29 May, 1995THE FEYNMAN REPRESENTATION FOR THE DIRACPROPAGATOR WITH A RADIALLY SYMMETRIC POTENTIALBrian JefferiesAbstract. The dynamical group associated with the Dirac equation with a radiallysymmetric potential in four space-time dimensions is represented in terms of integralswith respect to operator valued set functions associated with the free Dirac operator.In coordinates in which c = ~ = 1, the class of potentials treated includes Coulombpotentials �a=r with jaj < 1. IntroductionThis paper is a continuation of work commenced by the author in [J2], on therepresentation of the dynamical group U(t) associated with the Dirac equation witha radially symmetric potential as an integral with respect to an operator valued setfunctionMt, for each t > 0. The operator valued set functions hMtit�0, constructedfrom the free Dirac group S and a spectral measure Qr of multiplication by char-acteristic functions, measure the random events of an underlying one dimensionalprocess in place of a probability measure.For a radially symmetric potential V : x 7! q(jxj), x 2 R3 , the representationhas the form(0.1) U(t) = Z exp �� i Z t0 q(!(s)) ds�dMt(!); t > 0:The notation of the present work di�ers from that of [J2] in that the sample space is taken to be a set of paths ! : [0;1)! [0;1) rather than a collection of pathswith values in R3 . It is proved in [J2, Theorem 4.4] that the space may be takento be the collection of all paths ! : [0;1) ! [0;1) with speed �1 and at most�nitely many changes of direction in any bounded time interval. The feature of the\Zitterbewegung" of the Dirac particle remarked on elsewhere [I-T] is also presentin four space-time dimensions.The representation (0.1) is proved in [J2, Theorem 6.3] under the assumptionthat the function q is locally integrable on [0;1) and locally square integrable on(0;1), a result which excludes the case of a Coulomb potential q(r) = ��=r. Thedi�culty encountered in proving (0.1) for a Coulomb potential q is that for paths !which hit the origin during the time interval [0; t], the integral R t0 q(!(s)) ds divergeslogarithmically. The main result of the present work, Corollary 4.8, establishes thatthe set of such paths is negligible for Mt, contrary to the (unproved) assertion inthe �nal paragraph of [J2, p298], so at least the integrand in (0.1) makes sense1991 Mathematics Subject Classi�cation. Primary 81S40, 47D06; Secondary 28B05, 28C20,35L45. Typeset by AMS-TEX1

2 BRIAN JEFFERIESfor a full set of paths ! 2 . The notion of negligible sets for the unbounded setfunction Mt and an interpretation of the integral (0.1) are outlined in [J2, Section5].The representation (0.1) is proved for functions q : (0;1)! R which satisfy thebound(0.2) q = q(1) + q(2); supr>0 jq(1)(r)jr � �; supr>0 jq(2)(r)j <1;where � < p3=2. In particular, the result applies to Coulomb potentials q(r) =�a=r; r > 0 with jaj < p3=2. Here we are working in a coordinate system inwhich the speed of light c and Planck's constant ~ are equal to one, so that � < 1corresponds to Z < 137 in atomic units.Section 1 establishes some terminology for the type of processes this work em-ploys. In Section 2, the radially symmetric Dirac process is formally de�ned andsome results concerning the process proved in [J2] are mentioned again. Section3 reviews the notion, also developed in [J2], of integration with respect to theunbounded set functions associated with the radially symmetric Dirac process. De-pending on the reader's willingness to believe the author's assertions that are notherein proved, the present paper can be read independently of [J2]. The main ar-guments are in Section 4. The result that the set of paths which hit the origin isM+t -null is proved in Corollary 4.8 after obtaining, in Proposition 4.4, an explicitrepresentation by a perturbation series expansion, for regularisations M (")t ; " > 0,of Mt. The main result is given in Section 5. The representation (0.1) is estab-lished in Theorem 5.4 for potentials satisfying condition (0.2) with � < p3=2. Forpotentials q satisfying (0.2) with � < 1, and having constant sign near the origin,the dynamical group U(t) is represented in Theorem 5.6 as the limit of integralslike that on the right hand side of equation (0.1).1. (S;Q)-processesIn this section, we outline an abstract framework to deal with the processeswe wish to consider in the remainder of the paper. The framework indicates therelationship between what we consider a \process" here, and the more familiarnotion of a stochastic process with respect to a probability measure space. Let(�;B) be a measurable space and let S be a semigroup of continuous linear operatorsacting on a Banach space Y . The space L(Y ) of continuous linear operators actingon Y is equipped with the strong operator toplogy. Then S is a map from R+ intothe space L(Y ) such that S(s + t) = S(s)S(t) for all s; t � 0, and S(0) = Id, theidentity operator. No continuity assumption is made on S at this stage. Supposethat Q : B ! L(Y ) is a spectral measure, that is, Q is an operator valued measure,�-additive for the strong operator topology, such that Q(�) = Id and Q(A \B) =Q(A)Q(B) for all A;B 2 B.Suppose that is some non-empty collection of functions ! : R+ ! �: Lett > 0; n = 1; 2; : : : ; 0 < t1 < � � � < tn < t and suppose that B1; : : : ; Bn 2 B aresubsets of �: For each subset E of of the form(1.1) E = f! 2 : !(t1) 2 B1; : : : ; !(tn) 2 Bng;

FEYNMAN REPRESENTATION 3the operator Mt(E) 2 L (Y ) is de�ned by the formula(1.2) Mt(E) = S(t� tn)Q(Bn)S(tn � tn�1) � � �Q(B2)S(t2 � t1)Q(B1)S(t1):Provided that the set E is non-empty whenever each of the sets B1; : : : ; Bn isnot a Q-null set, it follows that as the times t1; : : : ; tn; the sets B1; : : : ; Bn andn = 1; 2; : : : vary, but t is �xed, the sets E form a semi-algebra St of subsets of and the expression (1.2) de�nes an additive operator valued set function Mt,de�ned on St and acting on Y: Furthermore, the additivity of the set function Mtensures that it has a unique extension, also denoted by Mt; to the algebra �(St)of subsets of generated by St: Set Xs(!) = !(s) for all s � 0. Elements of thesemi-algebra St are called elementary events before time t. Then the collection�; hStit�0; hMtit�0; hXtit�0� is called an (S;Q)-process. The same terminology isadopted in the case that the semi-algebras hStit�0 are replaced by the �-algebrasthat they generate and for each t > 0, Mt is an operator valued measure. TheBanach space Y is called the state space of the process. For a Markov processwith probabilities P x; x 2 Rd , the associated operator valued measures Wt : St !L�Lp(Rd)� are de�ned for A 2 St by the formula(1.3) hWt(A)�; i = ZRd P x��A:( �Xt)��(x) dx; � 2 Lp(Rd ); 2 Lp0(Rd ):The idea of associating an operator valued set function with an arbitrary semigroupand a spectral measure is due to I. Kluv�anek [Klu].If the range fMt(A) : A 2 �(St)g of the additive set function Mt on the algebra�(St) is bounded in the uniform operator norm of L(Y ) and = �R+, then undercertain mild conditions,Mt is actually the restriction to �(St) of an operator valuedmeasure de�ned on the �-algebra �(St) generated by St | the techniques of proba-bility theory are directly applicable to this case. If the range ofMt is unbounded on�(St) in the uniform operator norm of L(Y ), then integration with respect to Mtmay be controlled by a family of operator valued measures; this is the situation forthe radially symmetric Dirac process considered in the next section, and discussedin greater detail in Section 3.2. The radially symmetric Dirac process.The free Dirac operator is de�ned in L2(R3 ; C 4) by means of the di�erentialexpression D = cP3j=1 �jpj + �4mac2; where c > 0 is the velocity of light, ma >0 is the mass of the particle, pj = ~i @@xj ; and�j = � 0 �j�j 0 � ; j = 1; 2; 3; �4 = ��0 00 ��0 � :Here �1; �2; �2 are the Pauli matrices�1 = � 0 11 0� ; �2 = � 0 �ii 0 � ; �3 = � 1 00 �1�

4 BRIAN JEFFERIESand �0 = �1 00 1� is the 2� 2 identity matrix.Then D de�nes a selfadjoint operator, and so, a unitary group SD(t) = eiDt; t 2R of operators acting on L2(R3 ; C 4): For any n = 1; 2; : : : ; the space C n is assumedto be equipped with the inner product ha; bi =Pnj=1 ajbj for a = (a1; : : : ; an) andb = (b1; : : : ; bn) in C n . The space L2(R3 ; C 4) is a Hilbert space with the inner-product (f; g) = RR3hf(x); g(x)idx for f; g 2 L2(R3 ; C 4): We choose a coordinatesystem in which c = ~ = 1:The space of all smooth C 2 -valued functions on R++ = (0;1) with compactsupport is denoted by C1c (R++ ; C 2): The Borel �-algebra of a Hausdor� topologicalspace � is denoted by B(�):By virtue of the angular momentum decomposition of the Dirac operator, thereexists a family Hk;m; k = �1;�2; : : : and m = �jkj;�jkj+1; : : : ; jkj�1 of mutuallyorthogonal subspaces of L2(R3 ; C 4); such that L2(R3 ; C 4) is the Hilbert space directsum of Hk;m; each space Hk;m is a reducing subspace for the free Dirac operatorD and in each Hk;m; D is unitarily equivalent to the closure in L2(R+ ; C 2); of theoperator �k de�ned by(2.1) �k : g 7! � 0 �11 0 �� g01(r)g20(r)�+ � ma �kr�1�kr�1 �ma �� g1(r)g2(r)�on the space C1c (R++ ; C 2). The essential selfadjointness of the operator D with do-main C1c (R3 nf0g; C 4) serves to establish the essential selfadjointness in L2(R+ ; C 2)of the operator �k for each k = �1;�2; : : : . The details are laid out in [A]. Thus,for each k = �1;�2; : : : and m = �jkj;�jkj+1; : : : ; jkj�1; we have �k = Uk;m(D �Hk;m)U�k;m for a unitary operator Uk;m : Hk;m ! L2(R+ ; C 2) onto a closed sub-space of L2(R+ ; C 2). Here D � Hk;m is the restriction of D to the reducing subspaceHk;m:For each k 2 Z n f0g; let Tk be the unitary group of operators de�ned onL2(R+ ; C 2) by Tk(t) = ei�kt for all t 2 R: Let Q be the spectral measure, actingon L2(R+ ; C 2); of multiplication by the characteristic functions of Borel subsets ofR+ :Let be the set of all continuous functions ! : R+ ! R+ such that for eacht > 0, the sets f!0 = 1g and f!0 = �1g in (0; t) are the �nite union of openintervals, the union of which contains all except �nitely many points in (0; t). Lessformally, the paths ! 2 are continuous with velocity �1, and only �nitely manychanges in direction in any bounded time interval.For each k = �1;�2; : : : ; let �; hStit�0; R(k)t �t�0; hXtit�0� be the (Tk; Q)-process with state space L2(R+ ; C 2). For each t � 0, St is the algebra generatedby the family of all elementary events before time t , R(k)t : St ! Ls(L2(R+ ; C 2)) isan additive operator valued set function and Xt(!) = !(t); ! 2 is the evaluationmap.The (SD; Qr)-process with state space L2(R3 ; C 4) is denoted by�; hStit�0; hMtit�0; hXtit�0�:For each t � 0, Mt : St ! Ls(L2(R3 ; C 4)) is an additive operator valued setfunction. The (SD; Qr)-process is called the radially symmetric Dirac process. The

FEYNMAN REPRESENTATION 5notation adopted here di�ers from that of [J2, p317]; here we wish to emphasisethe one-dimensional character of the process.The radially symmetric Dirac process is written in terms of the processes�; hStit�0; R(k)t �t�0; hXtit�0�; k = �1;�2; : : :by setting(2.2) Mt(E) = 1Xjkj=1M0@ jkj�1Xm=�jkjMU�k;mR(k)t (E)Uk;m1A for all E 2 St:3. Integration with respect to Mt and M (")t �">0In order to de�ne integration with respect to the operator valued set functionsMt, we de�ne �-additive operator valued measures M (")t for all " > 0 by cutting o�the singularity in the right hand side of formula (2.1) at r = 0. The details of thefollowing construction are laid out in [J2].LetN = 1=p2�1 ii 1�. Then iN��0 �11 0�N = �1 00 �1�. Let D(�) be the collection of allfunctions f 2 L2(R+ ; C 2) such that f is absolutely continuous on all bounded subin-tervals of R+ , f 0 2 L2(R+ ; C 2) and (N�f)2 = 0. Let � : D(�)! L2(R+ ; C 2) be theoperator �0 �11 0�d=dx. Then i� is the generator of a C0-semigroup on L2(R+ ; C 2).For every " � 0, letUk;"(r) = 8>><>>:� ma �kr�1�kr�1 �ma � for all r > ";� ma �k"�1�k"�1 �ma � for all 0 < r � ":For each " > 0, Uk;" is a bounded function with values in the 2 � 2 hermitianmatrices and i�k;" = i(� + Uk;") is the generator of a C0-semigroup [K1, TheoremIX.2.1].The operator de�ned by (2.1) is essentially selfadjoint; its closure in L2(R+ ; C 2)is denoted by the same symbol. Then lim"!0 ei�k;"t = ei�kt in the strong operatortopology of L�L2(R+ ; C 2)� [J2, p309].Let Rt be the �-algebra generated by elementary events in before time t.It turns out ([J2, Theorem 4.3]; see also Corollary 4.6 below) that for each k =�1;�2; : : : and each " > 0, there exists an (ei�k;"�; Q)-process�; hRtit�0; hR(k;")t it�0; hXtit�0�for which R(k;")t : Rt ! L2(R+ ; C 2) is a �-additive operator valued measure foreach t � 0. The measures R(k;")t ; " > 0 will be used to control integration withrespect to R(k)t .Let " > 0. The �nite direct sum Pk;m2K"LU�k;m(� + Uk;")Uk;m over the setK" all integers k;m such that m = �jkj;�jkj+ 1; : : : ; jkj � 1 and 1 � jkj � 1=" is

6 BRIAN JEFFERIESdenoted by D". Then iD" is the generator of a C0-contraction semigroup SD" onL2(R4 ; C 4) satisfying(3.1) SD"(t) = Xk;m2K"MU�k;mei�k;"tUk;m; for all t � 0:The range K" of the values of the integers k;m in the direct sum is not mentionedexplicitly in [J2]; taking a �nite sum ensures that there is no di�culty with theexponentially increasing bound [J2, Lemma 3.1] on the diameters of the ranges ofR(k;")t for increasing jkj. Because of the boundary conditions imposed on � , theoperator D" is not selfadjoint. Nevertheless, as " ! 0+, the closed unboundedoperator D" converges in the strong resolvent sense to the free Dirac operator D.For each " > 0, there exists an (SD" ; Qr)-process�; hRtit�0; M (")t �t�0; hXtit�0�such that(3.2) M (")t (E) = Xk;m2K"MU�k;mR(k;")t (E)Uk;m for all E 2 Rt:Let t > 0. Rather than formally de�ne the idea of a function being integrablewith respect to Mt relative to the family M (")t �">0 of operator valued measures,we shall sketch the main idea and refer to [J2, Section 5], where the details are laidout explicitly.The space L1(M+t ) denotes the vector space of all (equivalence classes of) func-tions which are integrable with respect to each operator valued measureM (")t ; " > 0:A net converges in L1(M+t ) exactly when it converges in each space L1(M (")t ); " > 0:For a simple function s based on the semi-algebra St, the operator valued setfunction s �Mt : St ! L�L(R3 ; C 4)� | the inde�nite integral of s with respect toMt | is de�ned by linearity, in the obvious way; this is possible, becauseMt : St !L�L(R3 ; C 4)� is additive. Suppose that Tt is the locally convex topology de�nedon the simple functions such that a net hs�i�2I converges to a simple function sif and only if hs�i�2I converges to s in L1(M+t ), and for each A 2 St, the net�s� � M (")t �(A)��2I converges to the integral �s � M (")t �(A) of s with respect toM (")t over A, in the weak operator topology, uniformly for 0 < " � 1. Then given abounded Cauchy net hs�i�2I for the topology Tt, there exists f 2 L1(M+t ) such thaths�i�2I converges to f in L1(M+t ), and for each A 2 St, the net h�s� �M (")t �(A)i�2Iconverges to �f �M (")t �(A) in the weak operator topology, uniformly for 0 < " � 1.Moreover, ZA f(!) dMt(!) = �f �Mt�(A) := lim"!0+ �f �M (")t �(A)exists in the weak operator topology for each A 2 St. Then f � Mt : St !L�L2(R3 ; C 4)� is an additive operator valued set function de�ned on the semi-algebra St and the function f is said to be M+t -Mt-integrable. The space of all(equivalence classes of) M+t -Mt-integrable functions is denoted by L1(M+t ;Mt). Itis endowed with the topology associated with Tt. With this viewpoint, the null setsof Mt are just those subsets of which are M (")t -null sets for each " > 0.

FEYNMAN REPRESENTATION 7Remark. The idea of integration with respect to unbounded set functions Mtoutlined above applies to situations more general than the radially symmetric Diracprocess. The basic requirement is obviously thatMt should belong to the closure inthe topology of setwise convergence on the underlying semi-algebra of elementaryevents, of a set Mt = fM (")t : " > 0g of �-additive measures. In that situation,the scheme outlined in [J1] applies, and we obtain the notion of Mt-Mt-integrablefunctions. There may be many \regularisations" Mt of Mt | the appropriate oneis given by the problem at hand. For the radial parts of the Dirac operator, weeliminated the singularity at zero to obtain the family Mt above.By contrast, in non-relativistic quantum mechanics [J1] and, presumably, quan-tum �eld theory in Minkowski space, the unbounded set functionsMmt ;m 2 R;m 6=0 are the boundary values on the real axis of set functions Mzt with z in the upperhalf-plane P+. The family Mt = fMait : a > 0g of �-additive measures servesto de�ne Mt-M �t-integrable functions. Hence, the \regularisation" Mt of the setfunction M �t is obtained by analytic continuation Mzt ; z 2 P+ to the upper halfplane, followed by restriction to the positive imaginary axis.The technique of regularisation is familiar from other areas of analysis and isnaturally applied to the treatment of integration with respect to the unbounded setfunction arising in quantum physics.4. The support of the cut-off measures R(k;")t ; " > 0We shall require more information about the support of the operator valuedmeasures R(k;")t ; " > 0; k = �1;�2; : : : . To this end, the C0-semigroup ei(�+Uk;")tcan be written as a perturbation series ei(�+Uk;")t = P1j=0 ijV (")j (t), with V0(t) =ei�t and(4.1) V (")j (t) = Z t0 Z �j0 � � �Z �20 ei�(t��j)Uk;"ei�(�j��j�1) � � �ei�(�2��1)Uk;"ei��1 d�1 � � �d�jfor all t � 0 and all j = 1; 2; : : : : Here Uk;" is the operator of multiplication bythe matrix valued function r 7! Uk;"(r); r � 0. On applying this expansion to thede�nition of R(k;")t , we obtain R(k;")t (A) =P1J=0 iJR(k;")t;J (A), for all A 2 Rt, where(4.2) R(k;")t;J (E) = Xj0+���+jn=J V (")jn (t� tn)Q(Bn)V (")jn�1(tn � tn�1) � � �V (")j1 (t2 � t1)Q(B1)V (")j0 (t1);for all elementary events E of the form (1.1), and all J = 0; 1; 2; : : : . The sum isover integers j0; : : : ; jn � 0.It turns out that for each J = 0; 1; 2; : : : , R(k;")t;J is a �-additive operator valuedmeasure on Rt [J2, Lemma 4.2]. The set t denotes the collection of restrictionsof paths ! 2 to the interval [0; t]. It is natural to consider a set A 2 Rt as beinga subset of t, via the restriction map. We shall pass between the two viewpointswithout further mention.

8 BRIAN JEFFERIESThe aim in this section is to show that the set �t of all paths ! 2 which hitthe origin at some time in the interval [0; t] is an R(k;")t -null set for each " > 0. Tothis end, we shall show that �t is R(k;")t;J -null for each " > 0 and each J = 0; 1; 2; : : : .Let T be the in�nitesimal generator of the semigroup eTs; s � 0 which sends� 2 L2(R+ ; C 2) into the function�eTs��(x) = � �1(x+ s)�R+(x� s)�2(x� s)� ; for almost all x 2 R+ ;that is, the domain D(T ) of T is the set of all functions f 2 L2(R+ ; C 2) such thatf is absolutely continuous on all bounded subintervals of R+ , f 0 2 L2(R+ ; C 2) andf2(0) = 0. Then for the matrix N de�ned in section 3, the equalityi�k;" = N �� 1 00 �1� ddx + iN�Uk;"N�N� = N [T + iN�Uk;"N ]N�holds, D(�) = ND(T )N�, i� = NTN� andei�k;"s = Ne(T+iN�Uk;"N)sN�; for all s � 0:The operator in L�L2(R+ ; C 2)� of multiplication by iN�Uk;"N is bounded, so T +iN�Uk;"N is a bounded perturbation of the generator T of a C0-semigroup, henceit is the generator of a C0-semigroup which has a \Dyson" series expansion [K1,Theorem IX.2.1].The argument we are going to use is essentially combinatorial, so instead ofworking with the semigroups ei�s; s � 0 and ei�k;"s; s � 0, in formulae (4.1) and(4.2), we shall use eTs; s � 0 and e(T+U)s; s � 0, where U is a nonzero constantmatrix. The modi�cations required to replace U by a matrix valued multiplicationoperator will become apparent from the following discussion. The situation inwhich formulae (4.1) and (4.2) apply can then be covered by the application of thesimilarity transformation a 7! NaN� acting on L�L2(R+ ; C 2)�.To this end, set e(T+U)t =P1j=0 Vj(t), V0(t) = eTt and(4.3)Vj(t) = Z t0 Z �j0 � � �Z �20 eT (t��j)UeT (�j��j�1) � � � eT (�2��1)UeT�1 d�1 � � �d�jfor all t � 0 and all j = 1; 2; : : : : Here the same symbol U denotes the op-erator acting on L2(R+ ; C 2) of pointwise multiplication by the constant matrixU = �u11 u12u21 u22�. We de�ne Rt(A) =P1J=0Rt;J (A) for all A 2 Rt, where(4.4) Rt;J(E) = Xj0+���+jn=J Vjn(t� tn)Q(Bn)Vjn�1(tn � tn�1) � � �Vj2(t3 � t2)Q(B2)Vj1(t2 � t1)Q(B1)Vj0(t1);for all elementary events E of the form (1.1). That Rt;J is well-de�ned and additivefollows from the formula Pl+j=m Vl(u)Vj(v) = Vm(u + v) for m = 0; 1; : : : andu; v � 0 [J2, (4.2)]. The sum is over all non-negative integers.

FEYNMAN REPRESENTATION 9We shall compute Rt;0 and Rt;1 explicitly. The general nature of Rt;J and hence,of R(k;")t;J , will then be apparent for all J = 0; 1; 2; : : : ; a formal proof of Corollary4.8 by induction would be less than illuminating.First, for every set E of the form (1.1),Rt;0(E) = V0(t� tn)Q(Bn)V0(tn � tn�1) � � �V0(t2 � t1)Q(B1)V0(t1);so that Rt;0 is the �eT �; Q; t�-set function.Let � be the characteristic function of R+ . Then for every � 2 L2(R+ ; C 2), wehaveRt;0(E)� = eT (t�tn)Q(Bn)eT (tn�tn�1) � � � eT (t2�t1)� �B1(x)�1(x+ t1)�B1(x)�(x� t1)�2(x� t1)�... ...= � �Bn(x+ t� tn) � � ��B1(x+ t� t1)�1(x+ t)�B2(x� t+ tn) � � ��B1(x� t+ t1)�(x� t)�2(x� t)� :For convenience, we have written the composite function f � g as f(g(x)) on R+ .For each x � t let x;t : [0; t] ! R+ be the path de�ned by x;t(s) = x � t + sfor all 0 � s � t. For each x � 0 let �x;t : [0; t] ! R+ be the path de�ned by�x;t(s) = x + t � s for all 0 � s � t. Then in terms of the paths x;t and �x;t, wehave �Rt;0(E)��(x) = � �Bn(�x;t(tn)) � � ��B1(�x;t(t1))�1(�x;t(0)))�Bn( x;t(tn)) � � ��B1( x;t(t1))�( x;t(0))�2( x;t(0))� ;for almost all x � 0. Let � : Rt ! f0; 1g be the unit point mass at the path 2 t; as mentioned earlier, we consider elements of Rt to be subsets of t. Onappealing to the �-additivity of Rt;0, we have proved the followingProposition 4.1. For all A 2 Rt,(4.5)�Rt;0(A)��(x) = � ��x;t(A)�1(�x;t(0))� x;t(A)�( x;t(0))�2( x;t(0))� ; for almost all x > 0:If 0 � x � t, then(4.6) �Rt;0(A)��(x) = � ��x;t(A)�1(�x;t(0))0 � :Consequently, Rt;0 is supported by the set t;0 = f�x;t : x � 0g [ f x;t : x � tg.The two point set �t;0 = f�0;t; t;tg is the set of all paths ! 2 t;0 which hit theorigin in the interval [0; t]. The set �t;0 clearly has Rt;0-measure zero.For J = 1, there are n+ 1 terms in the sum de�ning Rt;J(E) in (4.4). The �rstterm is(4.7) V1(t� tn)Q(Bn)V0(tn � tn�1) � � �V0(t2 � t1)Q(B1)V0(t1)= Z t�tn0 eT (t�tn��)UeT�Q(Bn)eT (tn�tn�1) � � � eT (t2�t1)Q(B1)eTt1 d�= Z ttn eT (t��)UeT (��tn)Q(Bn)eT (tn�tn�1) � � � eT (t2�t1)Q(B1)eTt1 d�:

10 BRIAN JEFFERIESThe integrand of (4.7) is given byeT (t��)UeT (��tn)Q(Bn)eT (tn�tn�1) � � � eT (t2�t1)Q(B1)eTt1� =eT (t��) �u11 u12u21 u22�� �Bn(x+ �� tn) � � ��B1(x+ �� t1)�1(x+ �)�Bn(x� �+ tn) � � ��B1(x� �+ t1)�(x� �)�2(x� �)�= �u11 11 + u12 12u21 21 + u22 22�where 11 = �Bn(x+ t� tn) � � ��B1(x+ t� t1)�1(x+ t) 12 = �Bn(x+ (t� �)� (�� tn)) � � ��B1(x+ (t� �)� (�� t1))�(x+ (t� �)� �)�2(x+ (t� �)� �) 21 = �Bn(x� (t� �) + �� tn) � � ��B1(x� (t� �) + �� t1)��(x� (t� �))�1(x� (t� �) + �) 22 = �Bn(x� (t� tn)) � � ��B1(x� (t� t1))�(x� t))�2(x� t):If we write 11 in terms of the paths �x;t, we get 11(x) = �Bn(�x;t(tn)) � � ��B1(�x;t(t1))�1(�x;t(0)) = ��x;t(E)�1(�x;t(0));and for 22: 22(x) = �Bn( x;t(tn)) � � ��B1( x;t(t1))�( x;t(0))�1( x;t(0))= � x;t(E)�( x;t(0))�2( x;t(0)):We require a similar expression for the terms 12 and 21. To do this, we introducenew paths belonging to t which have a change in direction at 0 < � < t.Let x;t(�; � ) be the continuous path with x;t(�; t) = x, 0x;t(�; s) = �1 forall � < s < t and 0x;t(�; s) = 1 for 0 < s < �, so that x;t(�; � ) has a changeof direction at �. Then x;t(�; s) = x + (t � �) � (s � �) for all � � s � t and x;t(�; s) = x + (t � �)� (� � s) for all 0 � s � �. Then because 0 < t1 < � � � <tn � � � t 12(x) = �Bn( x;t(�; tn)) � � ��B1( x;t(�; t1))�( x;t(�; 0))�2( x;t(�; 0))= � x;t(�; � )(E)�( x;t(�; 0))�2( x;t(�; 0)):Similarly, let �x;t(�; � ) be the continuous path with �x;t(�; t) = x, �0x;t(�; s) = 1 forall � < s < t and �0x;t(�; s) = �1 for 0 < s < �, so that �x;t(�; � ) has a changeof direction at �. Then �x;t(�; s) = x � (t � �) + (s � �) for all � � s � t and�x;t(�; s) = x� (t� �) + (�� s) for all 0 � s � �, and we have 21(x) = �Bn(�x;t(�; tn)) � � ��B1(�x;t(�; t1))�(�x;t(�; �))�1(�x;t(�; 0))= ��x;t(�; � )(E)�(�x;t(�; �))�1(�x;t(�; 0)):

FEYNMAN REPRESENTATION 11Thus, for almost all x 2 R+ , the equality�eT (t��)UeT (��tn)Q(Bn)eT (tn�tn�1) � � � eT (t2�t1)Q(B1)eTt1�� (x)= � u11��x;t(E)�1(�x;t(0)) + u12� x;t(�; � )(E)�( x;t(�; 0))�2( x;t(�; 0))u21��x;t(�; � )(E)�(�x;t(�; �))�1(�x;t(�; 0)) + u22� x;t(E)�( x;t(0))�2( x;t(0))�holds. Applying (4.7) to � gives�Z ttn eT (t��)UeT (��tn)Q(Bn)eT (tn�tn�1) � � � eT (t2�t1)Q(B1)eTt1� d�� (x)= u11��x;t(E)��1(�x;t(0))0 � (t� tn)+ u12 Z ttn � x;t(�; � )(E)� �( x;t(�; 0))�2( x;t(�; 0))0 � d�+ u21 Z ttn ��x;t(�; � )(E)� 0�(�x;t(�; �))�1(�x;t(�; 0)� d�+ u22� x;t(E)� 0�( x;t(0))�2( x;t(0))� (t� tn):Adding the other n expressions in (4.4) yields the equality(4.8)

�Rt;1(E)��(x) = u11��x;t(E)��1(�x;t(0))0 � t+ u12 Z t0 � x;t(�; � )(E)� �( x;t(�; 0))�2( x;t(�; 0))0 � d�+ u21 Z t0 ��x;t(�; � )(E)� 0�(�x;t(�; �))�1(�x;t(�; 0)� d�+ u22� x;t(E)� 0�( x;t(0))�2( x;t(0))� t;a representation which is independent of the form of the elementary event E. Thefunction x;t(�; � ) takes its minimum value min(x; x;t(�; 0)) at one of the endpointsof the time interval [0; t]. The minimum value of �x;t(�; � ) is �x;t(�; �)).Let f x;t(�; � ) � 0g denote the set of all 0 � � � t for which the path x;t(�; � )belongs to t. Similarly for f�x;t(�; � ) � 0g. We have derived the following expres-sion for Rt;1:Proposition 4.2. For every A 2 Rt, we have(4.9)

�Rt;1(A)��(x) = u11��x;t(A)��1(�x;t(0))0 � t+ u12 Zf x;t(�; � )�0g � x;t(�; � )(A)��2( x;t(�; 0))0 � d�+ u21 Zf�x;t(�; � )�0g ��x;t(�; � )(A)� 0�1(�x;t(�; 0)� d�+ u22� x;t(A)� 0�( x;t(0))�2( x;t(0))� t:

12 BRIAN JEFFERIESIt follows that Rt;1 is concentrated on the set t;1 of paths ! 2 t with at mostone change of direction in the interval [0; t]. The set �t;1 of paths ! 2 t;1 whichhit the origin in the interval [0; t] isf�0;t; t;t; �x;t(x� t; � ); x;t((x+ t)=2; � ) : 0 � x � tg:It follows from Fubini's theorem that for any subset A 2 Rt contained in �t;1,the integral RR+ j�Rt;1(A)��(x)j dx is zero. Therefore, �t;1 is an Rt;1-null set.Having computed the cases J = 0; 1 explicitly, we show that there exists ananalogous representation in the cases J = 2; 3; : : : , although we shall not derive anexact formula in general. The main features are already apparent for J = 1.The terms in the sum (4.4) can be written as(4.10) Vjn(t� tn)Q(Bn)Vjn�1(tn � tn�1) � � �Vj1(t2 � t1)Q(B1)Vj0(t1) =ZW Pn(�j0+���+jn�1+1; : : : ; �j0+���+jn)�n�1Yk=0[Q(Bk+1)Pk(�j0+���+jk�1+1; : : : ; �j0+���+jk)] d�1 � � �d�JIf jk 6= 0, the operator valued function Pk(�j0+���+jk�1+1; : : : ; �j0+���+jk) is equal toeT (tk+1��j0+���+jk )UeT (�j0+���+jk��j0+���+jk�1) � � �: : : eT (�j0+���+jk�1+2��j0+���+jk�1+1)UeT (�j0+���+jk�1+1�tk);otherwise it is equal to the constant operator exp(T (tk+1 � tk)). The set W is theordered set of J points in the interval [0; t] such that there are exactly jk points inthe intervals [tk; tk+1], where k = 0; : : : ; n, t0 = 0 and tn+1 = t.The integrand in (4.10) can be expressed as a �nite linear combination of func-tionsZn(�j0+���+jn�1+1; : : : ; �j0+���+jn)�n�1Yk=1[Q(Bk+1)Zk(�j0+���+jk�1+1; : : : ; �j0+���+jk)]Q(B1)Z0(�1; : : : ; �j0):If jk 6= 0, the operator valued function Zk(�j0+���+jk�1+1; : : : ; �j0+���+jk) is equal to(4.11) eT (tk+1��j0+���+jk )Ej0+���+jkeT (�j0+���+jk��j0+���+jk�1) � � �� � � eT (�j0+���+jk�1+2��j0+���+jk�1+1)Ej0+���+jk�1+1eT (�j0+���+jk�1+1�tk);for some choice of matrices Ek, k = 1; : : : ; J from the standard basis E of the 2� 2matrices. If jk = 0, then it is equal to the constant operator exp (T (tk+1 � tk)).Let � : f1; : : : ; Jg ! f0; 1g and 0 < �1 < � � � < �J < t. Let +x;t(�1; : : : ; �J ; �; � )(respectively, �x;t(�1; : : : ; �J ; �; � )) denote the path ! : [0; t]! R, such that !(t) =x, !0(t) = 1 (respectively, !0(t) = �1) and for each j = 1; : : : ; J , there is a changein direction of ! at �j if and only if �(j) = 1; elsewhere, ! has velocity �1.

FEYNMAN REPRESENTATION 13In terms of the paths employed in Proposition 4.2 for the case J = 1, we have x;t = +x;t(�; 0; � ), �x;t = �x;t(�; 0; � ), x;t(�; � ) = �x;t(�; 1; � ), and �x;t(�; � ) = +x;t(�; 1; � ).We shall use the notation �x;t(�1; : : : ; �J ; �; � ) to denote the function whichassigns to each point x 2 R+ the path �x;t(�1; : : : ; �J ; �; � ). We use the terminologypath function here.Lemma 4.3. Let � 2 L2(R+ ; C 2). Then there exists a map � : f1; : : : ; Jg ! f0; 1g,a path function !�x = �x;t(�1; : : : ; �J ; �; � ) taking values in t, and a matrix M 2E [f0g, such that for every elementary event E � t of the form (1.1), the equality(4.12) Zn(�j0+���+jn�1+1; : : : ; �j0+���+jn)�[n�1Yk=1Q(Bk+1)Zk(�j0+���+jk�1+1; : : : ; �j0+���+jk)]Z0(�1; : : : ; �j0)�= �!�x (E)(M�)(!�x (0));holds almost everywhere. If EJ = "1 or EJ = "2, then the path function !�xappears, otherwise, it is !+x :Proof. Let"1 = � 1 00 0� ; "2 = � 0 10 0� ; "3 = � 0 01 0� ; "4 = � 0 00 1�be the standard basis elements of the 2� 2 matrices.The action of the operator eTs on 2 L2(R+ ; C 2) is to translate the graph ofthe �rst component 1 of to the left by an amount s, and the graph of the secondcomponent 2 to the right by an amount s, setting �eTs �2(x) equal to the zerofor 0 � x � s. However, on encountering a basis element "2 or "3, eTs translates acomponent in the opposite direction. For example, �eTs"2 �2 = 0 and the graphof �eTs"2 �1 is the translate of 2 to the left by an amount s.It is clear that the left hand side of (4.12) is some translate of the function M�,times a scalar function which takes the values zero or one, and depending on thesets B1; : : : ; Bn. Here the matrix M is the product EJ � � �E1 of the basis matricesappearing in (4.11).If the matrix M is the zero matrix, then � and the path function !�x can bechosen arbitrarily | both sides of (4.12) are identically zero.Suppose now that M 6= 0. For each j = 1; : : : ; J , set �(j) = 1 if Ej is equalto either "2 or "3, and set �(j) = 0 otherwise. Nonzero values of the function �correspond to changes in direction at the corresponding times �j. We shall proveformula (4.12) by induction on J . The case J = 1 was established for all t > 0 inthe course of proving Proposition 4.2.Suppose that for J = K, formula (4.12) is true with the choice above for the pathfunction !�x and the matrix M 6= 0, for all t > 0, all elementary events of the form(1.1), and all choices of non-negative integers j0; : : : ; jn such that j0+ � � �+ jn = K.

14 BRIAN JEFFERIESLet j0; : : : ; jn be non-negative integers such that j0 + � � � + jn = K + 1. Theexpression (4.11) for Zn(�j0+���+jn�1+1; : : : ; �K+1) can be written asexp(T (t� �K+1))EK+1Zn(�j0+���+jn�1+1; : : : ; �K)with the term Zn here de�ned by the expression (4.11) with tk+1 = �K+1: By theinduction hypothesis,Zn(�j0+���+jn�1+1; : : : ; �K)� = �!x(E)(M�)(!x(0));for the path function !x and matrix M de�ned above with the choice t = �K+1.ThenZn(�j0+���+jn�1+1; : : : ; �K+1)� = exp(T (t� �K+1))�!x(E)(EK+1M�)(!x(0)):We are only interested in the case EK+1M 6= 0. We examine the possible choicesof EK+1 2 E . If EK+1 is either "1 or "4, then EK+1M = M . In the case thatEK+1 = "1, then we can only have EK = "1 and �(K) = 0, or EK = "2 and�(K) = 1, otherwise EK+1M would be the zero matrix. In the either case,Zn(�j0+���+jn�1+1; : : : ; �K+1)� = �!x+(t��K+1)(E)(M�)(!x+(t��K+1)(0)):By the induction hypothesis, the path !x 2 �K+1 is equal to �x;�K+1(�1; : : : ; �K ; �; � ),for each x 2 R+ , because EK is equal to "1 or e2. Then !x+(t��K+1) is the path~!x = �x;t(�1; : : : ; �K+1; ~�; � ) with the function ~� : f1; : : : ; K +1g ! f0; 1g de�nedby ~�(j) = �(j) for all j = 1; : : : ; K and ~�(K + 1) = 0: if �(K) = 0, there has beenno change in direction in the path ~!x at �K , and if �(K) = 1, there has been achange in direction in the path ~!x at �K . In both cases, ~!x has derivative �1 inthe open interval (�K ; t). Thus,Zn(�j0+���+jn�1+1; : : : ; �K+1)� = �~!x(E)(M�)(~!x(0)):The negative sign appears in the path function ~!x in the case EK+1 = "1In the case EK+1 = "4, we obtainZn(�j0+���+jn�1+1; : : : ; �K+1)� =�(x� (t� �K+1))�!x�(t��K+1)(E)(M�)(!x�(t��K+1)(0)):By the induction hypothesis, the path !x 2 �K+1 is equal to +x;�K+1(�1; : : : ; �K ; �; � ),for each x 2 R+ , because EK is equal to "3 or e4. Then !x�(t��K+1) is the path~!x = +x;t(�1; : : : ; �K+1; ~�; � ) with the function ~� : f1; : : : ; K +1g ! f0; 1g de�nedby ~�(j) = �(j) for all j = 1; : : : ; K and ~�(K + 1) = 0: if �(K) = 0, there has beenno change in direction in the path ~!x at �K , and if �(K) = 1, there has been achange in direction in the path ~!x at �K . In both cases, ~!x has derivative +1 inthe open interval (�K ; t). If x < t � �K+1, then ~!x does not belong to the spacet of paths taking values in R+ , so �~!x(E) = 0. Thus,Zn(�j0+���+jn�1+1; : : : ; �K+1)� = �~!x(E)(M�)(~!x(0)):

FEYNMAN REPRESENTATION 15The positive sign appears in the path function ~!x in the case EK+1 = "4The other cases EK+1 = "2 and EK+1 = "3 correspond to changes in directionof the path !x; x 2 R+ , at time �K+1 with �K < �K+1 < t: The veri�cation of theinduction hypothesis for J = K + 1 is similar to that above, so the result followsby induction for all J = 1; 2; : : : . �The following statement is a generalisation of equation (4.9) to the case J > 1.Given x � 0, t > 0 and a map � : f1; : : : ; Jg ! f0; 1g, let f �x;t(�1; : : : ; �J ; �; � ) �0g denote the set of all J -tuples 0 < �1 < � � � < �J < t belonging to [0; t] for whichthe path �x;t(�1; : : : ; �J ; �; � ) belongs to t, that is, where it has its values in R+ .The set f �x;t(�1; : : : ; �J ; �; � ) � 0g is possibly empty.Proposition 4.4. Let � 2 L2(R+ ; C 2). The measure Rt;J� is a �nite linear com-bination of L2(R+ ; C 2)-valued measures(4.13) A 7! Zf �x;t(�1;:::;�J ;�; � )�0g � �x;t(�1;:::;�J ;�; � )(A)�(M�)( �x;t(�1; : : : ; �J ; �; 0)) d�1 � � �d�J ; A 2 Rt;for some maps � : f1; : : : ; Jg ! f0; 1g, path functions �x;t(�1; : : : ; �J ; �; � ) andmatrices M 2 E . In this notation, the element of L2(R+ ; C 2) on the right-handside of (4.13) is the function which assigns to each x 2 R+ , the element of C 2obtained by replacing x by x.Proof. The integrand in (4.10) can be expressed as a �nite linear combination ofthe expressions (4.12) by writing the matrix U as a linear combination of matricesfrom the standard basis E . The corresponding sum over the expression (4.10) forall non-negative integers j0+ � � �+ jn = J is a �nite linear combination of integralsof the expression (4.12) over the set 0 < �1 < � � � < �J < t in [0; t]J .Let x 2 R+ , � : f1; : : : ; Jg ! f0; 1g and A 2 Rt. Suppose that at some point0 < �1 < � � � < �J < t, the path �x;t(�1; : : : ; �J ; �; � ) takes on negative values inthe interval [0; t]. Then � �x;t(�1;:::;�J ;�; � )(A) = 0, because A is a subset of t whoseelements assume values only in the set R+ . The region of integration in which theintegrand is nonzero is therefore contained in the set f �x;t(�1; : : : ; �J ; �; � ) � 0g.The expression on the right hand side of (4.13) does not depend on the form ofan elementary event A like (1.1), and we have shown that for all elementary eventsA of the form (1.1), the operator valued set function de�ned by (4.4) is equal toa �nite linear combination of operator valued set functions de�ned by (4.13), soequality holds on the �-algebra Rt by �-additivity. �Let �t be the set of all paths ! 2 t which take the value zero at least once inthe interval [0; t]. Paths belonging to t are continuous, so �t 2 Rt by virtue ofthe equality �t = T1m=1S1j=1T1n=j Snk=1f! 2 t : !(tk=n) < 1=mg:Corollary 4.5. The set �t is an Rt-null set.Proof. The measure Rt is the setwise sum of the measures Rt;J ; J = 0; 1; 2 : : : onthe �-algebra Rt, in the strong operator topology of L �L2(R+ ; C 2)�, so it is enoughto show that �t is an Rt;J -null set, for each J = 0; 1; : : : . For J = 0; 1, we veri�ed

16 BRIAN JEFFERIESthis by writing down the set �t;J explicitly. In general, for almost all x 2 R+ , theset f(�1; : : : ; �J) : 0 < �1 < � � � < �J < t; 0 2 �x;t(�1; : : : ; �J ; �; [0; t])gis contained in the �nite union of hyperplanes in [0; t]J , so it has Lebesgue measurezero.For any subset A 2 Rt of �t, it follows that the integral on the right hand sideof expression (4.13) is zero at almost all points x 2 R+ , so Rt;J(A) = 0, provingthat �t is Rt;J -null. �Let t;J ; J = 0; 1; : : : be the set of all paths ! 2 t with at most J changesin direction in the interval [0; t]. Then t;J � t;J+1 and t = S1J=0t;J : Thefollowing corollary is proved in [J2, Lemma 4.2] without mentioning the formula(4.13) explicitly.Corollary 4.6. For each J = 0; 1; : : : , the measure Rt;J is supported by t;J .If the the matrix U is replaced by the bounded matrix valued function iN�Uk;"N ,then N�R(k;")t N may be represented as the sum of measures like (4.13), except thatthe integrand is multiplied by additional functions of �1; : : : �J and x associatedwith the translates of the components of iN�Uk;"N ; in the situation consideredabove, these were constants. It is clear that this change will not a�ect the conclu-sions of Corollary 4.5 with respect to the new measure N�R(k;")t N . We state thefollowing without writing down a formal proof:Proposition 4.7. The set �t is an R(k;")t -null set for all " > 0 and all nonzerointegers k.Corollary 4.8. The set �t is an M+t -null set.Proof. For all " > 0, the operator valued measureM (")t is the direct sum of measuresunitarily equivalent to R(k;")t , for k = �1;�2; : : : . If A 2 Rt is a subset of �t, thenR(k;")t (A) = 0 for all k = �1;�2; : : : , so thatM (")t (A) = 0. Hence, �t is anM+t -nullset. � 5. The Feynman representationIn this section, we use Corollary 4.8 to complete the proof of the Feynmanrepresentation (0.1) for a class of potentials q which includes a range of Coulombpotentials.Let q : R++ ! R be a locally square integrable function. Let D(�(q)) be theset of all functions � 2 L2(R+ ; C 2) which are absolutely continuous on all boundedsubintervals of R+ , such that (N��)2(0) = 0 and �(q)� := ���Q(q)� 2 L2(R+ ; C 2).It is proved in [J2, Proposition 2.3] that the operator i�(q) : D(�(q))! L2(R+ ; C 2)is the in�nitesimal generator of a C0-contraction semigroup ei�(q)t, t � 0. For eacht � 0, the operator ei�(q)t maps � 2 L2(R+ ; C 2) to the function given by(5.1) x 7! N � exp[�i R t0 q(x+ s) ds](N��)1(x+ t)�(x� t) exp[�i R t0 �(x� t)q(x� s) ds](N��)2(x� t)� ;

FEYNMAN REPRESENTATION 17for almost all x > 0.The function q need not be integrable in a neighbourhood of zero, although it isnecessarily integrable over all bounded intervals in R++ . Nevertheless, the formula(5.1) makes sense for almost all x > 0 and de�nes a function in L2(R+ ; C 2).Any bounded perturbation of �(q) is the generator of a C0-semigroup [K1, The-orem IX.2.1], so for all " > 0 and k 2 Z n f0g, �(q) + Uk;" is the generator of aC0-semigroup. By the notation �k;" :�q, we mean the operator with domain D(�(q))equal to �(q) + Uk;".The direct sumXk;m2K"MU�k;m(�k;" :�q)Uk;m � Xjkj>1="M� jkj�1Xm=�jkjMU�k;mQ(q)Uk;m�over the set K" all integers k;m such that m = �jkj;�jkj+ 1; : : : ; jkj � 1 and 1 �jkj � 1=" is denoted by D" :�Qr(q). Then i(D" :�Qr(q)) is the direct sum of in�nites-imal generators of C0-semigroups, so it is itself the in�nitesimal generators of a C0-semigroup. In the case that q is bounded and measurable, the operator D" :�Qr(q)so de�ned is, in fact, the di�erence between the unbounded operator D" and thebounded operator Qr(q), because Qr(q) =P1jkj=1L�Pjkj�1m=�jkjLU�k;mQ(q)Uk;m�.Lemma 5.1. Let q : R++ ! R be a locally integrable function and t > 0. Thenfor M+t -almost all ! 2 , the function s 7! q(!(s)) is Lebesgue integrable on [0; t].Proof. According to Corollary 4.8, the set �t of all paths ! 2 which hit theorigin at some time in the interval [0; t] is M (")t -null. If ! does not hit zero, thenthere exists �nitely many subintervals I of [0; t], for which there exist numbers0 < bI < aI , such that s 7! q(!(s)) is the restriction of one of the functionss 7! q(aI � s); s � bI or s 7! q(aI + s); s � 0 to I. Because q is assumed to belocally integrable on R++ , each of these functions is integrable over each intervalI, and so q � ! is integrable over [0; t] itself. �The semigroups ei(�k;" :�q)t; t � 0, k 2 Z; k 6= 0 and hence, ei(D" :�Qr(q))t; t � 0,may be represented as a perturbation series expansion in terms of the semigroupgiven by expression (5.1).Theorem 5.2. Let q : R++ ! R be a locally square integrable function and t > 0.Then for every " > 0, the function ! 7! e�i R t0 q(!(s)) ds de�ned for M+t -almost all! 2 is M (")t -integrable, and the equality(5.2) ei(D" :�Qr(q))t = Z e�i R t0 q(!(s)) ds dM (")t (!)is valid.Proof. Let qn = q�fjqj�ng for each n = 1; 2; : : : . Then by dominated conver-gence, for each ! 2 t n �t, the function qn � ! converges to q � ! almost every-where on [0; t] and R t0 qn(!(s)) ds converges to R t0 q(!(s)) ds. An appeal to domi-nated convergence for the operator valued measure M (")t proves that the operators

18 BRIAN JEFFERIESR e�i R t0 qn(!(s)) ds dM (")t (!); n = 1; 2; : : : converge in the strong operator topologyto the right hand side of (5.2).Another application of dominated convergence to the expression (5.1) and theperturbation expansion of e�(q)+Uk;" in terms of the bounded matrix multiplica-tion operator Uk;" [K1, Theorem IX.2.1] shows that the operators ei(�k;"�qn)t,n = 1; 2; : : : converge in the strong operator topology to ei(�k;" :�q)t, hence theoperators ei(D" :�Qr(qn))t; n = 1; 2; : : : converge in the strong operator topology toei(D" :�Qr(q))t. The equality (5.2) is true for bounded measurable functions q byvirtue of [J2, Lemma 6.2] and formula (3.2). Therefore, it is true for all locallysquare integrable functions q : R++ ! R. �The following result is the analogue of [J2, Lemma 2.5] in the present context.Lemma 5.3. Suppose that q : R++ ! R is a function such that the bound (0.2)holds with � < p3=2. Let qn = q�jqj�n for each n = 1; 2; : : : and let k be a nonzerointeger.Then for every � > 0, the function " 7! (� � i(�k;" � qn))�1; 0 � " � 1 iscontinuous on the strong operator topology of L �L2(R+ ; C 2)�, uniformly for alln = 1; 2; : : : .Furthermore, as " ! 0+, the operator ei(�k;"�qn)t converges to ei(�k�qn)t in thestrong operator topology of L �L2(R+ ; C 2)�, uniformly for all n = 1; 2; : : : , and allnumbers t in compact subsets of R+Proof. Let X be the Banach space of all continuous functions f from [0; 1] toL2(R+ ; C 2) with the uniform norm kfk1 = sup0�"�1 kf(")k2. For each t � 0,the operator S(t) : X ! X is de�ned by �S(t)f�(") = ei��;"f(") for f 2 X and0 � " � 1. Then, as in [J2, Lemma 2.5], S is a contraction C0-semigroup ofoperators acting on X.It follows by [K, Theorem IX.2.1] that i(�k;" :�q) is the generator of a contractionC0-semigroup on L2(R+ ; C 2), and the mapping "! ei(�k;" :�q)t; " > 0 is continuouson the open interval (0;1) in the strong operator topology of L �L2(R+ ; C 2)�,uniformly for t in compact subsets of R++ . The question of continuity at " = 0remains.A result originating with F. Rellich [G-R], [Ka, Theorem 5] ensures that �k � qis essentially selfadjoint on C1c (R++ ; C 2) if � < p3=2. Its closure in L2(R+ ; C 2)is denoted by the same symbol. Trotter's convergence theorem [T, Theorem 5.2]ensures that lim"!0+ ei(�k;"�q)t = ei(�k�q)t in L �L2(R+ ; C 2)�, uniformly for t incompact subsets of R+ . For each t � 0, and function q satisfying (0.2), the operatorSq(t) is de�ned for each f 2 X, by�Sq(t)f�(") = ei(�k;"�q)tf("); for all 0 � " � 1:Then Sq is a contraction C0-semigroup acting on X. Now C1c (R++ ; C 2) is a corefor �k � q, hence, C([0; 1]) C1c (R++ ; C 2) is a core for the in�nitesimal generator�k; � � q of Sq, because for any Banach space B, the linear space C([0; 1]) B isdense in the Banach space C([0; 1]; B) equipped with the uniform norm. Anotherappeal to Trotter's convergence result [T, Theorem 5.2] implies that as n ! 1,Sqn(t)! Sq(t) in L(X), uniformly for t in compact subsets of R+ .

FEYNMAN REPRESENTATION 19Let � > 0. For each h 2 L2(R+ ; C 2) and n = 1; 2; : : : , the functionZ 10 e��tSqn(t)(1 h) dtis just the continuous mapping " 7! (� � i(�k;" � qn))�1h, where the functionidentically equal to one on [0; 1] has been denoted by 1. As n!1, these functionsconverge uniformly on [0; 1] to the function " 7! (��i(�k;"�q))�1h, proving the �rstconclusion. The second follows from the existence of the limit limn!1 Sqn(t)(1h) = Sq(t)(1 h) in X. �In the following theorem, we �nally prove the validity of the Feynman represen-tation (0.1).Theorem 5.4. Suppose that q : R++ ! R is a function such that the bound (0.2)holds with � < p3=2. Let t > 0. Then the function ! 7! e�i R t0 q(!(s)) ds de�ned forM+t -almost all ! 2 is M+t -Mt-integrable.The operator D � Qr(q) is essentially selfadjoint on C1c (R3 n f0g; C 4). Let Hdenote the closure of �D +Qr(q) in L2(R3 ; C 4). Then the equality(5.3) e�iHt = Z e�i R t0 q(!(s))ds dMt(!)is valid.Proof. Let qn = q�jqj�n; n = 1; 2; : : : . By [J2, Lemma 6.2], for each n = 1; 2; : : : ;the function ! 7! e�i R t0 qn(!(s)) ds is M+t -Mt-integrable and(5.4) ei(D�Qr(qn))t = Z e�i R t0 qn(!(s)) ds dMt(!):We shall show that e�i R t0 qn�Xs ds converges to e�i R t0 q�Xs ds in L1(M+t ;Mt), asn!1.By Theorem 5.2, e�i R t0 q�Xs ds is M+t -integrable, and dominated convergenceensures that the function e�i R t0 qn�Xs ds converges to e�i R t0 q�Xs ds in L1(M+t ), asn ! 1. According to [J2, Proposition 5.7], it remains to prove that for eachE 2 St, the operators(5.5) ZE e�i R t0 qn�Xs ds dM (")t ; n = 1; 2; : : :converge in the weak operator topology, uniformly for " > 0.An appeal to Theorem 5.2 shows that for each elementary event E of the form(1.1), the operators (5.5) may be represented in the form (1.2), with S equal to thesemigroup e�i(D"�Qr(qn))� and with Q equal to the spectral measure Qr. An appealto Lemma 5.3 and the Banach-Steinhaus theorem shows that the convergence of theoperators (5.5) is actually in the strong operator topology, uniformly for " > 0. Herewe need to take the direct sum of operators unitarily equivalent to the operators

20 BRIAN JEFFERIEStreated in Lemma 5.3. Hence, e�i R t0 q�Xs ds is M+t -Mt-integrable, and the functione�i R t0 qn�Xs ds converges to e�i R t0 q�Xs ds in L1(M+t ;Mt), as n!1As in the proof of Lemma 5.3, D � Qr(q) is essentially selfadjoint on C1c (R3 nf0g; C 4) and the left hand side of (5.4) converges in the strong operator topologyto e�iHt as n!1. By [J2, Proposition 5.7], or more simply, from the de�nition ofconvergence in the space L1(M+t ;Mt), the right hand side of (5.4) converges in thestrong operator topology to R e�i R t0 q(!(s)) ds dMt(!) as n ! 1, thereby provingthe equality (5.3). �Corollary 5.5. The selfadjoint operator H above is the limit in the sense of strongresolvent convergence of D" :�Qr(q) as "! 0+.Proof. We proved in Lemma 5.3 that for each nonzero integer k, the operator �k;" :�qconverges to �k :�q in the sense of strong resolvent convergence. The result followsafter taking the direct sum of operators unitarily equivalent to these. �For the case p3=2 � � < 1 in the bound (0.2), we have the following result.Essential to the proof is a convergence result of R. W�ust [W�u] and T. Kato [K2,Theorem II] for the Dirac operator.Theorem 5.6. Suppose that q : R++ ! R is a function such that the bound (0.2)holds with � < 1, and suppose that q(1) � 0 near the origin. Let t > 0. Then thefunction ! 7! e�i� R t0 q(!(s)) ds de�ned forM+t -almost all ! 2 isM+t -Mt-integrablefor all � 2 C such that =(�) < 0. There exists " > 0 such that the map� 7! Z e�i� R t0 q(!(s))ds dMt(!); � 2 C ;=(�) < 0; j�j � 1 + ";is the restriction of a function continuous on the set f� 2 C : =(�) � 0; j�j � 1+"g,for the strong operator topology of L�L2(R3 ; C 4)�.Moreover, there exists a selfadjoint operator H such that the equality(5.6) e�iHt = lim=(�)<0;�!1 Z e�i� R t0 q(!(s))ds dMt(!)is valid. The selfadjoint operator H is characterised by the equalitye�iHt = limn!1 ei(D�Qr(qn))t;for all t 2 R, where qn = q�jqj�n for each n = 1; 2; : : : .If q(1) � 0 near the origin, then the analogous statement is valid for the region=(�) � 0.Proof. With the notation of the proof of Lemma 5.3, for each n = 1; 2; : : : ,thein�nitesimal generator of S�qn is a bounded perturbation of the generator of S, sothere exists �0 > 0 such that for all � > �0, the map � 7! R�qn(�) is the restrictionto the real axis of a function analytic in a neighbourhood of zero in C .Now suppose that there exists r0 > 0 such that q(1)(r) � 0 for all 0 < r < r0.Then for r � r0, we have jq(1)(r)j � 1=r0. It follows from the Trotter-Kato productformula that, for a = kq(2)k1, the bound(5.7) kei(�k;"��qn)tkL(L2(R+;C2 )) � e(j=(�)j=r0+a)t;

FEYNMAN REPRESENTATION 21holds for all � 2 C with =(�) � 0, all " > 0, all t � 0 and all n = 1; 2; : : : . Clearly,the bound kS�qn(t)kL(X) � e(j=(�)j=r0+a)t also holds for =(�) � 0, n = 1; 2; : : : andt � 0. Let �0 > a. Then for all � > �0 and n = 1; 2; : : : , the resolvent operatorR�qn(�) = Z 10 e��tS�qn(t) dtof S�qn has the property that � 7! R�qn(�) is analytic in the strip =(�) < 0; j=(�)j <(�� a)r0 and continuous for =(�) � 0; j=(�)j < (�� a)r0. Moreover, the bound(5.8) kR�qn(�)kL(X) � 1�� j=(�)j=r0 � aobtains for all =(k) � 0; j=(�)j � (�� a)r0 and n = 1; 2; : : : .According to [K2, Theorem I (iii)], in the case that =(�) � 0 and j�j < 1=2, thesubspace C1c (R++ ; C 2) is a core for �k��q, the operator i(�k��q) is the generatorof a C0-semigroup and the bound (5.7) holds with " = 0. Actually, the proof of[K2, Theorem I] refers to the direct sum H(�) of operators unitarily equivalent to�k � �q for every k = �1;�2; : : : , but we can take the restriction of H(�) to thereducing subspaces, because we are working with the radially symmetric potentialV : x 7! q(jxj); x 2 R3 .An argument analogous to that of Theorem 5.4 for the case of real � with j�j <p3=2, shows that for each � 2 C with =(�) � 0 and j�j < 1=2, the operator S�qn(t)converges as n ! 1 to the operator S�q(t) in L(X), uniformly for t in compactsubsets of R+ . An appeal to Vitali's convergence theorem [H-P, Theorem 3.14.1]ensures that S�qn(t); n = 1; 2; : : : converges in the strong operator topology of L(X),uniformly as � ranges over compact subsets of the region =(�) < 0. The limitingsemigroup is denoted by S�q for all complex numbers � satisfying =(�) < 0. As inthe proof of Theorem 5.4, the function ! 7! e�i� R t0 q(!(s)) ds is M+t -Mt-integrablefor all � 2 C such that =(�) < 0. Furthermore,(5.9) �S�q(t)(1 h)�(") = Z e�i� R t0 q(!(s))ds d(M (")t h)(!); 0 � " � 1;for all =(�) < 0 and h 2 L2(R+ ; C 2).By dominated convergence, for each � > a and � 2 C satisfying =(�) � 0and j�j < min(1=2; (� � a)r0), the resolvent operator R�qn(�) of S�qn convergesin the strong operator topology of L(X), as n ! 1, to the resolvent R�q(�) =R10 e��tS�q(t) dt of S�q.Set �0 = (�0 � a)r0. The bound (5.8) holds in the region ��0 � =(�) � 0,so another appeal to Vitali's convergence theorem shows that for each � > �0,the operator R�qn(�); n = 1; 2; : : : converges in L(X), uniformly as � ranges overcompact subsets of the set f� 2 C : ��0 < =(k) < 0g:Now suppose that �0 is chosen so that �0 = (�0 � a)r0 > 1. Then for � >�0, R�qn(�) converges in the strong operator topology to R�q(�) in the region��0 < =(�) < 0. Here, � 7! R�q(�), de�ned for ��0 < =(�) < 0, is the analyticcontinuation of � 7! R�q(�), de�ned as above, for =(�) � 0 and j�j � 1=2: Thus,

22 BRIAN JEFFERIESfor all � 2 C such that ��0 < =(�) < 0, R�q(�); � > �0 is the resolvent family ofthe C0-semigroup S�q.By [K2, Theorem I] and the assumption that � < 1, there exists " > 0 such thatfor each h 2 L2(R+ ; C 2), there exists an analytic continuation � 7! �R�q(�)(1 h)�(0) to all � 2 C such that j�j < 1 + ". Furthermore, by [K2, Theorem II (ii)],the analytic continuation has the property that for every h 2 L2(R+ ; C 2),(5.10) limn!1 �R�qn(�)(1 h)�(0) = �R�q(�)(1 h)�(0)in L2(R+ ; C 2), and there exists a selfadjoint operator H, such that the operatorh 7! �Rq(�)(1 h)�(0); h 2 L2(R+ ; C 2) is the resolvent (� + iH)�1 of �iH [K2,Theorem I (iv)]. Equation (5.10) and an appeal to strong resolvent convergence[K1, Theorem IX.2.16] shows that the operator H is characterised by the statedlimit.Now for each h 2 L2(R+ ; C 2), continuity ensures thatlim�!1 �R�q(�)(1 h)�(0) = �Rq(�)(1 h)�(0):By strong resolvent convergence,lim=(�)<0;�!1 �S�q(t)(1 h)�(0) = e�iHth:Combined with formula (5.9), this establishes the equality (5.6). �Remarks. (i) In the case of the inverse square potential for the Schr�odinger equa-tion, in the region where the associated Schr�odinger operator is not essentially self-adjoint but the Fredholm extension exists, Faris [Fa] establishes a representationsimilar to (5.6) with a double limit procedure.(ii) For the Coulomb potential q(r) = �a=r with a > 1, the limit on the righthand side of (5.6) exists in the strong operator topology, but it does not de�nea selfadjoint operator. In the context of the Schr�odinger equation, E. Nelson [N]interpreted this situation in terms of a possibility of collision with the centre ofattraction. For the Dirac equation, V. Popov [P] obtains a selfadjoint extension byimposing conditions at the nuclear boundary.Although the Feynman representation (5.3) reveals nothing about the dynamicsof the Dirac particle that cannot be deduced by operator-theoretic techniques, thestructure developed in the proof of the representation (5.3) may prove to be usefulin situations where the quantum dynamics is not readily constructed by traditionalmeans. References[A] M. Arai, On essential selfadjointness, distinguished selfadjoint extension and essentialspectrum of Dirac operators with matrix valued potentials, Publ. Res. Inst. Math. Sci.,Kyoto U. 19 (1983), 33{57.[Fa] W.G. Faris, The product formula for semigroups de�ned by Friedrichs extensions, Paci�c.J. Math. 22 (1967), 47{70.

FEYNMAN REPRESENTATION 23[G-R] K.E. Gustafson and P.A. Rejto, Some essentially self-adjoint Dirac operators with spheri-cally symmetric potentials, Israel J. Math. 14 (1973), 63{75.[H-P] E. Hille and R.S. Phillips, Functional Analysis and Semigroups, Amer. Math. Soc. Colloq.Publ. XXXI, Amer. Math. Soc., New York, 1957.[I-T] T. Ichinose and H. Tamura, Zitterbewegung of a Dirac particle in two-dimensional space-time, J. Math. Phys. 29 (1988), 103{109.[J1] B. Je�eries, Integration with respect to closable set functions, J. Funct. Anal. 66 (1986),381{405.[J2] , A process associated with the radially symmetric Dirac equation, Publ. Res. Inst.Math. Sci., Kyoto U. 30 (1994), 297{325.[Ka] H. Kalf, U.-W. Schmincke, J. Walter and R. W�ust, On the spectral theory of Schr�odingerand Dirac operators with strongly singular potentials, Proceedings of the Symposium onSpectral Theory and Di�erential Equations, University of Dundee, 1974, Lect. Notes inMath. 448, Springer-Verlag, Berlin, Heidelberg, New York, 1975, pp. 182{226.[K1] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.[K2] , Holomorphic families of Dirac operators, Math. Z. 183 (1983), 399-406.[Klu] I. Kluv�anek, Operator valued measures and perturbations of semi-groups, Arch. Rat. Mech.& Anal. 81 (1983), 161{180.[K-K] I. Kluv�anek and G. Knowles, Vector Measures and Control Systems, North Holland, Am-sterdam, 1976.[N] E. Nelson, Feynman integrals and the Schr�odinger equation, J. Math. Phys. 5 (1964),332{343.[P] V.S. Popov, \Collapse to the center" at Z > 137 and critical nuclear charge, Soviet J.Nuclear Phys. 12 (1971), 235{243.[T] H.F. Trotter, Approximation of semi-groups of operators, Paci�c J. Math. 8 (1958), 887{919.[W�u] R. W�ust, Distinguished self-adjoint extensions of Dirac operators constructed by means ofcut-o� potentials, Math. Z. 141 (1975), 93{98.School of Mathematics, UNSW, NSW 2052, AUSTRALIAE-mail address: B.Je�eries@unsw.edu.au