)

where ω0 = ~m` .

Blocking procedure

This result admits a different interpretation in terms of ablocking procedure, where the blocking is performed overtime.It is possible to interpret the renormalization flow as achange of variables in the path integral.In this case a simple change of variables involving a ωdependent scaling, z(ω) =


the commutative action into the noncommutative action.

This corresponds to the following blocking relation

z(t) =

∫ +∞

−∞dt ′

e−iω0(t ′−t)

√ω0(t ′ − t)3/2 z(t ′)

where ω0 = ~m` .

Blocking procedure

This result admits a different interpretation in terms of ablocking procedure, where the blocking is performed overtime.It is possible to interpret the renormalization flow as achange of variables in the path integral.In this case a simple change of variables involving a ωdependent scaling, z(ω) =



z(t) =

∫ +∞

−∞dt ′

e−iω0(t ′−t)

√ω0(t ′ − t)3/2 z(t ′)

where ω0 = ~m` .

This relation clearly reveals the fact that spatialnoncommutativity can be seen as a blocking procedureover time.

It is also reassuring to check, using the commutator of z(t)and z(t ′) (t 6= t ′)

[z(t), z(t ′)] = 2~(e−ieB(t−t ′)/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 originaltheory onto a noncommutative theory withnoncommutative parameter `.

It is important to note that this only happens if the coarsegraining relation above is used.



[z(t), z(t ′)] = 2~(e−ieB(t−t ′)/m − 1)/(eB)






[z(t), z(t ′)] = 2~(e−ieB(t−t ′)/m − 1)/(eB)






[z(t), z(t ′)] = 2~(e−ieB(t−t ′)/m − 1)/(eB)




Observations

We have constructed one parameter families of dualtheories using the ERGE for the Landau problem.

The precise choice made here for the kinetic energy termestablished these dual families to be noncommutativetheories.

We also observe a subtle link between noncommutativetheories and the blocking procedure in the ERG approach.

The latter may open up an avenue to explore theserelations in higher dimensional interacting field theories.

Observations





Observations





Observations





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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