## Abstract

The exchange phase for two spins is studied here from the point of view of the quantization of a fermion in the framework of Nelson’s stochastic mechanics. This introduces a direction vector attached to a space–time point depicting the spin degrees of freedom. In this formalism, a fermion appears as a scalar particle attached with a magnetic-flux quantum, and a quantum spin can be described in terms of an SU(2) gauge bundle. This helps us to recast the Berry–Robbins formalism where the exchange phase appears as an unfamiliar geometric phase arising out of the ‘exchange rotation’ in a transported spin basis in terms of gauge currents. However, for polarized fermions, the exchange phase is found to be given by the Berry phase.

## 1. Introduction

It is well known that fermions have a specific property that the situation corresponding to a 2*π* rotation is not described by the same wave function. Indeed, the wave function changes its sign in this case. In addition, for identical particles, exchanged wave functions differ from unexchanged wave functions by a sign. There are proofs of this specific spin-statistics relation that involve relativistic quantum field theory (Pauli 1940; Streater & Wightman 1964) or the existence of antiparticles (Tscheuschner 1989; Balachandran *et al.* 1993). There are also proofs that incorporate the concept of topological solitons (Finkelstein & Rubinstein 1968; Michelson 1984). Shaji & Sudarshan (2003) demonstrated this relation considering non-relativistic quantum field theory as opposed to usual quantum mechanics. Apart from these, Berry & Robbins (1997) furnished a proof that incorporates a ‘transported spin basis’ (an ** r**-dependent spin basis,

**being the position) by adapting the Schwinger representation of spin (Schwinger 1965) in terms of harmonic oscillators, and it is shown that this sign change effectively corresponds to an unfamiliar geometric phase. Anandan (1998) gave a relativistic generalization of this formalism. It may be noted that the familiar geometric phase, known as the Berry (1984) phase in the literature, corresponds to a cyclic evolution of a parameter. Simon (1983) pointed out that the geometrical meaning of this phase is precisely the holonomy in a Hermitian line bundle and naturally defines a U(1) connection. It is a geometrical phenomenon in which non-integrability causes some variables to fail to return to their original values when others that drive them are altered around a cycle through parallel transport. The U(1) bundle involves the first Chern class, where the Chern number is defined by the integral of the curvature which implies that a magnetic flux appears in the phase. However, the geometric phase that appears when two identical fermions are exchanged is associated with the first Stiefel–Whitney class (Milnor & Stasheff 1974; Nash & Sen 1983; Nakahara 1990), which is related to a real vector bundle and does not involve any curvature. In fact, the Chern class takes value in the**

*r**Z*-cohomology, whereas the Stiefel–Whitney class takes value in the

*Z*

_{2}-cohomology. The Stiefel–Whitney class

*W*

_{i}bears the important result that the existence of a spin structure in a manifold

*M*implies that

*W*

_{2}must vanish. Moreover, the vanishing of

*W*

_{1}implies that

*M*is orientable. It may be mentioned that, if

*W*

_{2}=0, then one also needs

*M*to be orientable, i.e.

*W*

_{1}=0. The exchange of two identical fermions is associated with the first Stiefel–Whitney class of the two-spin bundle for non-contractible loops in a non-simply connected space (Milnor & Stasheff 1974; Nash & Sen 1983; Nakahara 1990). This follows from the fact that the operation of exchange of the positions of two spins causes an obstruction for the orientability of the manifold.

In this paper, we shall try to address the problem from the viewpoint of the quantization of a fermion in the framework of Nelson’s stochastic mechanics (Nelson 1966, 1967). In an earlier paper (Bandyopadhyay & Hajra 1987; Hajra & Bandyopadhyay 1991), it has been pointed out that the relativistic generalization of Nelson’s stochastic mechanics, as well as the quantization of a fermion, can be achieved by introducing an internal variable that acts as a direction vector. In fact, this direction vector depicts the internal degrees of freedom representing spin. In this formalism, a spin can be described in terms of an SU(2) gauge bundle. This helps us to recast the Berry–Robbins formalism of the transported spin basis in terms of gauge currents. The exchange phase can be interpreted in terms of a geometric phase that does not involve any curvature and is consistent with the first Stiefel–Whitney class. However, for polarized fermions, we may relate the exchange phase with the familiar geometric phase, *viz*. the Berry phase. Indeed, for a chiral spinor, we have only a left-handed or right-handed fermion with a specific spin polarization that involves the breaking of the reflection symmetry (*Z*_{2}-symmetry). This helps us to avoid the Stiefel–Whitney class that takes the value in the *Z*_{2}-cohomology. In fact, a polarized fermion can be viewed as a fermion having no spin degree of freedom. The exchange phase in this case can be interpreted in terms of the Berry phase. It may be noted that chiral symmetry breaking leads to chiral anomaly, which is associated with this phase (Banerjee & Bandyopadhyay 1992).

In §2, we shall recapitulate certain features associated with the quantization of a fermion and the representation of a quantum spin in terms of an SU(2) gauge bundle. In §3, we shall formulate the exchange phase in terms of the transported spin basis corresponding to the gauge currents. In §4, we shall consider the case of polarized fermions, and it is shown that the exchange phase in this case is related to the familiar Berry phase.

## 2. Quantization of fermion, spin structure and SU(2) gauge bundle

In earlier papers (Bandyopadhyay & Hajra 1987; Hajra & Bandyopadhyay 1991), it was pointed out that, in the framework of Nelson’s (1966, 1967) stochastic mechanics, the quantization of a fermion can be achieved when we introduce an internal variable that appears as a direction vector. Indeed, the relativistic generalization of Nelson’s stochastic quantization procedure is attained when we introduce Brownian-motion processes both in the internal space as well as in the external observable space. The quantization of a Fermi field is achieved when we introduce an anisotropic feature in the internal space so that the internal variable appears as a direction vector. This gives rise to two internal helicities that correspond to spin degrees of freedom. A space–time manifold *E* carries a spin structure if the second Stiefel–Whitney class of *E* vanishes (Milnor & Stasheff 1974; Nash & Sen 1983). Geroch (1968) demonstrated that this condition for non-compact space–time is equivalent to the existence of a global field of orthonormal tetrads on *E*. Chevalley (1954) and Crumeyrolle (1969) formulated the definition of a spinor structure based on Clifford algebra. In this approach, a space- and time-orientable space–time manifold carries a spinor structure if and only if the structure group of the bundle of orthonormal tetrads over *E* is reducible to a group , which is now known as the Crumeyrolle group. Bugaska (1980) has studied the Crumeyrolle group of four-dimensional space–time, and it has been shown that the complexification of the Lie algebra of the group is a spinor space. This enables us to associate the spinor space to each space–time point in a continuous way. The metric tensor field *g* on *E* allows us to construct the Clifford algebra *C*(*E*,*g*) over *E*. Let (*e*_{0},*e*_{1},*e*_{2},*e*_{3}) be a base of the tangent space *T*_{m}(*E*) at a point *m* of *E*. It has been explicitly shown by Bugaska that the Lie algebra of the Crumeyrolle group is spanned by2.1It is noted that [*x*_{1},*x*_{2}]=0. If we now define2.2and write , , we note that and generate the same (complexified) algebra as *x*_{1} and *x*_{2}. The Iwasawa decomposition of the Lorentz group SO(3,1) has the form2.3where *K* is the maximal compact subgroup SO(3), *A* the Abelian one-parameter subgroup generated by the acceleration *L*_{03} and *N* the nilpotent Abelian two-dimensional subgroup generated by and . As the Lie algebra of the Crumeyrolle group is spanned by *x*_{1} and *x*_{2} given by equation (2.1), we note that the two-dimensional group *N* generated by and represents the Crumeyrolle group . Thus, the condition for the existence of a spin structure over space–time *E*, i.e. the reducibility of the Lorentz structure to the Crumeyrolle group , can be regarded as a feasibility of setting up at each point of *E* two spinors *u* and *v* associated with the generators and of the Lie algebra of . It is observed that when we go over to the complexified group SL(2,*C*) (SU(2)), which is the covering group of SO(3,1) (SO(3)), and noting that *N* represents the Crumeyrolle group , the decomposition (2.3) implies that can be parametrized by the elements , of a two-dimensional complex space that transform as spinors (Bugaska 1980). Thus, these degrees of freedom lead to a richer structure of *E* with spinorial parameters. The degrees of freedom related to these spinorial parameters have a direct correspondence with the direction vector that depicts the internal helicity states. In fact, when we introduce a direction vector attached to a space–time point, the configuration space can be depicted by a complex coordinate *z*_{μ}=*x*_{μ}+*iξ*_{μ}, where the ‘direction vector’ *ξ*_{μ} is attached to the space–time point *x*_{μ}. The ‘direction vector’ *ξ*_{μ} can be taken to depict the helicity states when we consider the complex coordinate *z*_{μ} in the form , where *θ*^{α} is a two-component spinorial variable and , for a specific space–time index *μ*, is an arbitrary two-component vector. We now replace the chiral coordinates by matrices so that we write2.4whereandWe can now define the helicity operator2.5representing a spin up or down state where is the spinorial variable corresponding to the four-momentum *p*_{μ} (the conjugate of *x*_{μ}). In fact, we can replace the momentum 4-vector *p*_{μ} by a matrix *p*^{AA′} in analogy to the space–time variable and we can write2.6where is a spinorial variable. It is noted that this matrix representation necessarily implies so that the particle will have its mass due to the non-vanishing value of the quantity (Bandyopadhyay & Hajra 1987; Bandyopadhyay 2000). It is observed that the complex conjugate of the chiral coordinate will give rise to the opposite helicity state. We can define the upper half-plane *D*^{−} where the coordinate *z*_{μ}=*x*_{μ}+*iξ*_{μ} is such that *ξ*_{μ} belongs to the interior of the forward light-cone *ξ*≫0 with the conditions det *ξ*^{AA′}>0 and Tr *ξ*^{AA′}>0. The lower half-plane *D*^{+} is given by the set of coordinates *z*_{μ}=*x*_{μ}−*iξ*_{μ} with *ξ*_{μ} in the interior of the backward light-cone *ξ*≪0. The map sends the upper half-plane to the lower half-plane. The space of the null plane is the boundary so that a function holomorphic in *D*^{−} (*D*^{+}) is determined by the boundary value. This implies that, in the null plane representing massless spinors, the helicity corresponding to the up (down) spin may be taken to be the limiting value of the helicity in the upper (lower) half-plane that represents massive spinors.

In the complexified space–time exhibiting the helicity states, we can now write the metric in the form . It was shown (Bandyopadhyay & Hajra 1987) that this metric structure gives rise to SL(2,*C*) gauge theory and the field strength is given by2.7where is the vector potential belonging to SL(2,*C*). Demanding hermiticity, we may take belonging to the unitary group SU(2). It may be mentioned that as the spinorial variable is associated with the homogeneous space , where is the Crumeyrolle group, a spin at a certain space point can be represented in terms of the SU(2) gauge bundle when the base space is the (3+1)-dimensional space–time. This suggests that when the space–time coordinates are written in the complexified form incorporating the direction vector, this effectively represents a gauge theoretical extension of the space–time coordinates. Analogously, we also have a gauge theoretical extension of the momentum coordinates. In fact, we can now write the extended space–time coordinate as well as the momentum as the gauge covariant operator acting on functions in phase space,2.8with and *q*_{μ}(*p*_{μ}) denoting the mean position (momentum) of the external observable space. The internal degree of freedom associated with the spin can now be depicted in terms of this SU(2) gauge field. It is observed here that the space–time coordinates as well as the momentum variables given by equation (2.8) represent non-commutative geometry as their components do not commute, and the non-commutativity parameter is given by2.9The functional dependence of this non-commutativity parameter effectively corresponds to the existence of monopoles (Jackiw 1985; Bèrard & Mohrbach 2004). In particular, the spatial components of the momentum variable can be taken to satisfy2.10where *μ* appears as a monopole strength. The angular momentum of a charged particle in the field of a magnetic monopole is given by2.11where *μ* takes the value 0,±1/2,±1,… . This implies that a fermion can now be depicted as a scalar particle having orbital momentum 1/2 in this space. As *μ*=1/2 corresponds to one flux quantum, a fermion is described by a scalar particle attached with one magnetic-flux quantum (Wilczek 1982).

It may be mentioned that in (2+1) dimensions, a boson–fermion transformation is achieved through the introduction of a Chern–Simons field. In a three-dimensional manifold, the non-Abelian Chern–Simons action is given by2.12where *k* is an integer and is the non-Abelian Chern–Simons gauge field. The Chern–Simons invariant is related to the Pontryagin index through the relation2.13where is the two-form related to the field strength. The magnetic monopole strength *μ* is related to the Pontryagin index *q* associated with this four-dimensional integral, and we have the relation *q*=2*μ*. In view of this, we note that this formalism of boson–fermion transformation through the attachment of a magnetic-flux quantum to a scalar particle is analogous to the procedure of boson–fermion transformation in (2+1) dimensions through the introduction of a Chern–Simons field (Bandyopadhyay 1996). In this scheme, a neutral massive elementary fermion may be depicted as a scalar particle attached with a vortex line corresponding to the ‘direction vector’ *ξ*_{μ}. As a vortex line is topologically equivalent to a magnetic-flux line, we can utilize this formalism in this case too. In fact, a non-commutative manifold has an inherent anisotropy and the factor *μ* in expression (2.10) measures the degree of anisotropy. In this anisotropic space, a particle can move with orbital momentum *l*=1/2, which is analogous to the angular momentum of a charged particle in the field of a magnetic monopole. In view of this, we can associate a fictitious magnetic field in this case. The phase generated by this angular momentum is given by *e*^{i2πj} with *j*=1/2, which suggests that a 2*π* rotation will give rise to the phase *e*^{iπ} implying a change in the sign of the wave function. This formalism suggests that a massive fermion appears as a soliton (Skyrmion; Bandyopadhyay & Hajra 1987).

## 3. Exchange phase of two fermions in transported spin basis

Berry & Robbins (1997) showed that the exchange phase of two identical fermions can be considered as a geometrical phase arising out of the exchange rotation when we consider a spin in a transported basis. A transported basis for a spin operator can be formulated by introducing an ** r**-dependence (position dependence) in the definition of a spin. In their formalism, the transported basis has been formulated in terms of harmonic oscillators using Schwinger’s (1965) representation of spin. The spin states of two fermions with the quantum numbers

*m*

_{1}and

*m*

_{2}representing the

*z*-component of their spin can be defined by3.1where represents the exchanged state of spin. In the transported basis, we can write3.2where

**is the relative position of spins**

*r***=**

*r*

*r*_{2}−

*r*_{1},(

*r*_{1},

*r*_{2}) are the positions of spins and

*U*(

**) is a unitary operator. In the transported basis, the spin operators**

*r***(**

*S***)=(**

*r*

*S*_{1},

*S*_{2}) can be written as3.3where

*S*_{fixed}denotes the usual fixed-spin operator. The exchange phase is determined from the relation3.4with

*K*being an integer. The parallel transport rule for the exchange of spins is governed by the relation3.5implying that there is no usual local gauge potential in this case, which gives rise to the familiar geometric phase (Berry phase). It is noted that the parallel transport relation (3.5) here rules out the possibility of constructing operators

*U*(

**) that generates exchange by using the usual spin operators**

*r*

*S*_{1}and

*S*_{2}, as this implies

*U*(

**) is constant. In view of this, Berry & Robbins have considered the Schwinger representation of spin that involves the introduction of harmonic oscillators, and a spin is described as3.6where**

*r**a*and

*b*are the two oscillators and

**denotes Pauli matrices. In this formalism,**

*σ**S*

_{z}is given by3.7Berry & Robbins have shown that in this formalism, the exchange phase appears as an unfamiliar geometric phase arising out of the exchange rotation. This does not involve any curvature consistent with the fact that the exchange phase is associated with the first Stiefel–Whitney class of the two-spin bundle for non-contractible loops in a non-simply connected space. From the quantization point of view, as discussed in the previous section, a spinor is described by a magnetic-flux quantum attached to a scalar particle and the up or down spin is associated with the magnetic-flux line of opposite orientations. This helps us to formulate a transported basis for the spin in terms of a topological current constructed from SU(2) gauge fields, and the transverse currents in two opposite directions correspond to up- and down-spin states. Indeed, in an earlier paper (Goswami & Bandyopadhyay 1993), it was pointed out that in two spatial dimensions, a spin system on a lattice can be described in terms of the SU(2) gauge field when the gauge current represents the spin where the gauge field lies on the bond. The system can be associated with Chern–Simons topology. It is noted that when the internal coordinate

*ξ*

_{μ}is attached to the space–time point

*x*

_{μ}, we should take into account the polar coordinates

*r*,

*θ*and

*ϕ*for the space component of the vector

*x*

_{μ}and the angle

*χ*to specify the rotational orientation around the direction vector

*ξ*

_{μ}. The eigenvalue of the operator

*i*∂/∂

*χ*corresponds to the internal helicity. The spherical harmonics incorporating these angles

*θ*,

*ϕ*and

*χ*can be written as , with

*m*and

*μ*being the eigenvalues of the operators

*i*∂/∂

*ϕ*and

*i*∂/∂

*χ*, respectively (Hurst 1968). This, in fact, represents the monopole harmonics as it corresponds to the angular momentum of a charged particle in the field of a magnetic monopole. The doublet , with3.8corresponds to a two-component spinor and the charge conjugate state is given by , with3.9A spin operator in the Lie algebra of SU(2) can be constructed using oscillators. These could be either fermionic or bosonic. In terms of fermionic oscillators with creation and destruction operators given by and

*ψ*

_{α}(

*x*), respectively, which satisfy the algebra3.10the spin operator in the Lie algebra of SU(2) is given by3.11where

*σ*_{αβ}is the (

*αβ*) component of the matrices of the Pauli vector

**. The components of**

*σ***satisfy the algebra3.12The components of the spin operator**

*σ***obey3.13**

*S*We here construct the spin operator ** S** taking into account the fermionic oscillator

*ψ*

_{α}(

*x*) represented by the spherical harmonics , with

*m*=±1/2,

*μ*=±1/2. A unit vector

**can be constructed as3.14withandwhere**

*n***is the vector of Pauli matrices. This helps us to construct the spin operator**

*σ***in terms of the angular variables**

*S**θ*,

*ϕ*and

*χ*as3.15with

*ψ*

_{1}and

*ψ*

_{2}given by equation (3.8). Evidently, the operator

**here is position dependent**

*S***(**

*S**x*), where

*x*is the position and represents a transported spin basis. We consider a unit 4-vector

*n*

_{μ}with

*μ*=0,1,2,3 in (3+1) dimensions incorporating the 3-vector

**given by equation (3.14). In fact, we can write the 4-vector**

*n**n*

_{μ}as3.16with

*σ*

_{0}=

*I*, where

*I*is the identity matrix and

**(**

*σ**σ*

_{1},

*σ*

_{2},

*σ*

_{3}) are the Pauli matrices. We now construct the topological current3.17where (

*a*,

*b*,

*c*,

*d*) correspond to (0,1,2,3) and (

*μ*,

*ν*,

*λ*,

*σ*) represent space–time indices. The current (3.17) can be written in the form3.18where

*g*=

*n*

_{0}

*I*+

*i*

**⋅**

*n***belongs to the group SU(2) (Abanov & Wiegmann 2000). It is noted from equations (3.14) and (3.15) that the position-dependent spin vector**

*σ***(**

*S**x*) corresponds to the unit vector

**(**

*n**x*) and these are constructed from the spinorial fields

*ψ*

_{1}and

*ψ*

_{2}. The current

*J*

_{μ}given by equation (3.17) incorporates the unit vector

**(**

*n**x*) through the 4-vector

*n*

_{μ}, which can be transcribed in terms of SU(2) gauge fields as given by equation (3.18). In view of this, the spin may be associated with the SU(2) gauge current. It may be mentioned that we can consider the topological Lagrangian in terms of the SU(2) gauge fields in affine space,3.19This gives rise to the topological current (Carmeli & Malin 1977)3.20where we have taken the SU(2) gauge field and the corresponding field strength as3.21If we consider Euclidean four-dimensional space–time and demand that at all points on the boundary

*S*

^{3}of a certain volume

*V*

^{4}inside which , then the gauge potential tends to a pure gauge in the limiting case towards the boundary, and we can write3.22where

*g*is an element of SU(2) group. This helps us to write the topological current (3.18) as3.23with given by equation (3.22). From this, it appears that the spin operator

**(**

*S**x*) can be depicted by the topological current

*J*_{μ}given by equation (3.20) written in terms of the SU(2) gauge fields. The topological charge3.24corresponds to the winding number associated with the homotopy

*π*

_{3}(

*S*

^{3})=

*Z*and can be written as3.25It is noted that

*q*, which is an integer, represents the Pontryagin index and the relation

*q*=2

*μ*implies that

*μ*corresponds to the magnetic monopole strength and can take the value

*μ*=0,±1/2,±1,±3/2,…. As the Pontryagin index

*q*=2

*μ*corresponds to the charge associated with the SU(2) gauge current

*J*

_{μ}related to spin, we are effectively taking into account the magnetic moment associated with the spin. From the association of the spin with the current given by equation (3.20), we note that we can take any specific component of

*J*_{μ}, to represent the

*z*-component of the spin. Writing this specific component as3.26we note that this satisfies the relation3.27where

*η*

^{μ}is a constant axial vector derived from the contraction of the epsilon tensors and

*ϕ*(

*x*) is a scalar function. This follows from the fact that

*f*

_{λσ}is an antisymmetric function and so, for a fixed

*μ*, we can replace ∂

_{ν}

*f*

_{λσ}by

*ϵ*

_{νλσ}

*ϕ*(

*x*), where

*ϵ*

_{νλσ}is the Levi–Civita antisymmetric tensor. Now from the conservation of the current ∂

_{μ}

*J*

_{μ}=0, we note that

*ϕ*(

*x*) is a constant and so

*J*

_{μ}(

*x*) effectively represents a constant axial vector at the point

*x*. In view of this, we write the two-spin states

*M*(

**)=|**

*r**m*

_{1}

*m*

_{2}(

**)〉, where**

*r***is the relative position, in the form3.28It is noted that the conservation of the current governed by the relation ∂**

*r*_{μ}

*J*

_{μ}=0 ensures the parallel transport rule given by equation (3.5). Now to consider the exchange of two spins, we have to find out the relation between and , where

**is the relative position. If we consider that the constant axial vector is at the position**

*r*

*r*_{1}and at the position

*r*_{2}so that

**=**

*r*

*r*_{2}−

*r*_{1}, the very axial vector nature of

*η*

^{μ}implies the relation3.29Thus, the exchange phase −1 appears here as a geometric phase when the axial vectors and at the positions

*r*_{1}and

*r*_{2}are parallel transported so that they interchange their positions.

## 4. Exchange phase and polarized fermions

Here, we consider the case when we split the Dirac spinor into chiral form so that the two helicities correspond to left-handed and right-handed spinors. As is well known, the chiral-symmetry breaking leads to chiral anomaly. In this case, the divergence of the topological current *J*_{μ} given by equation (3.20) does not vanish. In fact, it satisfies the relation (Roy & Bandyopadhyay 1989)4.1where is the second component of the gauge current and is the axial vector current associated with the chiral spinor field *ψ*. In addition, the three components of the gauge current satisfy the relation4.2

It is noted that when the divergence of the current (*a*=1,2,3 corresponding to the group index) does not vanish, then the parallel transport rule given by equation (3.5) is modified and satisfies the relation4.3where the spin state is described in terms of the gauge current. Indeed the relation (4.3) follows from the condition . This essentially implies the existence of a *U*(1) gauge field and the Berry phase is the holonomy element of this connection. It is noted that when a chiral spinor interacts with a gauge field, the divergence of the axial vector current is given by (Fujikawa 1979; Jackiw 1984)4.4where is the Hodge dual,4.5So, from the relation (4.1), we have4.6and from equation (4.2) we note that and can be related with this term. It is observed that the divergence of the gauge current given by equation (4.6) is related to the topological Lagrangian (3.19). Indeed, the anisotropic feature associated with the ‘direction vector’ that is responsible for the quantization of a fermion in (3+1) dimensions appears as the main geometrical feature underlying this topological term. For chiral fermions, this anisotropic feature gives rise to chiral anomaly, which is associated with the Berry phase. In case this topological term given by equation (3.19) is incorporated in the Lagrangian corresponding to a non-Abelian gauge theory, this introduces certain topologically non-trivial Abelian background gauge fields in the configuration space. In fact, this term leads to a vortex line in gauge orbit space in (3+1)-dimensional space–time (Wu & Zee 1985; Sen & Bandyopadhyay 1994). The hidden Abelian gauge field related to the vortex line in a non-Abelian gauge theory may be viewed as if, in the gauge orbit space, the position of a particle indicated by (non-Abelian gauge potential) moves in the space *U* of non-Abelian gauge potentials under the influence of an Abelian gauge field. In the language of differential forms, we can write4.7where and *a*=*a*_{μ} d*x*^{μ}. The gauge orbit space , where denotes the space of local gauge transformations *g*(*x*) consists of the points *a*(*x*). It may be mentioned that as *U* is the space of all non-Abelian gauge potentials, it is contractible. Noting that for all simple non-Abelian groups and for all *n*, we have4.8In (3+1) dimensions, we have4.9In fact, when we take the *x*-space as a compactified three-sphere *S*^{3}, *g*(*x*) defines a map . The equality follows from the condition that the gauge transformation *g*(*x*) approaches a constant independent of the direction of *x* as . Thus, is multiply connected and has the topology of a ring indicating that there is a vortex line which is topologically equivalent to a magnetic-flux line. It is now pointed out that a gauge orbit space effectively represents a loop space when a loop can be visualized as an orbit (Loll 1992). Thus, the above result suggests that when the topological term given by equation (3.19) is incorporated in the theory, this implies that a magnetic flux line is enclosed by the loop. When a scalar particle encircles the loop enclosing the magnetic-flux line, the particle acquires a geometric phase (Berry phase) given by *e*^{i2πμ}, where *μ* is the monopole strength and *μ*=1/2 corresponds to one magnetic-flux line (Banerjee & Bandyopadhyay 1992). So, the particle acquires the phase *e*^{iπ}, which is the phase associated with a fermion when it traverses a closed circuit representing a 2*π* rotation. Evidently, the system represents a fermion and the orientation of the magnetic-flux line corresponds to an up- or down-spin state. So, when a magnetic-flux line with a specific orientation is enclosed by the loop, the scalar particle encircling the loop represents a polarized fermion. Now we note that when a polarized fermion is described by a scalar particle moving around a magnetic-flux line with a specific orientation attached to a point *r*_{1} inside one loop and for another fermion with the magnetic-flux line having the same orientation is attached to the point *r*_{2} inside the other loop, we may view it as if two spins are attached to the points *r*_{1} and *r*_{2}. If we denote the exchange of two spins by the rotation of both particles in a half-circle as envisaged by Feynman (1987), this is equivalently described by the rotation of either particle in a full circle when one magnetic flux line is enclosed by it. This will give rise to the geometric phase *e*^{i2πμ} with *μ*=1/2. Thus, the exchange phase when two spins change their positions can be witnessed through the Berry phase. It may be pointed out that in the spherical harmonics , *μ* is related to the angle *χ*, which corresponds to the rotational orientation around the direction vector *ξ*_{μ}. The introduction of this angle *χ* effectively takes care of the extension of the canonical system with certain internal structure. For compactified space, this enlarges the configuration space and the angle *χ* acts like *U*(1) gauge degrees of freedom, which represents Hopf fibration of *S*^{2}. The angular part of the spherical harmonics associated with the angle *χ* is given by *e*^{−iμχ} (Hurst 1968). So, from the relation4.10we note that when *χ* is changed to *χ*+*δχ*, we have the relation4.11This implies that the wave function will acquire an extra factor *e*^{iμδχ} due to the infinitesimal change of the angle. For one complete rotation, the phase is4.12which is the Berry phase (Banerjee & Bandyopadhyay 1992). For exchange rotation when both particles traverse a half-circle, the associated phase is given by4.13So, for polarized fermions where the corresponding magnetic-flux lines have the same orientation implying *μ*=±1, the phase is *e*^{iπ}. This is equivalent to the rotation in a full circle by either particle encircling one magnetic-flux line that generates the geometric phase4.14Thus, the exchange phase here is found to be a manifestation of the celebrated Berry phase. It may be mentioned that the exchange phase of two spins is associated with the first Stiefel–Whitney class, which does not involve any curvature, and so the parallel transport rule (3.5) is not associated with any local gauge field. However, the Berry phase is given by the holonomy in a Hermitian line bundle, which involves the first Chern class, and the Chern number is given by the integral of the curvature. Now the relationship we have achieved here for the exchange phase of two polarized spins in terms of the Berry phase can be understood from the consideration that, for a polarized fermion, the *Z*_{2}-symmetry breaks down, which is related to the chiral-symmetry breaking. Hence, the Stiefel–Whitney class, which is associated with the cohomology group with *Z*_{2}-coefficients, will not be operative here. Indeed, the *Z*_{2}-symmetry breaking for chiral fermions corresponds to chiral symmetry breaking leading to chiral anomaly, which is related to the Berry phase.

## 5. Discussion

We have considered here the exchange phase of two fermions in the transported spin basis when the spin is associated with the SU(2) gauge bundle. The transported spin basis is formulated by associating a spin with the gauge current when up and down spins correspond to transverse currents. It is found that the axial vector nature of the gauge current leads to the Pauli sign of the exchange phase. However, when we consider two polarized fermions, we can relate the exchange phase with the celebrated Berry phase. It is known that the exchange phase is associated with the first Stiefel–Whitney class of the two-spin bundle for non-contractible loops in a non-simply connected space. However, the Berry phase is associated with the holonomy in a Hermitian line bundle corresponding to the first Chern class. The fact that we have achieved the exchange phase for polarized fermions in terms of the Berry phase can be understood from the consideration that, in this case, we have *Z*_{2}-symmetry breaking leading to chiral-symmetry breaking. Indeed, as the Stiefel–Whitney class is associated with the cohomology group with *Z*_{2}-coefficients, this is not operative in this case. Finally, we may mention that the picture we have adopted here for a spin arising out of the quantization procedure of a fermion suggests that a fermion is depicted as a scalar particle attached with a magnetic-flux line. This helps us to consider a massive fermion as a soliton (Skyrmion; Bandyopadhyay & Hajra 1987). It is noted that when we have a magnetic-flux tube as an ‘external’ system, a charged particle acquires the familiar Aharanov–Bohm (A–B) phase. However, in this solitonic picture of a fermion, the magnetic-flux line is incorporated as an internal degree of freedom and is responsible for the Berry phase associated with the chiral anomaly. Thus, the relationship between the Berry phase and the statistical phase for polarized fermions suggests that the A–B phase, Berry phase and the statistical phase are interrelated to each other.

## Footnotes

- Received January 27, 2010.
- Accepted April 6, 2010.

- © 2010 The Royal Society