aa r X i v : . [ phy s i c s . c l a ss - ph ] J a n DIP 1901-01
Another Counter-Example to Dirac’sConjecture
Takayuki Hori ∗ Doyo-kai Institute of Physics, 3-46-4, Kitanodai,Hacniouji-shi, Tokyo 192-0913, Japan
Abstract
Another counter-example to Dirac’s Conjecture is presented, which resembles theCawley model but is so modified as to include second class constraints. The arbitraryfunction in the general solution to the defining equations of momenta satisfies a non-linear differential equation. Dirac’s conjecture is examined for some solutions to theequation.
The problem of Dirac’s conjecture has been sometimes argued by various authors since Diracstated it[1]. One group of them agrees that it holds, while another group does not. The reasonof the disagreement seems to be in the setting of the conjecture and the procedure leading tothe canonical formalism.The original statement by Dirac himself can be expressed as that every transformationgenerated by first class constraint, which we call Dirac transformation, maps a state to itsphysically equivalent one. According to an interpretation the above statement is equivalent tothat the appropriate hamiltonian is the sum of the canonical hamiltonian and linear combina-tion of all first class constraints including the secondary ones. Sugano et al.[2] argued that theabove hamiltonian does not work in some models. However, in his canonical formalism, thereseems to be no rigorous proof of the equivalence to the lagrangian formalism. On the otherhand Deriglazov et al.[3] showed that there is an extended lagrangian, which is obtained byadding auxiliary variables and is classically equivalent to the original lagrangian, where theDirac transformation of the original lagrangian is the physically equivalent transformation inthe theory described by the extended lagrangian.A system may be described by various lagrangians which are classically equivalent butdifferent from each other in auxiliary or unphysical variables. The constraint structures ofthe various lagrangians are different and the validity of Dirac’s conjecture depends on theform of each lagrangian. Indeed the gauge symmetry itself is an artifact emerging from thedegrees of freedom of the unphysical variables. Hence the meaning of Dirac’s conjecture inDeriglazovs’ formalism seems to be different from that of original Dirac’s one. ∗ email: [email protected] general solution for the velocityvariables, u = ˆ U ( q, π ), to the defining equations of momenta, π = W def = ∂L/∂u . Theprocedure of the method is logically clear compared with that of Lagrange multiplier byDirac. The latter is mysterious according to Deriglazov[5].For the model with only first class constraints the condition for the generating functionof the Dirac transformation to be the physically equivalent one was given in [4]. The Cawleymodel[6, 7] is one of the models in which Dirac’s conjecture does not hold. In order togeneralize the formalism of [4] to the models with second class constraints, we present here amodel with such constraint, where Dirac’s conjecture does not hold. A new aspect which isabsent in the model with only first class constraints is that the arbitrary functions appearedin the general solution to the defining equations of momenta are restricted. In order to illustrate the canonical formalism of [4], let us start with the Cawley model[6, 7].The action of the Cawley model[6] is S Cawley = Z dτ L, L = u u + 12 q ( q ) , (1)where q A , ( A = 1 , ,
3) are the coordinate variables and u A , ( A = 1 , ,
3) are the correspondingvelocity variables. The action is invariant under the transformation δq A = δ A ǫ + δ A ddτ (cid:18) ˙ ǫq (cid:19) δu A = ddτ ( δq A ) , ( A = 1 , ,
3) (2)where ǫ is an arbitrary infinitesimal parameter.The Euler-Lagrange equations obtained by varying q A are[EL] = ˙ u = 0 , [EL] = ˙ u − q q = 0 , [EL] = q = 0 . (3)Since the lagrangian does not contain u , the time development of u (and q ) is not deter-mined, so we put ˙ u = c , where c is an arbitrary function of q ’s and u ’s. We regard q as anunphysical variables. The consistency of the Euler-Lagrange equations requires ℓ = q = 0 , ℓ = u = 0 , (4)which are called lagrangian constraints[8], and should be satisfied in the initial condition forthe differential equations (3). Equation ℓ = 0 is the third eqation of (3) and is called firstorder lagrangian constraint, while ℓ = 0 is obtained by time derivative of ℓ = 0 and is calledsecond order one.Let us proceed to the canonical theory according to [4]. Denote the canonical conjugateof q A as π A . Since the Hessian matrix, M AB = ∂W B /∂u A , where W B def = ∂L/∂u B , is singular,the defining equations of momenta, π A = W A ( q, u ), have not unique solution for the velocity2ariables. Hence we consider the general solution to the equation and denote it as u A =ˆ U A ( q, π ). The following functions on the coordinate-velocity space play an impotant role: U A pb ( q, u ) def = ˆ U A ( q, W ( q, u )) . (5)We call a function A ( q, u ) the pull-back of a function ˆ A ( q, π ) if A ( q, u ) = ˆ A ( q, W ( q, u )), anddenote A = ˆ A PB . For example , U A pb ( q, u ) are the pull-back of ˆ U A ( q, π ). The functions ˆ U ’scontain some arbitrary functions, which play the similar role as the Lagrange multipliers inthe Dirac theory. In the present model they are ˆ U = π , ˆ U = π , ˆ U = ˆ v ( q, π ) , where ˆ v isan arbitrary function of the canonical variables.The hamiltonian in the present formalism is defined by H = π A ˆ U A − L ( q, ˆ U ) , (6)and nothing is added[4]. In the usual approach the hamiltonian is defined by using ˙ q ’sinstead of ˆ U ’s. But the meaning of ˙ q ’s is obscure, since we are arguing on the phase space.Kamimura[8] developed a generalized canonical formalism on the space spanned by ( q, ˙ q, π ).He introduced the concept of generalized canonical quantity which plays a role of canonicalvariables in the usual formalism.Now the primary cinstraint is ϕ def = π = 0. Hamiltonian of the present model is expressedas H = vϕ + q χ + π χ , χ = −
12 ( q ) , χ = π . (7)The first order and the second order secondary constraints are χ = 0 and χ = 0, respectively.From the definition of the hamiltonian we seeˆ U A = ∂H∂π A , mod ϕ, (8)since we have π = W ( q, ˆ U ( q, π )) on the constrained sub-space. An orbit O in the velocity-coordinate space is mapped by Φ : ( q, u ) ( q, π = W ( q, u )) to an orbit ˆO in the phase space.Since π = W ( q, ˆ U ( q, π )) on ˆO, by differentiating it with respect to τ along the orbit we have˙ π A = − ∂H∂q A + (cid:20) [EL] A + ( ˙ q B − u B ) ∂W A ∂q B (cid:21) u = ˆ U , mod ϕ. (9)At this stage, however, the solution orbit of the Euler-Lagrange equations in the velocity-coordinate space has no counterpart in the phase space which is obtained by a full set ofcanonical equations of motion. In the Dirac recipe they are obtained by the canonical vari-ational principle which imposes the hamiltonian action, ˙ q A π A − H T , to be stationary, where H T is the canonical hamiltonian plus the Lagrange multiplier terms assuring the constraints.In the present method we do not adopt the Dirac recipe, and we have only the relations (8)and (9).In order to get the canonical equations of motion, we need relations which express ˙ q A in terms of canonical variables. We determine the relation by imposing that the resulting3anonical equations are equivalent to the Euler-Lagrange equations. The correct choice turnsout to be ˙ q A = ˆ U A ( q, π ) . (10)In fact from eqs.(8)-(10) we see that the canonical equations of motion, ˙ q A = ∂H/∂π A , ˙ π A = − ∂H/∂q A , are equivalent to [EL] A ( U pb ) = ˙ q A − U A pb = 0. These equations are equivalent tothe second order Euler-Lagrange equations obtained by eliminating u -variables. As for thesecondary constraints, the pull-back of the k -th order secondary constraints are shown to bethe k -th order lagrangian constraints where u ’s are replaced by U pb ’s [4]. (The pull-back ofthe primary constraints are of cause identity.) Thus the pull-backed theory from the canonicaltheory is completely described by the original lagrangian with the notational change u → U pb .The canonical equation of motion for a function ˆ F ( q, π ) is expressed as ddτ ˆ F = { ˆ F , H } , (11)where the Poisson bracket is defined by { ˆ F , ˆ G } def = ∂ ˆ F∂q A ∂ ˆ G∂π A − ∂ ˆ F∂π A ∂ ˆ G∂q A . (12)Now let us examine Dirac’s conjecture. The transformation generated by a linear combi-nation of first class constraints with arbitrary parameters is called Dirac transformation[4].Dirac’s conjecture claims that Dirac transformation map a state to its physically equivalentone. The all of the Poisson brackets among ϕ, χ and χ vanish, and they constitute firstclass constraints. Hence the Dirac transformation is generated by Q = ˆ ǫ ϕ + ˆ ǫ i χ i , (sum over i = 1 ,
2) (13)where ˆ ǫ n , ( n = 0 , ,
2) are arbitrary infinitesimal quantities, and the transformation is δ Q q A = { q A , Q } = δ A ˆ ǫ + δ A ˆ ǫ + { q A , ˆ ǫ } χ + { q A , ˆ ǫ } χ , (14)where ǫ n ( q, u ) def = ˆ ǫ n ( q, W ( q, u )) , ( n = 0 , , ǫ ’s, which do notdepend on π , then the pull-back transformation is δ D q A = δ A ǫ + δ A ǫ , δu A = ddτ ( δq A ) . (15)Under the above variations the Euler-Lagrange equation [EL] = 0 varies as δ D [EL] = ¨ ǫ − ǫ q , (16)which cannot vanish identically at the point where Euler-Lagrange equations and the la-grangian constraints hold. This means the breakdown of Dirac’s conjecture.4or a function ˆ F ( q, π, τ ) let us denoteˆ F ∼ def = ∂ ˆ F∂τ + { ˆ F , H } . (17)We can prove[4] that˙ F ( q, U pb ) = ˆ F ∼ ( q, W ( q, u )) + [EL] A ( U pb ) " ∂ ˆ F∂π A PB , (18)where F ( q, u, τ ) = ˆ F ( q, W ( q, u ) , τ ). Let us define the function E ( τ ) by E ( τ ) def = F ( q sol , ˙ q sol ),where q A sol is the solution to [EL] A = 0. Then it satisfies ˙ E ( τ ) = ˆ F ∼ (cid:12)(cid:12)(cid:12) PB ( q sol , ˙ q sol ). If the gener-ating function Q of the Dirac transformation satisfies Q ∼ = 0 mod ϕ, then the transformationis a gauge transformation. This can be proved by a general theorem[4], i.e. , Q ∼ = δ D L (cid:12)(cid:12)(cid:12) u = ˆ U +total derivative, where δ D stands for the pull-back of the Dirac transformation. In the presentmodel we have Q ∼ = (ˆ ǫ ∼ − ˆ ǫ ) χ + (cid:0) ˆ ǫ ∼ − q ˆ ǫ (cid:1) χ mod ϕ. (19)Hence the Dirac transformation is gauge transformation if ˆ ǫ = ˆ ǫ ∼ , ˆ ǫ = ˆ ǫ ∼ /q . The pull-backed transformation is eqs.(2) with ǫ ( τ ) = ǫ ( q sol , ˙ q sol ), which keeps the action be invariant.Since ˆ ǫ is an arbitrary function of ( q, π ), the parameter ǫ ( τ ) is sufficiently arbitrary. In factthe action is invariant for completely arbitrary ǫ ( τ ). The Cawley model contains only the first class constraints. Let us examine whether modelwith second class constraints has the similar property or it needs modification of the generaltheory[4], where it assumed there is no second class constraint.Consider the lagrangian L = u u + q (cid:18) q − q (cid:19) . (20)The action is invariant under the following transformation δq A = δ A ǫ − δ A ˙ ǫu q − q , (21)where ǫ is an arbitrary infinitesimal quantity. The Euler-Lagrange equations are[EL] = ˙ u = 0 , [EL] = ˙ u − q = 0 , [EL] = − ( q − q ) = 0 . (22)Since the time development of u is not determined by the Euler-Lagrange equations, q isregarded as unphysical variable, and we set ˙ u = c , where c is tentatively an arbitrary functionof ( q, u )’s. 5he first and the second order lagrangian constraints are ℓ = q − q = 0 , ℓ = u − u = 0 . (23)Instead of the third order lagrangian constraint we have the condition c = 0. It is importantto note that the unphysical variable q must be of the form q = aτ + b , with constant a and b ,though the action has the gauge degrees of freedom. Otherwise the Euler-Lagrange equationshave no solution. In most of the gauge model this is not the case, i.e. , the origin of the gaugesymmetry is the arbitrariness of the unphysical variables. It would be thought that the abovecuriosity comes from the fact that the gauge transformation is singular at the point satisfying ℓ = 0. However, in the Cawley model the unphysical variable q is completely arbitrary,though the gauge transformation is singular at ℓ = 0. The above property in the presentmodel is related to the existence of second class constraint as is shown in the canonical theory.The Hessian matrix and W = ∂L/∂u are M ij = , W = u , W = u , W = 0 . (24)The primary constraint is ϕ def = π = 0, and the general solution for u ’s to the definingequation of momenta, π = W , isˆ U ( q, π ) = π , ˆ U ( q, π ) = π , ˆ U ( q, π ) = ˆ v ( q, π ) , (25)where ˆ v is an arbitrary function of the canonical variables. We see U , = u , , U = v ( q, u ) , (26)where v ( q, u ) def = ˆ v ( q, W ( q, u )).The hamiltonian is H = ˆ vϕ + π π − q (cid:18) q − q (cid:19) = H + ˆ vϕ − q χ + π χ , H = ˆ vπ −
12 ( q ) , (27)where χ = q − q , χ = π − ˆ v. (28)The first and the second order secondary constraints are χ = 0 and χ = 0, respectively.There is no third order secondary constraints, but we have the condition for ˆ v :ˆ v ∼ = 0 mod ( ϕ, χ , χ ) . (29)The pull-back of the secondary constraints, χ = χ = 0, are the lagrangian constraints, ℓ ( U pb ) = ℓ ( U pb ) = 0, and that of the condition (29) corresponds to the condition c = 0 inthe lagrangian formalism. 6ntroducing the new constraints by φ = χ − { χ , ˆ v } ϕ + { ϕ, ˆ v } χ = 0 , (30)the Poisson brackets among ( φ , φ , φ ) = ( ϕ, χ , φ ) are { φ n , φ m } = K nm def = − mod ( φ , φ ) , ( n, m = 0 , , . (31)Hence we have two second class constraints, φ , φ and one first class constraint, φ . Thecondition (29) comes from the fact that χ contains ˆ v , which is a consequence of { φ , φ } = 1.In general the emergence of the condition for the arbitrary function in the general solution tothe defining equations of momenta is a characteristic feature in the model with second classconstraints.Since we have only one firat class constraint, φ = 0, the Dirac transformation is generatedby Q = ˆ ǫφ , with arbitrary function ˆ ǫ ( q, π ). Let us obtain the condition for ˆ ǫ that the Diractransformtion is gauge transformation. The criterion is Q ∼ = 0 mod ϕ .ˆ ǫ ∼ = ˆ ǫ (cid:18) ∂ ˆ v∂q + c (cid:19) , c + c ∂ ˆ v∂q + ∂ ˆ v ∼ ∂q = 0 , mod ϕ, (32)where c i are defined by ˆ v ∼ def = c χ + c χ mod ϕ. (33)The pull-back of the Dirac transformation is δ D q = ∂∂u ( ǫF ) , δ D q = ∂∂u ( ǫF ) , δ D q = ǫK + ǫ F, (34)where K def = ∂φ ∂π (cid:12)(cid:12)(cid:12) PB , F def = φ (cid:12)(cid:12)(cid:12) PB . (35)A choice for the arbitrary function, ˆ v , partially determines not only the canonical formal-ism but restricts the lagrangian formalism. Hence, if one choses one of the general sotutions,then the pull-back of it restricts the unphysical variables expressed in terms of U pb in thelagrangian theory. This is illustrated below.The condition (29) restricts the function form of ˆ v , and is written as (cid:26) ˆ v, π π − q (cid:18) q − q (cid:19)(cid:27) + { ˆ v, π } ˆ v = 0 mod ( ϕ, χ , χ ) (36)which is a non-linear differential equation for ˆ v . It seems difficult to obtain the general solutionto (36), but we find some solutions:ˆ v = 0 , ϕ, χ , π , ( q ) π . (37)7or different choices of ˆ v , the gauge structure and the Dirac transformations become different.In the case ˆ v = π there is no first class constraint, and the gauge symmetries of the canonicalformalism are absent. Let us examine the remaining four cases separately.(1) ˆ v = 0 , ϕ, χ :In these cases φ = π , K = 0 , F = u , and the pull-back of the Dirac transformation is δ D q A = δ A ( ǫ + u ǫ ) + δ A u ǫ + δ A ǫ u , (38)where ǫ A = [ ∂ ˆ ǫ/∂π A ] PB . For simplicity let us take ǫ = ǫ = 0, then the Euler-Lagrangeequation, [EL] = 0, varies under the transformation as δ D [EL] = ¨ ǫ − ǫ u , (39)which cannot vanish for any choice of the unphysical variable, q . Hence Dirac’s conjecturedoes not hold.The second term in (32) is satisfied in these cases, and the condition for the Dirac trans-formation to be gauge transformation becomes ˙ ǫ = [EL] A ǫ A . In fact lagrangian varies underthe Dirac transformation as δ D L = u ( ˙ ǫ − [EL] A ǫ A ) + T . D . . If one choose the parameter ǫ ( τ ) = ǫ ( q sol , ˙ q sol ) as in the case of the Cawley model, then we see ˙ ǫ = 0. This exhibits nogauge symmetry. Instead, we choose the parameter as ∂ǫ/∂u A = 0. Here ∂ǫ/∂u = 0 is au-tomatically satisfied since ∂W A /∂u = 0. Then the condition (32) becomes ˙ ǫ = − ( q − q ) ǫ , and the gauge Dirac transformation is expressed as eq.(21).(2) ˆ v = ( q ) / π :In this case we have ˆ v ∼ = 0 mod ϕ . In eq.(27) we see H = 0, so the hamiltonian itself is alinear combination of constraint functions. In fact, by direct calculations we have φ = H/π .For simplicity let us express the Dirac transformation in the case of ∂ ˆ ǫ/∂π A = 0: δ D q = ǫ, δ D q = ǫq (2 q − q )2( u ) , δ D q = ǫ (cid:18) q u (cid:19) + ǫ H PB u . (40)The Euler-Lagrange equations, [EL] = [EL] = 0, vary as δ D [EL] = d dτ ( δ D q ) , δ D [EL] = ¨ ǫ − δ D q , (41)which cannot vanish simultaneously for any choice of the unphysical variable, q . HenceDirac’s conjecture does not hold.The condition for the Dirac transformation to be gauge transformation is˙ ǫ − [EL] A ǫ A = ǫ q u . (42)We set again ǫ , = 0, then we have ǫ = ( ǫq − ˙ ǫu ) /u ( q − q ) , . Substituting it into (40),we have the gauge Dirac transformation under which the action is invariant.Finally, let us comment on the relation of the second class constraints and the Diracbracket. The Poisson brackets among φ , φ are written as { φ i , φ j } = A ij def = (cid:18) − (cid:19) , ( i, j = 0 , . (43)8sing A ij , the hamiltonian is expressed as H = H ′ − φ i ( A − ) ij { φ j , H ′ } , H ′ def = −
12 ( q ) + π ( φ + ˆ v ) . (44)In the above expression the squares of the constraint functions are omitted, which are noeffect on the canonical equations of motion, ˙ˆ F = { ˆ F , H ′ } D , here the Dirac bracket is definedby { ˆ F , ˆ G } D def = { ˆ F , ˆ G } − { ˆ F , φ i } ( A − ) ij { φ j , G } . (45) The arbitrary functions, ˆ U ’s, emerging in the general solution to the defining equations ofmomenta play a role similar as the Lagrange multipliers, λ ’s, in the Dirac recipe, thoughthe ways of the appearance of them are entirely different. λ ’s are independent variables, andsatisfy linear algebraic equations. The solution to the linear equations is a sum of a specialsolution and the general solution to a homogeneous equation. The former is a consequenceof the presence of the second class constraints and the latter corresponds to the first classones. In the present approach ˆ U ’s generally satisfy non-linear differential equations due tothe presence of the second class constraints.The Dirac recipe may be easy to treat compared with that of the present paper. However,logical validity of the method of Lagrange multiplier is obscure. In fact the method maycontain contradiction in a simple model[9]. On the other hand the non-linear differentialequations emerged in the presence of the second class constraints may cause a technicaldifficulty in constructing a general theory with such constraints. Acknowledgement
The author is grateful to Dr. Alexei Deriglazov for valuable discussions.
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