REND. SEM. MAT. UNIVERS. 0OLITECN. TORINO

Vol.390, 2 (1981)

W.M. Tulczyjew

CLASSICAL AND QUANTUM MECHANICS OF PARTICLES IN EXTERNAL GAUGE FIELDS

Summary: Gauge invariant formulations of the dynamics of relativistic particles interacting with external gauge fields are reviewed. Relations with wave mechanics are discussed.

A number of publications on classical dynamics of particles interacting with external gauge fields have appeared recently [5], [11], [19.]. The authors of these publications concentrate on the construction of a phase space for such particles. The proposed constructions are applicable to both Abelian and non-Abelian gauge theories and have have clear analogies in the wave mecha­nics of particles interacting with gauge fields. Weinstein's construction [19] is the classical version of the principal bundle formulation of wave mechanics and Sternberg [11] uses analogy with the associated bundle formulation. Weinstein's construction makes the symplectic structure of the pahse space appear in a more natural way. Dynamics of particles interacting with the electromagnetic field was formulated in references [7], [8], [12J. Important features of this formulation are lost in generalizations to non-Abelian gauge. To recover these features drastic modification of the theory of internal degrees of freedom seems to be necessary. A quantum theory of systems with inter­nal degrees of freedom was proposed in [16].

Both Sternberg and Weinstein imply that the phase space is to be used to formulate ghe Hamiltonian dynamics of particles. Arguments based on cor­respondence between classical mechanics and quantum theory show that

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generalized Hamiltonian systems [3] must be used to describe the dynamics of relativistic particles. The purpose of this lecture is to review with only minor additions the material on generalised Hamiltonian dynamics and gau­ge fileds presented in references [1], [7], [8], [12], [13], [14] and [15]. ,

1. Hamiltonian systems and generalizations.

Let Q be a manifold. We denote by T*(Q) the cotangent bundle and by

7 r o : r * (0 ) ->g (l .D

the cotangent bundle projection. The canonical 1-form on T*(Q) will be denoted by 6Q. The manifold T*(Q) together with the 2-form CJOQ^CIBQ

form a symplectic manifold. /

Let

X: T*(Q)^T(T*(Q)) (1.2)

be a Hamiltonian vector field. For the sake of simplicity we assume that X is globally Hamiltonian: there is a Hamiltonian

H:T*jQ)^R (1.3)

such that

XJuQ = -dH. (1.4)

We also assume that X is complete and denote by

4>:RX T*(Q)^T*(Q) (1.5)

the flow of X. Mappings

<l>t:T*(Q)-+T*(Q):p^<l>(t,p) (1.6)

are symplectomorphisms. Let ir^Q denote the 1-form on T(T*(Q)) denifed by

<w,iTojQ> = <TTiT*(Q))(w) A TTT*{Q)(W), COQ> , (1.7)

where

TT(T*(Q)): T(T(T*(e)))->T(T*(0)) (1.8)

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is the tangent bundle projection and

TTT*{Q):T(T(T*(Q)))-+T(T*(Q)) (1.9)

is the tangent mapping of the tangent bundle projection

r r . ( o ) : T ( r * ( 0 ) ) - > r * ( e ) . (1.10)

The manifold T(T*(Q)) together with the form

dTcoQ~diT(jOQ (1.11)

form a symplectic manifold. Let WQGCOQ be the 2-form

CJQ 0COQ =pr2*coQ -prt*coQ, (1.12)

where

pr1:T*(Q)X T*(Q)^T*(Q) (1.13)

and

pr2 : T*(Q) X T*(Q) -+ T*(Q) (1.14)

are the canonical projections of T*(Q) X T*(Q) onto the right factor and the left factor respectively. The manifold T*(Q) X T*(Q) together with cog 0 toQ form a symplectic manifold.

The following observations about Hamiltonian systems provide a basis for their generalization:

The image D' of X : T*(Q)-+T(T*(Q)) is a Lagrangian submanifold of the symplectic manifold (T(T*(Q)),dToo0). For each tGR the graph Dt of the mapping </>t: T*(Q)-+T*(Q) is a Lagrangian submanifold of (T*(Q)X T*(Q), CJ0 B o ; 0 ) and (p2,pi)€Dt if and only if there is a curve 17: [0, t] -• T*(0) such that T?(0) =p! , 77(f) = p 2

a nd vectors tangent to T? belong to D'. Let ^7'0Q be the 1-form

dTdQ=diT6Q+iTddQl (1.15)

where i r 0£ is the function

iTeQ:T(T*(Q))^R:w^<w,dQ>. (1.16)

Then

' drojQ=ddT0Q. (1.17)

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It is known that there are diffeomprphisms

OLQ:T(T*(Q))-+T*(T(Q))

and

PQ:T(T*(Q))^T*(T*(Q))

such that

and the diagram

(1.18)

(1.19)

(1.20)

(1.21)

T*(T(Q)) T(T*(Q)) T*(T*(Q))

KT*(Q) (1-22)

is commutative. The diffeomorphism j3g is the vector bundle isomorphism

PQ: T(T*(Q))-+T*(T*(Q)): V»+WJ<OQ: (1.23)

The diffeomorphism a^ is harder to define directly. Indirectly it is com­pletely characterized by formula (1.20).

It is easily seen that

X = jSQ1 - ( - i f f ) . (1.24)

Hence, ~H is a generating function of D' in the sense used in [9]. The submanifold D' may also be generated by a Lagrangian

L.T(Q)^R

in the sense that D' is the image of a section

( V -dL :T(Q)^T(T*(Q)Y

There is an obvious diffeomorphism

(1.25)

(1.26)

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7Q : T*(Q) X T*(Q) "* T*(Q X 0) (1.27)

such that

7 O * 0 Q X Q = 0 Q ©0<? (1.28)

= pr2*0Q-prl*dQ

and

^OXQ '7(? =7T(? X 7TQ,. (1.29)

Typically the Lagrangian submanifolds Dt are generated (at least local­ly) by the Hamilton principal function

W .RXQX Q-+R (1.30)

A submanifold Dt is the image of

Tg'1 *dWtt (1.31) where

Hamiltonian systems are used to represent time independent dynamics of nonrelativistic particles. The following definitions generalize the concept of a Hamiltonian vector field and the associated flow in a manner useful for representing time dependent dynamics and dynamics of relativistic particles.

Let M be a manifold, let ICR be a neighbourhood of 0 6 / ? and let rj:I -» T*(M) be a differentiable curve. We denote by rj.I -> T(T*(M)) the map­ping associating with £ E / the vector tangent to r\ at TJ(?).

Definition 1.1. A Lagrangian submanifold D' of (T(T*(M)), i rcoM) is called an infinitesimal symplectic relation if for each w€.D' there is a curve r\:I-* -+T*(M) such that the image of rj is contained in D' and r?(0) = w.

Definition 1.2. For each tER let D, C T*{M) X r*(Af) be the set defined by : ( p 2 , p i ) ^ A if there is a curve rj:[Of t]^-T*(M) such that 7?(0)=/?,, W(t)=p2 and the image of 77 is contained in an infinitesimal symplectic rela­tion D\ If Dt is a closed Lagrangian submanifold for each t then D' is said to be complete and the family {Z)J is called the integral of D'.

If D' is not complete suitable defined local integrals may be considered.

Esample 1.1. Generalized Hamiltonian dynamics of Dirac [3], [6], [7]. Let C C T*(M) be a coisotropic submanifold and let H:C -» R be a function con-

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stant on leaves of the characteristic foliation of co^ |C, The set

D' = {w e T(T*(Mh wETCwJ(uM\C)=-dH} (1.33)

is an infinitesimal symplectic relation. If the reduced symplectic manifold exists then D' is complete.

Example 1.2. Homogenous systems. With H = 0 a generalized Hamiltonian system is called a homogenous system and denoted by (T*(M), coM\C). The infinitesimal symplectic relation associated with a homogenous system is the characteristic distribution of co^lC The integral of D' consists of only one symplectic relation

D= {(p2,Pi)GT*(M)X T*(M); Pi and p2 belong to the ( 1 J 4 )

same leaf of the characteristic foliation of toM\C}.

Example 1.3. Homogenous formulation of nonrelativistic dynamics. Let 44 =

= R X Q represent the space-time manifold of a particle. The cotangent bun­dle T*(M) isomorphic to R2 X T*(Q) represents the space-time-energy-mo­mentum manifold of the particle. We denote by t the projection of M on R and by t and e the two projections of T*(M) on R. If H:T*(Q)^>R is a Hamiltonian then the equation e = H defines a coisotropic submanifold of (T*(Af), CJM)- The leaves of the characteristic foliation of coM\C are exactly the space-time-energy-momentum trajectories of the particle. Hence, the dy­namics of the particle is correctly represented by the homogenous system

The homogenous formulation of dynamics offers several advantages. Ex­tension to time dependent dynamics is immediate. If H.R X T*(Q) -* R is a time dependent Hamiltonian then the homogenous system .(T*(M), co^; C) with C defined by e = H again correctly represents the dynamics. The homo­genous formulation is closely related to the Poincare-Cartan formulation [17] and provides a basis for a geometric interpretation of the Hamilton-Jacobi method [2]. Homogenous systems have been used to describe relati­vistic dynamics [8]. A more accurate formulation of relativistic dynamics is given in the next section.

2. Dynamics of relativistic particles.

Let M be a differential manifold of dimension 4 representing the space-time manifold of a relativistic particle. The gravitational field is represented

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by a pseudo-Riemannian covariant metric tensor g: T(M) X T(M)-+R. The sa­me symbol g will also denote the contravariant metric tensor g:T*(M) X X T*(M)^>R. Formulae

<uyg(v, •)>=£(*>,«), (2.1)

<g(p, -),'r>=g(pfr) (2.2)

define expressions g(v, •) and g(p, •)• The cotangent bundle T*(M) represents the space-time-energy-momen­

tum manifold of the particle. The following statement summarizes what is known about space-time-energy-momentum trajectories of particles:

A trajectory of a particle of mass m is an oriented one-dimensional sub-manifold t] of T*(M) contained in the mass shell

C={peT*(M);g(p,p) = m2}. (2.3)

The projection £ of 7? to M is an oriented time-like geodesic. At each point p Er? the vector g(p, •) is tangent to % and compatible with the orienta­tion.

It is easily seen that space-time-energy-momentum trajectories of a par­ticle are integral manifolds of the infinitesimal symplectic relation [15]

D' = {w e T(T*(M)hp = TT*{MM3 C> V =

T*M<V>) is time-like: g(v, v) > 0, mv -\/g(v, v) g(p, •), w is horizontal in the sense of the Riemannian connection in T(T*(M))}.

The function

L:T(M)^R:v-^mVg(v,v) (2.5)

defined for time-like vectors v has always been regarded as the Lagrangian of a relativistic particle. This function generates the infinitesimal relation D' in the sense that D' is the image of the mapping

*MX -dL:T(M)^>T(T*(M)). (2.6)

Hamiltonian description of relativistic dynamics is obtained by using the Morse family [18]

F:R+ X T*(M) -* R :(X, p) -+ \(y/g{p7F) ~ m) (2.7)

defined for g(p, p) > 0. The Lagrangian submanifold D' is the inverse image

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by PM of t n e Lagrangian submanifold of (T*(T*(M)), OJT*(M)) generated by -F.

If M is sufficiently regular (for example flat) then there are symplectic relations D+ and D- such that Dt~D+ for £ > 0 , Dt = D- for £ < 0 and,D0 is empty. The relation D+ is generated by a function

W:AfXM-*K (2.8)

representing the geodesic distance between two time separated points of M multiplied by m. The relation D- is generated by'— W.

The infinitesimal symplectic relation Dr is not the image of a Hamilto-nian vector field, it is close to a homogenous system. The set D' consists of elements of the characteristic distribution of coM\C compatible with the na­tural orientation. The non-relativistic limit of D* is an infinitesimal symplectic relation representing the dynamics of a non-relativistic particle in Newtonian space-time. /

Wave mechanics of a relativistic particle is based on the Klein-Gordon equation. Statistical interpretation of this equation encounters serious dif­ficulties. The most successful attempt at such interpretation was made by Feynman [4] and was based on the choice of a Green's function. The formu­lation of classical mechanics presented here is in a natural correspondence with the quantum theory proposed by Feynman. The Klein-Gordon equation cor­responds to the mass shell constraint and Feynman's choice of the Green's function corresponds to our choice of the natural orientation [15].

The Hamiltonian system associated with the Hamiltonian

H:T*(M)->R:p^—g(p,p) (2.9) 2m

has been used to describe dynamics of relativistic particles. This system is only loosely related to the infinitesimal symplectic relation D'. It is neither the classical limit of a quantum theory nor does it have the expected non-re­lativistic limit. Information contained in this system is incomplete without the mass shell constraint and the mass shell by itself containes complete in­formation on dynamics.

3. Dynamics of charged particles.

It is believed that dynamical laws governing the behaviour of physical systems can be modified to include interaction with the electromagnetic

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field by applying the principle of minimal coupling. The geometric interpreta­tion of electromagnetism associated with this principle is valuable although examples show that strict insistence on minimality of couplings could lead to contradictory results. Wave mecanics of relativistic particles with spin can be equally well described by first order equations (Dirac) as by second order equations (Klein-Gordon with additional relations). Both descriptions can be extended to include coupling with the electromagnetic field consistent with experiment. Only in the case of first order equations is this coupling minimal. Recent research has improved the understanding of the geometric structure of electromagnetism [7], [8], [10], [12]. We give a summary of different geometric formulations of the dynamics of relativistic charged particles.

The relativistic dynamics of the preceding section can be modified to include interaction with the electromagnetic field by the use of the sub­stitution

^A:T*(M)^T*(M):p^p-eA(7rM(p)\ (3.1)

where e is the charge of the particle and A is a 1-form on M representing the electromagnetic potential. The modified mass shell

<M = I M C ) (3.2)

generates the infinitesimal symplectic relation

D'A = {to G T(T*(M)); w G T(CA), w J (a>M\CA )= 0, • _ (3.3)

mv=Vg(v, v)g(p - eA(7TM(p)), -),v = TirM(w),p = TT*{M)(W)}.

Integral manifolds of D'A are the space-time-energy-momentum trajectories of charged particles. The substitution (3.1) is exactly what is recommended by the principle of minimal coupling. A quantum version of this substitution leads to the well known modification of the Klein-Gordon equation. This formulation of classical dynamics and the corresponding wave mecanics are strongly gauge dependent. Moreover the energy-momentum of the particle is represented by p ~ eA(7TM(p)) tather than directly by p G T*(M).

An alternate formulation of dynamics is obtained by preserving the mass shell C and modifying instead the symplectic structure of T*(M). Let 6M be the modified canonical 1-form

^=^A1reM=eM^eirM*A (3.4)

and let oo% denote the 2-form

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dO&=>wM-eirM*F, (3.5)

where F = -dA. The form (oj& is gauge independent and is defined even if 4 exists only locally. The infinitesimal symplectic relation

D'F = {w G T(T*(M))-, w e TC, w J (co& |C) = 0, . (3.6)

mv =Vg(v, v)g(p, •),*> = 7XM(W),p = Tr*(yVf)(w)}

provides a gauge invariant formulation of dynamics. The energy-momentum of the particle is represented in this formulation directly by elements of T*(M). Unfortunately only the symplectic 2-form coj is gauge invariant. The 1-form and consequently at least the Lagrangian description of dynamics will depend on the choice of A.

A truly gauge independent formulation of dynamics of charged particles is given in terms of the principal fibre bundle of electromagnetic gauge fra­mes. This formulation is equivalent to the formulation based on Kaluza theory [8], [12], [20], [21]. Let ?:Z->Af be a principal fibre bundle with the additive group of real numbers for the structural group. Let X.Z^T(Z) denote the fundamental vertical vector field. The electromagnetic field is represented by a connection in the bundle Z. The connection form will be denoted by a. The cotangent bundle T*(Z) represents the space-time-phase -energy-momentum-charge manifold of a charged particle. The charge of the particle is the vertical component^of r^T*(Z) and energy-momentum is the horizontal component. The vertical component of r can be represented by the number q = <X, r> and the horizontal component can be represented by a covector p E T*(M) at f (irz(r)) denoted by hor(r). Trajectories in T*(Z) of a particle of mass m and charge e must be two-dimensional and must be contained in the coisotropic submanifold

C={reT*(Zh<X,r>=e1g(p1p) = m2ip = hor(r)l (3.7)

The infinitesimal symplectic relation

D' = {w e T(T*(Z)h w G T(C\ w J (cozlC) = 0, . (3.8)

mv =Vg(v, v)g(p, •), v = TZ(Tnz(w)), p = hor(TT*iZ)(w))}

describes the dynamics of the particle. The infinitesimal relation D' is ge­nerated by the Lagrangian

L:T(Z)-+R:w-*m Vg(v,v) + e<w,-a> (3.9)

defined for g(v, v) > 0, v — T$(w). Hamiltonian description of D' is obtained

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by using the Morse family

F.R+XRX T*(Z)^R:(\)i,r)^\(\/g(pyp)-m)+ ( . + H«X,r>-e)

defined for g(p, p) > 0, p = hor(r). The above formulation of classical dynamics in T*(Z) is in good cor­

respondence with the principal bundle formulation of wave mechanics of charged particles [12]. Gauge dependent formulations of classical dynamics in T*(M) are obtained from the formulation in T*(Z) by reduction [7], [12].

Let K denote the coisotropic submanifold

{rGT*(Z);<X,r> = e}. (3.11)

Leaves of the characteristic foliation of c*)z\K are the orbits belonging to K of the symplectic action of the structural group in T*(Z). The reduced symplectic manifold (5, a) is what Weinstein [19] calls the "universal phase space". The infinitesimal symplectic relation Df reduces to an infinitesimal symplectic relation D' CT(S). Although S is canonically diffeomorphic to T*(M) the symplectic manifold (5, a) is symplectomorphic to (T*(M), oiM) only locally and in a gauge dependent fashion.

The canonical diffeomorphism

p.S^T*{M) (3.12)

is defined by

<v,p(s)> = <w,r>, (3.13)

where rGKC T*(Z) is a representative of s and w is the the horizontal lift of v G T(M) to T(Z) at z = 7TZ(r). Let F be the curvature form on Af representing the electromagnetic field. Then

(pl)*o = uM-eirM*F = u5i. (3.14)

It follows that (5, a) is not symplectomorphic to (T*(M), OJM) but can be i-dentified with (T*(M), coj&). The image Tp(D') of the infinitesimal relation D' is the relation Dp. The mapping

hor:T*(Z)->T*(Af):r->hor(r) (3.15)

restricted to K is constant on leaves of the characteristic foliation and in­duces exactly the diffeomorphism p. If the universal phase space is identified

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with (T*(M), coj&) then its elements represent the energy-momentum of the particle.

Let A be the potential corresponding to a local trivalization of Z. The formula

<z>, pA (s) >= <w, r > (3.16)

defines a local sumplectomorphism

pA:S-+T*(M) (3.17)

if r is a representative of s and W is a lift of z> tangent to the trivializing section through z = irz(r). The diffeomorphisms p and p^ are related by

P = \IJA'PA- (3.18)

The infinitesimal relation D'A is the image TpA(D') of the relation D'CT(S).

4. Generalizations.

The principal bundle interpretation of electromagnetism has been ex­tended to gauge fields interacting with extended objects [22]. Although gauge independent dynamics of extended objects has not been formulated it is obvious that it can be done by~~imitating the gauge independent dynamics of charged particles. Generalizations to non-Abelian gauge theories is less obvious. Two versions have been formulated. Both are based on the assump­tion that the correct phase space for isotopic spin (non-Abelian charge) is an orbit of the coadjoint representation of the gauge group.

Let f:Z-*Af be a principal bundle with structural group G. This bundle induces in a natural way a principal bundle f:Z-*T*(fl4) (the pull back of f by ^M). If E is an orbit of the coadjoint representation of G representing the phase space of isotopic spin then the phase space of isotopic spin com­bined with space-time degrees of freedom is the associated bundle Z XGE. Sternberg [11] constructs a symplectic form Q, for this phase space using the symplectic structure of T*(Af) and the connection in Z representing the gau­ge field. Let 0:Z XGE;-• T*(M) be the bundle projection. If C C T*(M) is the mass shell then the characteristic distribution of 12 restricted to 0_1(C) with the canonical orientation taken into account provides an infinitesimal symplectic relation describing the dynamics of particles interacting with the gauge field.

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An alternate construction of a phase space for particles interacting with a gauge field was given by Weinstein [19]. This construction consists in redu­cing the symplectic manifold 7*(Z) X E with respect to a coisotropic sub-manifold K defined in terms of a momentum mapping. A connection in Z establishes an isomorphism of the abstract reduced space with the associated bundle ZXGE. The procedure is a generalization of the procedure formulated in [7], [12] and described in the preceding section.

An orbit E of the coadjoint representation is not, as a rule, isomorphic to a cotangent bundle. It follows that neither T*(Z) X E nor ZXGE are iso­morphic to cotangent bundles. Consequently dynamics of particles formulated in Z"XGE can not be described in Lagrangian terms and the Hamilton-Jacobi theory can not be formulated. Since the Hamilton-Jacobi theory is an im­portant link between wave mechanics and classical dynamics the present un­derstanding of dynamics interacting with non-Abelian gauge fields can not be considered entirely satisfactory. It is my conviction that the dynamics of such particles should be formulated in the cotangent bundle T*(Z). This conviction is founded upon a different approach to the quantum description of internal degrees of freedom such as spin or isotopic spin [16]. In this description the group space G plays the part of the configuration space. Func­tions on G represent quantum states of isotopic spin. Classical dynamics has not been developed. It is clear however that T*(G) should play the part of the phase space for isotopic spin.

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W.M. TULCZYJEW, Istituto di Fisica Matematica dell'Universita di Torino, Italia.

Lavoro pervenuto in redazione il: 10-VII-1981