i(a (a' (a x ia').886 physics: j. schwinger proc. n. a. s. is infinitesimal tothefirst...

UNITARY TRANSFORMATIONS AND THE ACTION PRINCIPLE*

BY JULIAN SCHWINGERHARVARD UNIVERSITY

Communicated April 25, 1960

Some earlier notes' have been dedicated to the analysis of the basic empiricalfact of microscopic measurement, the existence of incompatible physical proper-ties. From these considerations has emerged the general mathematical structureof quantum mechanics, as an operator algebra, and as a geometry. A further im-plication is the classification of quantum degrees of freedom by irreducible pairs ofcomplementary physical properties, which possess optimum incompatibility andgenerate a complete operator basis. The unitary operators that dominate thelatter discussion continue to be the objects of interest here. We shall consider, inparticular, infinitesimal unitary transformations and their composition to formfinite transformations. In the special example of a particular type of quantumdegree of freedom, for which the complementary pair of properties have continuousspectra, this leads to a powerful tool for the construction of transformation func-tions-the action principle.The automorphisms of the unitary geometry of states are produced by the unitary

transformations

=4 U, .*=U-1, X=U-'XU

applied to every vector and operator, where the unitary operator U obeys

U-= U-1.

All algebraic relations and adjoint connections among vectors and operators arepreserved by this transformation. Two successive unitary transformations form aunitary transformation, and the inverse of a unitary transformation is unitary-unitary transformations form a group. The application of a unitary transforma-tion to the orthonormal basis vectors of the a-description, which are characterizedby the eigenvector equation

( a' (A -a') = 0

yields orthonormal vectors

(a't = (a'IUthat obey the eigenvector equation

(a' I(A -a') = 0.

Hence the (a' are the states of a new description associated with quantities Athat possess the same eigenvalue spectrum as the properties A. Since all rela-tions among operators and vectors are preserved by the transformation, we have

(a't a , ') =(a X Ia').883

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884 PHYSICS: J. SCHWINGER PROC. N. A. S.

The equivalent forms

(a'lX la") = (a'|UXU-1Ia11)(and'I' = (a'IU1, brtd') = 4U-Ia')

exhibit the a-representatives of operators and vectors as the a-representatives ofassociated operators and vectors.As we have already remarked, the basis vectors of any two descriptions, with

each set placed in a definite order, are connected by a unitary operator. The trans-formation function relating the a- and b-representations can thereby be exhibitedas a matrix referring entirely to the a- or the b-representations,

(a lb') = (ak Uba la') = (bk |Uba Ib')and all quantities of the b-representation can be expressed as a-representatives ofassociated operators and vectors,

(bkfX Ibl) = (ak IUabXUba la')(bk It = (ak| Uab*L, 1 lbk) = 4.Uba lak).

If the two sets of properties A and B possess the same spectrum of values, the opera-tors A and B are also connected by a unitary transformation. With the orderingof basis vectors established by corresponding eigenvalues we have

B = E bk bk) (bk| = Ei akUba lak) Kak |Uab = UbaAUab.

The definition of a unitary operator, when expressed as

(U - l)t(U - 1) + (U - 1) + (U - 1)t = 0,

shows that a unitary operator differing infinitesimally from unity has the generalform

U = 1 + iG, Ut = U- = 1 -iG,

where G is an infinitesimal Hermitian operator. The coordinate vector transforma-tion described by this operator is indicated by

beat|= (at| - (a' = (a' iG

a Ia') = l')- a') = -iG la').Now, a change of coordinate system, in its effect upon the representatives of opera-tors and vectors, is equivalent to a corresponding change of the operators and vec-tors relative to the original coordinate system. Hence

S(a' lXa") = (a'X Id") - (a' lXa")- (a' I6X Ia")

and

b(a' 'I' = (a'lt5, 65 D la') = bIa')

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VOL. 46, 1960 PHYSICS: J. SCHWINGER 895

where5* = (U - 1)t = iGI

&D= 4'(U-1 -1) =-iG,and

aX= UXU-1-X=I [XG].

The rectangular bracket represents the commutator

[A, B] = AB-BA.

Since all algebraic relations are preserved, the operator and vector variations aregoverned by rules of the type

5(XY) = 6XY + X6Y

(X*) = ax* + Xs*.

One must distinguish betweenX + AX and

X = U-1XU = X-AX;

the latter is the operator that exhibits the same properties relative to the a-descrip-tion that X possesses in the a-description. Thus the basis vectors (a' are theeigenvectors of A -AA with the eigenvalues a'.

In discussing successive unitary transformations, it must be recognized that atransformation which is specified by an array of numerical coefficients is symbolizedby a unitary operator that depends upon the coordinate system to which -it isapplied. Thus, let U1 and U2 be the operators describing two different transforma-tions on the same coordinate system. When the first transformation has beenapplied, the operator that symbolizes the second transformation, in its effect uponthe coordinate system that has resulted from the initial transformation is

LI2 = Ui1_U2U1.

Hence the operator that produces the complete transformation is

U1U2 = U2U1..

The same form with the operators of successive transformations multiplied fromright to left, applies to any number of transformations. In particular, if one followstwo transformations, applied in one order, by the inverse of the successive trans-formations in the opposite order, the unitary operator for the resulting transforma-tion is

U[121 = '(U1U2)-1U2U1 = U[21 -1.

When both transformations are infinitesimal,

U1,2= 1+iG1,2,

the combined transformation described by

U[121 = 1 + iG[121

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is infinitesimal to the first order in each of the individual transformations,

G[12= [G1, G2]

=-G[211,The infinitesimal change that the latter transformation produces in an operator is

6[12]X = 6261X -6162X,

which, expressed in terms of commutators, yields the operator identity

[[X, G1],G2]- [[XI, G2],G1] = [XI,[G1, G2]].The continual repetition of an infinitesimal unitary transformation generates

a finite unitary transformation. On writing the infinitesimal Hermitian operatorG, the generator of the unitary transformation, as arG1, we find that the applicationof the infinitesimal transformation a number of times expressed by r/5r yields, inthe limit Tr -O.0

U(r) = Lim (1 + i rGi)T/5T = e

These operators form a one-parameter continuous group of unitary transformations,

U(Ti) U(r2) = U(ri + r2), U(T)'1 = U(-r), U(O) = 1.

A number of finite Hermitian operators G . .. G,,, generates an n-parameter con-tinuous group of unitary transformations if they form a linear basis for an operatorring that is closed under the unitary transformations of the group. This requiresthat all commutators [Ga, Gb] be linear combination of the generating operators.To discuss in more detail the conditions for the generation of a continuous group

of unitary transformations from its infinitesimal elements, we consider the neigh-boring operators U(T), U(r + dT) and the two related infinitesimal unitary operators

nU(Tr) 1U(r + dT) = 1 + iibo Ta(T)Ga

a

and

U(T + dr)U(T)1 = 1 + iE 1ra(r)Ga.a

Heren

'Ta(7)= E rak(T)dTA

k= 1

5ra(r) = E lak(T)dlkk

are two sets of inexact differentials that specify, in alternative ways, the additionalinfinitesimal transformation needed to convert U(r) into U(r + diT),

U(r + dT) = U(r) [1 + iE 5r7aGa]

= [1 + iE 61TaGa]U(T).

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In the necessary equivalence of these forms we have the statement that unitarytransformations of the group applied to the operators Ga must yield linear com-binations of these operators. Hence there exist real n-dimensional matrices cU(T)and 9ii(r), such that

U(T)GbU(T)1 = YGqa'ab(T)a

U(T) GaU(r) = ZcLtab(T)Gb,b

and which also represent the underlying abstract group in the sense that

U(Tr)U(72) = U(T)

implies

c1L(Tr1)cU('2) = CIL(T)

and

4l( ri)' (Tm) = 9( )

The connection between the two types of differentials is expressed by

lac(T) = Z'lLat,(T)rbk(T)b

rbkr(T) = ZUhab(T)lak(T),a

andA= (quT)-1

When U(r) is the infinitesimal transformation

U(Or) = 1 + ij3raGa,we have, correspondingly,

U.(5T) = 1 + iE 6raga'L(6T) 1 + iE67agax

whereTgaa=_ga

and thus

[Gb, Gc] = EGagaoca

with

gabc = (9b)ac = -gzacbFurthermore, the n-dimensional imaginary matrices g9 and 9a must possess the samegroup multiplication properties as the operators Ga, which is to say that

[9bgelc] = 2gagabc,a

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or

Z [geafgfbc + gebfgca + 9ecf9fab] 0.

f

The equations that define the inexact differentials b'Ta and 3'Ta provide differentialequations for the U(T) as functions of the group parameters Tk,

.- U(T) = U(T)Z Garak(T)aTk a

= EGalak(T) U(T).a

In order that a unique transformation be associated with every set of group param-eters, the operator differential equations must obey the integrability conditions

[Tk ,U(r)

which imply the two sets of numerical equations,

ram - rak + i £ gabel bkrcm 0aTk bTm bc

and

lam lak - iEtZabclbklcm = 0.a)Tk aTm be

These equations, in turn, must satisfy integrability conditions that, for example,express the vanishing of

a~k taTI ram-T raz)

when the two additional terms obtained by cyclic permutation of klm are added.It will be verified that the resulting restrictions are just the quadratic equationsobeyed by the elements gabc. Thus, the chain of integrability conditions terminateswith the algebraic equations that express the isomorphism of the commutation rela-tions obeyed by the n-dimensional matrices ga with those of the operators Ga.Incidentally, if the integrability conditions for U(r) are expressed with the aid ofboth left and right differential forms, we infer that

lam(T) = v 1tab(T) - rbk(T).bTk b 53Tm

The multiplication law of the group is described by the relations among theparameters in the unitary operator product

U(T) = U(1) U(r2)

7 = T(Tj, T2).

An infinitesimal change of the set Ti induces a corresponding change in T, and

(1 + ieZ'Ta(T)Ga)U(T) = (1 + iEZ Ta(Ti)Ga)U(Tj)U(72)

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whenceEYTa(T) = a Ta(TO)

The resulting system of differential equations (Maurer-Cartan)

Elakl(T)dTk = Elak(Tj)dTik,k k

in combination with the initial condition (which employs the convention U(O) = 1)

0: T = T22

serves to determine the composition properties of the group parameters. The samefunction is performed by the second set of differentials

EST7a(T) = 5T'a(T2)

and the initial condition

T2 = 0: T = Ti.

The correspondence between operators Ga and matrices ga persists under a changeof operator basis, to within the freedom of matrix transformations that preservealgebraic relations. Thus, the nonsingular transformation

Gb ZE GaXaaba

induces

9b= X'(Z 9akab)X,a

which follows from

and

Ta = ZaEab'6T7b.b

The choice of parameters is also arbitrary within the class of nonsingular transforma-tions r -a r'. Such a change does not affect the inexact differentials,

56a = bra'

and thus

67~klam(T') = E lak(T)

for example. Through these possibilities of basis and parameter transformation,one could achieve the identity of the inexact and exact differentials in the neigh-borhood of the unit operator. This would be expressed by adding the initial condi-tions

rab(0) = lab (0) = at,

to the corresponding differential equations.

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890 PHYSICS: J. SCHWINGEIR PROC. N. A. S.

A special set of parameters ta, called canonical, maintains this identity alongany straight line drawn from the origin of the t-parameter space. That is to say,

a Ita = arta = dt,when

dta = dXta.

The significance of these parameters emerges from

U((X + dX)t) = (1 + idXEtaGa)U(Xt)= UT(Xt)(1 + idXEtaGa)

for, with continuously variable X, we have a one-parameter subgroup that includesany chosen unitary transformation of the group, which is thereby given the ex-ponential form

U(t) = exp [iE taGa].

The canonical parameters are consistent with the general requirements of thetheory and permit an explicit evaluation of rab(t) and lab(t). Note that the identity

of art and a It for dt = dAt is expressed by

E 'ltac(t)tc = taC

where

qt(t) = exp [iEtbgb]-b

The necessary and sufficient condition for this to hold is

o = E(Ztbgb)actc = Zgabctbtcc b bc

and that result is assured by the antisymmetry of gab, in the last pair of indices.The basic property of the canonical parameters is conveyed by

Z rab(t)tb = Z lab(t)tb = ta-b b

These statements can be usefully combined with the differential equation

lab(t) = Z Clac(t) at rce(t)

by multiplication of the latter with te followed by a summation over this index.The result,

( t a+ 1) lab(t) = cqab(t)

can also be presented as the matrix equation

[Xl(Xt)] = 'a(Xt) = eigt

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in an evident notation, which gives the explicit construction

1(t) = fj dAei>t = (eiut - 1)/igt.

Similarly,

r(t) = e-tl(t) - (1 -e-'g')/tgt

and the last property is characteristic of the canonical parameters, for which

U(0)- = U(-t)

and

t = t(tl, Q2 = -t(-t2,-ti) .

A simple illustration of these considerations is provided by a group of threeparameters that is defined by the commutation relations

[G1, G2] = iG3, [G1,g3] = [G2, G3] = 0.

The only nonvanishing g elements are

9312 = -9321 =

and the quadratic equations for gab, are satisfied, particularly since

9a9b = 0

while g3 is the null matrix. It follows from these algebraic properties that

cL(t) = 1 + i(gltl + g2t2)and therefore

1(t) = 1 + 1/2i(gltl + 92t2),

or equivalently

5 t1 = dt1, a5t2 = dt2

51t3 = dt3 - 1/2(tidt2 - t2dt1).

The resulting group parameter composition law, in the notation t = t(t', t"), isgiven by

tl = tl' + tl, t2 = t2' + t2

t3 = W~+ 6 , - 1/2(t1't2" - t2 l),which expresses the unitary operator multiplication property

eia eiGt' = exp [iG(t' + t') - 1/2iG3(t1't2 - t2'tl')].The additional G3 term of the latter exponential can also be written as a multiplica-tive factor, since this operator is commutative with the others.

It is precisely this example that is supplied by the unitary operators associated

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with the quantum degree of freedom labeled v = A. The transition to the limitof infinite v is accomplished by writing

U = ei Vq V = e'fP e = (27r/ )l/2

and, with the notation

q' = nE, -p' =me,

the operator basis elements

e(ri/v)mnUmi,;rn = e-(yi/v)mnVnumbecome

e/2ipq' eipt e-ip'q = 1-2ip/ -ip'q eipq' (pq- p'q

The last version, the validity of which will be seen shortly, refers only to the limitv -A , where the numbers q' and p' become continuously variable. The identityof the first two unitary operators asserts that

e-ipq e- ip'q eipq' = e- ip'(q-q')and

eipq ei q' e - e

or equivalently for v = ,

e - qeP= qq'

eip q pe ip'q= p _pa

and the limit of infinitesimal q' and p' in these unitary transformations implies that

[q, p] = - [p, q] = i.

Thus, q, p, 1 or p, - q, 1 possess the properties ascribed to G1,2,3, in the preceding ex-ample. If the general unitary operator of this group2 is written

U(q'p' ') = exp [i((p' + pq' - p'q)the group multiplication law reads

U(q'p'p')U(q"p'tp') = U(q' + q", p' + p", p' + s"` + ,1/2(p'q" - pqN))

and the previously mentioned equivalence of unitary operators appears as the spe-cialization

U(q'p'O) = U(q'OO) U(Op'O) U(OO, 1/2p'q')= U(Op'O)U(q'OO)U(OO, -1/2p'q')

The elementary unitary transformations that compose the group, which we termthe special canonical group, are the phase factors exp(isc/), which leave operatorsunaltered, the transformation

U(q') = ep

which induces

= q - q',P =P

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and

U(p') =e-iP'

q2q, P - P'The infinitesimal versions of the last two transformations are

U(5q) = 1 + ipaq, U(6p) = 1 - ibpq,

so that p and - q can be identified as the Hermitian operators that generate unitchanges in q and p, respectively. These infinitesimal generators, G. and G., canbe regarded as members of the class of generators

Gx = Xpbq - (1 - X)Opq.that produce changes in q and p of amounts X6q and (1 -)6p, respectively. Thedifference of two such generators is

Gx-Gx, = (X - X')(pbq + apq) = (X -X')5[pq]where a refers to the infinitesimal transformation that alters q and p by 6q and 6p,respectively. In particular, for X = 1/2we have

Ggr p= 1/2(paq - bpq)which symmetrically generates changes in q and p of l/26q and 1/26p. Note that

Gq-Gq, p Gq.p G-p = 6[1/2pqIand

Gq-G = 6[pq].It can be remarked here that the discussion of one degree of freedom of type v = o

is extended to n such continuous degrees of freedom by the systematic notationalinterpretation:

n

pq = Pkqk.k=1

We are now going to examine the construction of finite unitary transformationsfrom infinitesimal ones for a physical system of n continuous degrees of freedom.Thus, all operators are functions of the n pairs of complementary variables qk, Pk,which we denote collectively by x. Let us consider a continuous set of unitaryoperators labeled by a single parameter, U(r). The change from T to r + dr is theinfinitesimal transformation

1 + idrG(x, r),

which includes a possible explicit T dependence of the generator, and

U(r + dr) = [1 + idrG(x, r)]U(T)= U('r)[1 + idTG(X(T), r)]

wherex (r) = U(T)'XU(T)

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are the fundamental quantum variables of the system for the description producedby the transformation U(T). The accompanying state transformations are indi-cated by

(a'lU(T) = (a'TU(T) 1lb') = b T).

A useful representation of the unitary transformation is giv-en by the transformationfunction

(a'T, Ib T2) = (a' U(T1) U(T2) 1 bV)which includes

(a'T Ib O) = (a' E (T) Ib),the matrix of U(T) in the arbitrary ab representation. The relation between in-finitesimally neighboring values of T is indicated by

(a T + dT Ib T) = (a'T [1 + idTG(x(T), T) ]bT)

= (a' I [1 + idTG(x, T) ]I b').The general discussion of transformation functions indicates that the most com-

pact characterization is a differential one. Accordingly, we replace this explicitstatement of the transformation function (a'T + dT b'T) by a differential descrip-tion in which the guiding principle will be the maintenance of generality by avoidingconsiderations that refer to specific choices of the states a' and b'. Wenote first that the transformation function depends upon the parameters T, T +dT and upon the form of the generator G(x, T). Infinitesimal changes in theseaspects [8"] induce the alteration

6"(T + dT IT) = i( I6 [dTG(x, T) I)= i(r + dT j6"[dTG(x(T), T)] I T),

where the omission of the labels a', b' emphasizes the absence of explicit referenceto these states. Yet some variation of the states must be introduced if a sufficientlycomplete characterization of the transformation function is to be obtained. Forthis purpose we use the infinitesimal transformations of the special canonical group,performed independently on the states associated with paramete -s T and T + dT [8].Thus

8'(T + dT| = i(T + dT IG,(T + dT)6 T) = -iG,(T) T)

in which the infinitesimal generators are constructed from the operators appropriateto the description employed for the corresponding vectors, namely x(T + dT) andx(T). It is convenient to use the symmetrical generator Gq, p which produceschanges of the variables x by 1126x. Then

GJ(T) = 1/2P(T)6q(T) - bp(T)q(T)which, with the similar expression for GJ(T + dT), gives

V'(T + dTIT) = i(T + dT [G,(T + dT) - G,(T)] T),

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where

GJ(T + dr) - G8(r) = '/2[P(T + dT)6q(T + dT) + bp(T)q(T) -

p(T)6q(T) - 6p(T + dT)q(T + dT)],

and the 5X(T), 6X(T + dT) are independent arbitrary infinitesimal numbers uponwhich we impose the requirement of continuity in T.The infinitesimal unitary transformation that relates x(r) and X(T + dT) is ob-

tained from

X(7 + dT) = U(T + dT) 1XU(T + dT)= [1 - idTG(x(r), T)] x(r)[1 + idTG(x(T), T)]

asx(T + dT) = X(T) - (1/i) [x(T), dTG(x(T), T)].

Accordingly, one can write

GS(r + d) - GJ(T) = 1/2[p(T)6q(r + dr) + bp(T)q(r + dT) -

p(r + dri)6q(T) - bp(i + dT)q(T) ] - (1/i) [p(T)6q(T) - 6p(T)q((T),dTG(x(T),T))] - 6'['/2(P(T)q(T + dT) - p(r + dT)q(T)) ] +

(1/i) [dTG(x(T), T), Gq(T) + GP(T)]or

G3(T + dT) - G.(T) 6' [1/2(P(r)q(i + dT) - P(T + dT)q(T)) + dTG(X(T), T) ]

in which 6' is used here to describe the change of q, p by 6q and 5p, occurring inde-pendently but continuously at T and T + dT. The two species of variation cannow be united: 6 = 6' + 6, and

6(T + dT |IT) = i(T + dTI16[W] I-)where

W(T + dT, T) = 1/2(P(T)q(T + dT) - P(T + dT)q(T)) + dTG(X(T), T)

= 1/2(pdq - dpq) + dTG.Our result is a specialization of the general differential characterization of trans-formation functions whereby, for a class of alterations, the infinitesimal operator6W is derived as the variation of a single operator W. This is a quantum actionprinciple3 and W is the action operator associated with the transformation.We can now proceed directly to the action principle that describes a finite uni-

tary transformation,6(Tr IT2) = i(T1 I6[WV12] IT2)

for multiplicative composition of the individual infinitesimal transformation func-tions is expressed by addition of the corresponding action operators

T1

W12 = E W(T + diT, T)72

= 1['/2(p(T)dq(T) - dp(T)q(i)) + dTG(x(r), T)].

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As written, this action operator depends upon all operators X(T) in the T intervalbetween Ti and T2. But the transformations of the special canonical group, appliedto (Ti T2), give

a' ( T T2) = i(Tl [G,(ri) - G8(r2)] T2)which is to say that 6'W12 does not contain operators referring to values of T in theopen interval between Ti and T2, or that W12 is stationary with respect to the specialvariations of x(T) in that interval. Indeed, this principle of stationary action,the condition that a finite unitary transformation emerge from the infinitesimalones, asserts of q(T), p(T) that

dq YG dp _ GdT Up' dt aq'

which are immediate implications of the various infinitesimal generators.The use of a single parameter in this discussion is not restrictive. We have only

to write

G(x, r) - Z Gk(X, )dT)k=1d'

with each dTk/dT given as an arbitrary function of T, and then regard the transforma-tion as one with p parameters, conducted along a particular path in the parameterspace that is specified by the p functions of a path parameter, T&(T). Now

W12= fr2 [1/2(pdq - dpq) + E Gk(X(T), T)dTk]

is the action operator for a transformation referring to a prescribed path and gener-ally depends upon that path. If we consider an infinitesimal path variation withfixed end points we find that

bpathW12 f=21 E 1/2(bTkdTl - 5TldTk)Rkl

where

RkI GI a- k [Gk, GI ]

=-Rlk.

The vanishing of each of these operators is demanded if the transformation is to beindependent of path. When the operators Gk(x, T) can be expressed as a linear com-bination of an equal number of operators that are not explicit functions of the param-eters, Ga(x), the requirement of path independence yields the previously con-sidered conditions for the formation of a group.We now have the foundations for a general theory of quantum dynamics and

canonical transformations, at least for systems with continuous degrees of freedom.The question is thus posed whether other types of quantum variables can also beemployed in a quantum action principle.

* Publication assisted by the Office of Scientific Research, United States Air Force, under con-

tract number AF49(638)-589.

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VOL. 46, 1960 PHYSIOLOGY: CONWAY AND SAKAI 897

l These PROCEEDINGS, 45, 1542 (1959); 46, 257 (1960); and 46, 570 (1960).2 The group can be obtained directly as the limit of the finite order group associated with each

v. That group, of order 0, is generated by U, V, and the vth root of unity given by VUV-'U-1.Some aspects of the latter group are worthy of note. There are 2 + v - 1 classes, the commu-tator subgroup is of order vi, and the order of the corresponding quotient group, 02, is the numberof inequivalent one-dimensional representations. The remaining v' - 1 matrix representationsmust be of dimensionality v, v3 = V2 + (V- 1)v2, and differ only in the choice of the generatingvth root of unity. That choice is already made in the statement of operator properties for U andV and there can be only one irreducible matrix representation of these operators, to within thefreedom of unitary transformation.

3 In earlier work of the author, for example Phys. Rev., 91, 713 (1953), the quantum actionprinciple has been postulated rather than derived.

CAFFEINE CONTRACTURE*

BY DOROTHY CONWAYt AND TOSHIO SAKAI

THE ROCKEFELLER INSTITUTE

Communicated by George W. Corner, April 25, 1960

Introduction. A number of observations show that the activation of the myo-plasm is preceded by depolarization of the excitable membrane. This intimateassociation of two characteristic features of muscular activity led to the conclusionthat "depolarization is the essential thing" in myoplasmic activation.1

Recent experiments, however, suggest that the membrane potential change is a"priming step" only,2 and it has been shown that an electrical current,3 2 calcium,4and iodide' can activate depolarized resting muscle without membrane potentialchange. These observations were interpreted to mean2 that a calcium step is moredirectly linked to myoplasmic activation than is depolarization. Similar viewsare implied in the articles of Sandow6 and Frank.7On the other hand it has been shown that a variety of pharmacological agents

lose their characteristic "activity promoting" effect on the smooth muscle of theuterus if either its excitable membrane is drastically depolarized8' 2 or it is rendered"Ca-deficient."9 This observation and also the fact that current, calcium, andiodide only activate muscle after the priming effect of depolarization suggest that themembrane potential change is an essential step in normal activation even if it isinsufficient in itself to evoke myoplasmic activity, or if it is not directly linked tosuch activity.These conclusions are challenged by Evans, Schild, and Thesleff,10 who claim

that characteristic pharmacological effects can be induced on completely depolarizedsmooth muscles. Csapo2 pointed out that strictly isometric recording carried outfor a long period bf time (after the administration of the drug) should cancel thedifferences in the findings. Only isometric and prolonged recording brings out thefact that the normal muscle responds to activity-promoting agents with a series ofcontractions repeated over a period of several hours, whereas the depolarized muscledevelops only an abortive contracture of small amplitude. However, Axelsson andThesleff1 reported that caffeine activates cross-striated muscle which has beencompletely depolarized with isotonic K2SO4 for several minutes, and that caffeine

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