Twistor formulation of a massive particle with rigidity

A massive rigid particle model in $(3+1)$ dimensions is reformulated in terms of twistors. Beginning with a first-order Lagrangian, we establish a twistor representation of the Lagrangian for a massive particle with rigidity. The twistorial Lagrangian derived in this way remains invariant under a local $U(1) \times U(1)$ transformation of the twistor and other relevant variables. Considering this fact, we carry out a partial gauge-fixing so as to make our analysis simple and clear. We develop the canonical Hamiltonian formalism based on the gauge-fixed Lagrangian and perform the canonical quantization procedure of the Hamiltonian system. Also, we obtain an arbitrary-rank massive spinor field in $(3+1)$ dimensions via the Penrose transform of a twistor function defined in the quantization procedure. Then we prove, in a twistorial fashion, that the spin quantum number of a massive particle with rigidity can take only non-negative integer values, which result is in agreement with the one shown earlier by Plyushchay. Interestingly, the mass of the spinor field is determined depending on the spin quantum number.


I. INTRODUCTION
A model of a relativistic point particle with rigidity (or simply a rigid particle model) has first been proposed by Pisarski [1] about 30 years ago as a 1-dimensional analog of the rigid string model presented by Polyakov [2]. The action of the rigid particle model contains the extrinsic curvature, K, of a world-line traced out by a particle, in addition to the ordinary term that is identified as the arc-length of the world-line. By choosing the arc-length, l, to be a world-line parameter along the world-line, the action takes the following form: where m is a mass parameter and k is a dimensionless real constant (in units such that = c = 1). Pisarski demonstrated, using the back-ground field method in sufficiently large Euclidean dimensions, that the renormalized version of k −1 behaves asymptotically free. Not long after that, Plyushchay investigated the (3 + 1)-dimensional model governed by the action S in both cases of m = 0 and m = 0 [3][4][5][6]. 1 He studied the canonical Hamiltonian formalism based on S and the subsequent canonical quantization of the system. Then it was clarified that the massless rigid particle model, specified by m = 0, describes a massless spinning particle of helicity k [4,5], which can take both integer and half-integer values [5]. Also, it was shown that the massive rigid particle model, specified by m = 0, describes a massive spinning particle whose spin quantum number can take only non-negative integer values [3,6]. On the other hand, Deriglazov and Nersessian have recently claimed that the massive rigid particle model can yield the Dirac equation and hence can describe a massive spinning particle of spin one-half [17]. This statement is inconsistent with that of Plyushchay. It is therefore necessary to clarify which statement is correct.
Recently, the massless rigid particle model has been reformulated in terms of twistors [18]. It was demonstrated in Ref. [18] that the action S with m = 0 is equivalent to the gauged Shirafuji action [19][20][21] (rather than the original Shirafuji action [22]) that governs a twistor model of a massless spinning particle of helicity k propagating in 4dimensional Minkowski space, M. The gauged Shirafuji action can thus be regarded as a twistor representation of the action for a massless particle with rigidity. Upon canonical quantization of the twistor model, the allowed values of k are restricted to either integer or half-integer values, which are in agreement with the allowed values obtained in Ref. [5]. At the same time, an arbitrary-rank spinor field in complexified Minkowski space, CM, can be elegantly derived via the Penrose transform [23][24][25]. It can also be shown that this field satisfies generalized Weyl equations.
Since the twistor formulation of a massless particle with rigidity has been well established, it is quite natural to next consider the twistor formulation of a massive particle with rigidity. Twistor approaches to massive particle systems were investigated independently by Penrose, Perjés, and Hughston about 40 years ago [26][27][28][29][30][31]. After a long while, Lagrangian mechanics of a massive spinning particle has been formulated until recently in terms of two twistors [32][33][34][35][36][37][38][39][40][41][42]. 2 In fact, the Shirafuji action for a massless spinning particle has been generalized in various ways to describe a massive spinning particle in M [32,34,35,[37][38][39][40]42]. Among the generalized Shirafuji actions, the gauged generalized Shirafuji (GGS) action presented in Ref. [42] is one of the most desirable actions, because it yields just sufficient constraints among the twistor variables in a systematic and consistent manner. All the constraints, excluding the mass-shell condition of Fedoruk-Lukierski type [38,39], are indeed derived on the basis of the fact that the GGS action remains invariant under the reparametrization of the world-line parameter and under the local U(1) × SU (2) transformation of the twistor and other relevant variables.
In light of the equivalence between the action S with m = 0 and the gauged Shirafuji action, we can expect that the action S with m = 0 is equivalent to the GGS action or its analog. One of the purposes of the present paper is to confirm this expectation by reformulating the massive rigid particle model in terms of twistors. To this end, following the procedure developed in Ref. [18], we first provide an appropriate first-order Lagrangian and demonstrate that it is equivalent to the Lagrangian L K (l) := −m − |k|K found in Eq. (1.1). (The constant k contained in S is replaced by |k| for the sake of consistency, as will be seen in Sec. 2.) The first-order Lagrangian eventually gives five constraints among dynamical variables. We simultaneously solve two of the five constraints by using two (commutative) 2-component spinors [44] without spoiling compatibility with the other three constraints. Substituting the solution obtained there into the first-order Lagrangian, we have a Lagrangian written in terms of the spacetime coordinate variables and the spinor variables. Then, using the two (novel) twistors defined from these variables, we express the Lagrangian in a twistorial form. Each of these twistors satisfies the so-called null twistor condition [23][24][25]. We incorporate the null twistor conditions for the two twistors into the Lagrangian with the aid of Lagrange multipliers so that all the twistor components can be treated as independent fundamental variables. In addition, we slightly modify the Lagrangian so that it can immediately give the mass-shell condition of Fedoruk-Lukierski type. The modified Lagrangian is actually equivalent to the one before modification and hence to L K (l). In this way, we can elaborate an appropriate twistor representation of L K (l). The action defined with the modified Lagrangian is not the GGS action itself but its analog. It is thus confirmed that the action S = l 1 l 0 dl L K (l) with m = 0 is equivalent to an analog of the GGS action. Since the modified Lagrangian governs a novel twistor model that has not been studied yet, we need to investigate classical and quantum mechanical properties of this model in detail. For this purpose, we carry out a partial gauge-fixing for the local U(1) × U(1) symmetry of the twistor model by adding a gauge-fixing term and its associated term to the modified Lagrangian. (The modified Lagrangian remains invariant under the local U(1) × U(1) transformation rather than the local U(1) × SU (2) transformation.) The gauge-fixing condition and its associated condition are chosen so as to make our analysis simple and clear. With the gauge-fixed Lagrangian, we study the canonical Hamiltonian formalism of the twistor model by completely following the Dirac algorithm for Hamiltonian systems with constraints [45][46][47]. We see that by virtue of the appropriate gauge-fixing procedure, the Dirac brackets between the twistor variables take the form of canonical bracket relations. Also, we can obtain manageable first-class constraints. The subsequent canonical quantization of the twistor model is performed with the aid of the commutation relations between the operator versions of the twistor and other canonical variables. In the quantization procedure, the first-class constraints turn into the conditions imposed on a physical state vector. It is pointed out that the physical state vector can be chosen so as to be an eigenvector of the spin Casimir operator. Among a total of six physical state conditions, five are represented as simultaneous differential equations for a function of the canonical coordinate variables, while the remainder turns into the (algebraic) mass-shell condition. The five differential equations eventually reduce to two differential equations for a twistor function, a holomorphic function only of the twistor variables (excluding the dual twistor variables). We solve one of these equations by applying the method of separation of variables, finding a certain twistor function as its general solution. The other differential equation is used in some other stage.
After completing the quantization procedure, we consider the Penrose transform of the above-mentioned twistor function to obtain an arbitrary-rank massive spinor field in CM. The spinor field obtained has extra upper and lower indices in addition to dotted and undotted spinor indices. Because of the structure of the Penrose transform, the number of upper (lower) extra indices is equal to the number of undotted (dotted) spinor indices. We can find the allowed values of the spin quantum number of a massive particle with rigidity by evaluating the total number of extra indices of the spinor field in CM. Using a useful relation proven in Appendix B, we indeed prove that the spin quantum number can take only non-negative integer values. This result is in agreement with the one shown earlier by Plyushchay [3,6] and contradicts the recent statement of Deriglazov and Nersessian [17].
We also demonstrate, by using the mass-shell condition, that the spinor field satisfies generalized Dirac-Fierz-Pauli (DFP) equations with extra indices. In addition, we show that the spinor field symmetrized totally with respect to the extra indices satisfies the (ordinary) DFP equations [48][49][50][51]. A physical mass parameter included in both the generalized and ordinary DFP equations is identical to the one found by Plyushchay [3,6] and turns out to be dependent on the spin quantum number. We thus see that the physical mass of the spinor field is determined depending on its rank. This paper is organized as follows: In section 2, we provide a first-order Lagrangian for a massive particle with rigidity. In section 3, beginning with the first-order Lagrangian, we elaborate an appropriate twistor representation of the Lagrangian for a massive particle with rigidity. A partial gauge-fixing is carried out for this twistor representation. The canonical Hamiltonian formalism based on the gauge-fixed Lagrangian is investigated in section 4, and the subsequent canonical quantization procedure is performed in section 5. In section 6, we derive an arbitrary-rank spinor field in CM via the Penrose transform of a twistor function and demonstrate that this spinor field satisfies generalized DFP equations with extra indices. We also show that the allowed values of the spin quantum number are restricted to arbitrary non-negative integers. Section 7 is devoted to a summary and discussion. In Appendix A, we focus on the Pauli-Lubanski pseudovector and find a specific form of the physical mass parameter. In Appendix B, we prove a useful relation. In Appendix C, we extract some part of Plyushchay's noncovariant formulation from the twistor formulation developed in this paper.
From the Lagrangian (2.3), the Euler-Lagrange equations for x µ , q µ , p µ , r µ , e, and β are derived, respectively, asṗ constraints. It can be seen from Eq. (2.5e) that the sign of k is determined depending on which sign is chosen in ± −q 2 r 2 . Taking the derivative of Eq. (2.5e) with respect to τ and using Eqs. (2.5b), (2.5d), and (2.5f), we have q 2 r µ p µ = 0. Since q 2 > 0 has been postulated, it follows that Taking the derivative of Eq. (2.5f) with respect to τ and using Eqs. (2.5b) and (2.5d), we have The derivative of Eq. (2.7) with respect to τ is identically satisfied with the use of Eqs.
Using Eq. (2.5d), we can eliminate the auxiliary field r µ from the Lagrangian (2.3) to obtain Here, e is no longer an independent auxiliary field and is determined from Eq. (2.5d) as follows: Contracting Eq. (2.5d) with q µ and using Eq. (2.5f), we have where qq := q µq µ . Then Eq. (2.5d) leads to whereq 2 ⊥ is defined byq 2 ⊥ :=q ⊥µq µ ⊥ withq µ ⊥ :=q µ −q µ (qq)/q 2 . In this way, e is determined to be e = ± 1 2 −q 2 ⊥ /q 2 . Because q 2 > 0, the inequalityq 2 ⊥ = q 2q2 − (qq) 2 /q 2 ≤ 0 holds, 3 and hence e still remains purely real. The Lagrangian (2.9) can be written explicitly as which becomes by eliminating q µ and p µ with the use of Eq. (2.5c). In Eqs. (2.12) and (2.13), one of the signs in the symbol ∓ is chosen so that the Lagrangian can be negative definite even in the limit m ↓ 0, whether k is positive or negative. As a result, ∓k = −|k| is realized and the Lagrangian (2.13) reads This is exactly the Lagrangian for a massive particle with rigidity represented as a function of τ : L(τ ) = √ẋ 2 L K = √ẋ 2 (−m − |k|K) [1,3,6]. (Note that ∓k = −|k| is compatible with Eq. (2.5e); in fact, it leads to a consistent result −q 2 r 2 = |k|.) The conditioṅ x 2 > 0 implies that the particle moves at a speed less than the speed of light. As expected, the reparametrization invariance of the action S = τ 1 τ 0 dτ L is maintained with the Lagrangian (2.14). In our present approach, the Lagrangian (2.14) has been found from the Lagrangian (2.3) by eliminating the auxiliary fields p µ , r µ , e, and β, and furthermore the field q µ . The Lagrangian (2.3) is thus established as a first-order Lagrangian for a massive particle with rigidity, being considered the fact that the Lagrangian (2.3) is first order inẋ µ andq µ .

III. TWISTOR REPRESENTATION OF THE LAGRANGIAN
In this section, we derive a twistor representation of the Lagrangian (2.14) by following the method developed in Refs. [18,22]. A partial gauge-fixing for a local gauge symmetry 3 Let u µ be an arbitrary timelike vector in M and v µ an arbitrary vector in M. Since u µ is timelike, we can choose the rest frame such that u i = 0 (i = 1, 2, 3). In this frame, u 2 = u µ u µ and uv = u µ v µ reduce to u 2 = (u 0 ) 2 and uv = u 0 v 0 , respectively. Then it can readily be shown that Because u 2 , v 2 , and uv are Lorentz scalars, u 2 v 2 ≤ (uv) 2 holds true in arbitrary reference frames. of the twistor representation is also considered.
Then it follows that Eqs. (3.3a), (3.3b), and (3.3c) are compatible with the transformation rules (2.4b), (2.4a), and (2.4c), respectively. For maintaining the reparametrization symmetry in the solution (3.3), it is necessary to introduce scalar-density fields such as f and g. We see that p αα , q αα , and r αα in Eq.
. Hereafter, we refer to the local U(1) transformation with θ i as the U(1) i transformation. Its corresponding gauge group is simply denoted as U(1) i .
With Eqs. (3.9) and (3.10), the Lagrangian (3.7) can be written as which can be expressed more concisely as in terms of the (novel) twistors defined by Z A i := ω α i , π iα and their dual twistorsZ i A := π i α ,ω iα (A = 0, 1, 2, 3). Equation (3.12) can now be expressed as These are precisely the null twistor conditions, and hence Z A i turn out to be null twistors [23][24][25]. The transformation rules (3.5a) and (3.11) can be combined into If we regard ω α i andω iα as primary independent variables, without referring to Eqs. (3.9) and (3.10), then the Lagrangian (3.14) itself does not remain invariant under the U(1) 1 ×U(1) 2 transformation owing to the existence of derivatives with respect to τ . This Lagrangian becomes invariant only if the null twistor conditions in Eq. (3.15) are used. Now we incorporate the null twistor conditions into the Lagrangian (3.14) with the aid of real Lagrange multiplier fields, a i = a i (τ ), on T , so that all the twistor components can be treated as independent variables at the Lagrangian level. The Lagrangian (3.14) is thus improved as follows: From this Lagrangian, the null twistor conditions can be derived as the Euler-Lagrange equations for a 1 and a 2 . The field a i is assumed to transform, under the proper reparametrization, as so as to maintain the reparametrization invariance of the action S = τ 1 τ 0 dτ L with Eq. (3.17). Also, we assume that under the U(1) i transformation, a i transforms as Thereby, the U(1) 1 × U(1) 2 invariance of the Lagrangian (3.7) is recovered in the Lagrangian (3.17). If we substitute Eqs. (3.9) and (3.10) into Eq. (3.17), it reduces to Eq.
(This condition can also be derived from the Lagrangian (3.7).) Equation (3.23) is equiv- Here it is assumed that ϕ = ϕ(τ ) is a real scalar field on T and transforms, under the Accordingly, it follows that Eq.
(3.24) remains unchanged under both of the reparametrization and the U(1) 1 × U(1) 2 transformation. In order to obtain Eq. (3.24) immediately, without referring to Eq. (3.23), we modify the Lagrangian (3.20) as and obeys the transformation rule The action S = τ 1 τ 0 dτ L with the modified Lagrangian (3.26) has a form very similar to the GGS action found in Ref. [42]. Clearly, the action S with Eq. (3.26) is invariant under the reparametrization and the U(1) 1 × U(1) 2 transformation. At this place,   (3.20) and is established as a twistor representation of the Lagrangian for a massive particle with rigidity. Now, we carry out a partial gauge-fixing for later convenience by imposing the condition This condition partially breaks the U(1) 1 × U(1) 2 invariance of the Lagrangian (3.26) so as to maintain its invariance under the restricted gauge transformation with the gauge functions θ i such that θ 1 + θ 2 = 0. (The condition (3.30) is equivalent to the Lorenz-type gauge conditionȧ 1 +ȧ 2 = 0, provided that this condition is reparametrization-covariant.) We also impose the additional condition is a Nakanishi-Lautrup real scalar field on T , and ζ = ζ(τ ) is a real field on T obeying the transformation rule The reparametrization invariance of the action S with Eq.   .27) gives h = (f − iζ)/2. We thus see that −ζ/2 can be identified as the imaginary part of h.

IV. CANONICAL FORMALISM
In this section, we study the canonical Hamiltonian formalism of the twistor model governed by the Lagrangian (3.32). The canonical momenta conjugate to the canonical coordinates The canonical Hamiltonian corresponding to L is defined by the Legendre transform of L, The equal-time Poisson brackets between the canonical variables are given by which can be used for calculating the Poisson bracket between two arbitrary analytic functions of the canonical variables.
Equations (4.1a)-(4.1j) are read as the primary constraints where the symbol "≈" denotes the weak equality. Now, we apply the Dirac algorithm for constrained Hamiltonian systems [45][46][47] to develop the canonical formalism of the present model. Using Eq. (4.3), the Poisson brackets between the primary constraint functions φ's are found to be The Poisson brackets between H C and the primary constraint functions can be calculated to obtain where I AB and I AB are the so-called infinity twistors [23,24], defined by Equations (4.10a) and (4.10b) determineū i A and u A i , respectively, as follows: Equations (4.10i) and (4.10j) reduce to ϕ ≈ 0 and e ≈ 0, respectively. Taking into account ϕ ≈ 0 and e ≈ 0, we see that Eqs. (4.10c)-(4.10j) yield the secondary constraints All the Poisson brackets between H C and the secondary constraint functions χ's vanish. The Poisson brackets between the primary and secondary constraint functions are found to be all others = 0 .
All the Poisson brackets between the secondary constraint functions vanish.
Next we investigate the time evolution of the secondary constraint functions using Eqs. (4.9) and (4.13). The time evolution of χ (a)i can be calculated aṡ (no sum with respect to i) by using Eqs. (4.11a), (4.11b), (4.12d), (4.12e), (4.12f), (4.12g), and (4.12h), together with the formulas Then the conditionχ (a)i ≈ 0 determines u (b) to be Mζ. The time evolution of χ (b) and χ (e) is found to beχ The time evolution of χ (h) is evaluated aṡ by using I AB I AC = 0 and Eqs. (4.11b), (4.15a), (4.12d), and (4.12b). Hence the conditioṅ χ (h) ≈ 0 is identically fulfilled. Similarly, it can be shown thatχ (h) ≈ 0 is identically fulfilled. The time evolution of χ (ϕ) , χ (ζ) , and χ (g) is found to bė (4.18c) The conditionsχ (b) ≈ 0,χ (e) ≈ 0, andχ (ϕ) ≈ 0, respectively, give From the conditionsχ (ζ) ≈ 0 andχ (g) ≈ 0, the Lagrange multipliers u (ϕ) and u (e) are determined to be zero. As can be seen from the above analysis, no any further constraints are derived, and hence the procedure for finding secondary constraints is now complete. We have seen that u (a)1 + u (a)2 , u (ϕ) , and u (e) vanish and u A i ,ū i A , u (b) , u (ζ) , and u (g) are determined to be what are written in terms of other variables such as the canonical coordinates. In contrast, u (a)1 −u (a)2 , u (h) , and u (h) still remain as undetermined functions of τ .
We have obtained all the Poisson brackets between the constraint functions, as in Eqs. (4.5) and (4.13). However, it is difficult to classify the constraints in Eqs. (4.4) and (4.12) into first and second classes on the basis of Eqs. (4.5) and (4.13) together with the vanishing Poisson brackets between the secondary constraint functions. To find simpler forms of the relevant Poisson brackets, we first definẽ and furthermore define It is easy to see that the set of all the constraints given in Eqs. (4.4) and (4.12), i.e., is equivalent to the new set of constraints We can show that except for
In this manner, we obtain the canonical Dirac bracket relations appropriate for the subsequent quantization procedure. In fact, the Dirac brackets in Eqs. (4.30a) and (4.30b) immediately lead to the twistor quantization procedure.

V. CANONICAL QUANTIZATION
In this section, we perform the canonical quantization of the Hamiltonian system investigated in Sec. 4. To this end, in accordance with Dirac's method of quantization, we introduce the operatorsF andĜ corresponding to the functions F and G, respectively, and impose the commutation relation all others = 0 .
The commutation relations in Eqs. (5.2a) and (5.2b) govern together so-called twistor quantization [23,24]. The operatorsẐ A i andẐ i A , referred to as the twistor operators, can be expressed in terms of their spinor components asẐ A i = ω α i ,π iα andẐ i A = π i α ,ω iα . Accordingly, Eq. (5.2a) can be decomposed as follows: In the canonical quantization procedure, the first-class constraints are transformed into the conditions for specifying physical states, after the replacement of the first-class constraint functions by the corresponding operators. In the present model, the first-class constraints (4.28a)-(4.28f) lead to the following physical state conditions imposed on a physical state vector |F : Now we introduce the bra-vector with a reference bra-vector 0| satisfying Using the commutation relations in Eq. (5.2), we can show that Also, it is easy to see that Here, Eq. (5.13) is obtained under the natural condition F J = 0. Equations (5.14a)-(5.14c) imply that F J does not depend on a − , h, andh. Hence it follows that F J is a function only of the twistors Z A i (i = 1, 2). Such a holomorphic function of Z A i is often called a twistor function. As can be seen immediately, Eqs. (5.13) and (5.14e) are, respectively, equivalent to We now apply the method of separation of variables to Eq. (5.14d) to find its general solution. Substituting the factorized function into Eq. (5.14d), we can separate it into the two equations where s * is a constant and F being particular solutions of Eq. (5.14d). Here, C Js * are complex coefficients.

VI. PENROSE TRANSFORM
In this section, we obtain an arbitrary-rank spinor field in complexified Minkowski space CM via the Penrose transform of F J (Z). Then it is shown that the allowed values of the spin quantum number J are restricted to arbitrary non-negative integers. We also demonstrate that the spinor field satisfies generalized DFP equations with extra indices. Furthermore we mention the total symmetrization of the spinor field and a braket formalism of the Penrose transform.
Now we suppose that among the indices i 1 , . . . , i p , the number of 1's is p 1 and the number of 2's is p 2 (= p − p 1 ). We also suppose that among the indices j 1 , . . . , j q , the number of 1's is q 1 and the number of 2's is q 2 (= q − q 1 ). Since F Js * Z is a homogeneous twistor function of degree −2s * − 2 with respect to each of Z A 1 = ω α 1 , π 1α and Z A 2 = ω α 2 , π 2α , only one integral in the infinite sum in Eq. (6.4) can remain nonvanishing, provided that the following two conditions are simultaneously satisfied in this integral: with s * satisfying Eq. (6.5). Equations (6.5a) and (6.5b) yield q 1 − p 1 = q 2 − p 2 . Using this, it can be shown that N + : = q + p = 2(q 1 + p 2 ) = 2(q 2 + p 1 ) , (6.7a) Thus we see that the rank of Ψ , denoted by N + , is even, and that the difference between the numbers of the dotted and undotted spinor indices of Ψ , denoted by N − , is also even. In Appendix B, it is proven that 7 Since N + is non-negative and even, it follows from Eq. (6.8) that J takes only non-negative integer values: J = 0, 1, 2, . . . . Therefore we can conclude that the massive particle with 7 Some readers may think that the proof of Eq. (6.8) can simply be accomplished within the framework of group theory. However, the spin quantum number J at present is a quantum number that appears in the context of treating the Pauli-Lubanski pseudovector of a massive particle with rigidity. For this reason, the proof of Eq. (6.8) must be achieved within the present framework by taking into account the origin of J.
rigidity is allowed to possess only integer spin. This result is in agreement with the one obtained by Plyushchay [3,6] (see also Appendix C).
Next, we demonstrate that Ψ satisfies generalized DFP equations with extra indices. To this end, it is useful to exploit that The derivative of Ψ with respect to z ββ can be calculated by using Eqs. (6.1) and (6.9) as follows: Contracting over the indicesβ andα 1 in Eq. (6.10) and using Eq. (5.15a), we have  This makes it clear that Ψ is a field of mass M J . Thus, taking into account Eq. (6.8), we can conclude that the spinor field of rank 2J with mass M J has been obtained by means of the Penrose transform (6.1).
Before closing this section, we make mention of the total symmetrization of Ψ and a bra-ket formalism of the Penrose transform. We thus see that Ψ (S) satisfies the (ordinary) DFP equations [48][49][50][51]. This result is consistent with the totally symmetric property of Ψ (S) with respect to the spinor indices. It is evident from Eqs. (6.15) and (6.16) that Ψ (S) fulfills the Klein-Gordon equation with the mass parameter M J .
B. Bra-ket formalism of the Penrose transform In the same manner as the F J Z, a − , h,h = Z, a − , h,h F J given under Eq. (5.13), we can express the twistor function F J Z as where (6.20) [see Eq. (5.7)]. Using Eq. (6.20), it is easily shown that Z π iα = π iα Z , (6.21a) with the ket and bra vectors This operator is abbreviated in Appendix B asP (S) .

VII. SUMMARY AND DISCUSSION
In this paper, we have reformulated the massive rigid particle model, defined by the action (1.1) with m = 0, in terms of twistors and have investigated both classical and quantum mechanical properties of the twistor model established in the reformulation.
We first presented a first-order Lagrangian for a massive particle with rigidity, given in Eq. (2.3), and verified that it is indeed equivalent to the original Lagrangian (2.14). We subsequently elaborated a twistor representation of the Lagrangian (2.14), given in Eq. (3.26), via the first-order Lagrangian by following the procedure developed in the massless case [18]. It was pointed out that the action with the twistorial Lagrangian (3.26) has a form very similar to the GGS action for a massive spinning particle [42]. The Lagrangian (3.26) remains invariant under the U(1) 1 ×U(1) 2 transformation. Considering this, we carried out a partial gauge-fixing for the U(1) 1 × U(1) 2 symmetry by adding the gauge-fixing term b(a 1 + a 2 ) and its associated term Mζϕ to the Lagrangian (3.26). The canonical Hamiltonian formalism of the twistor model was studied on the basis of the gauge-fixed Lagrangian (3.32). We were able to immediately obtain the canonical Dirac bracket relations between twistor variables, given in Eq. (4.30), by virtue of choosing the appropriate gauge-fixing condition a 1 + a 2 = 0. In addition, the associated condition ϕ = 0 made our formulation simple and clear. 8 As a result of this classical mechanical treatment, the canonical commutation relations between twistor operators, given in Eq. (5.2), were obtained, so that the canonical quantization of the twistor model was properly accomplished. In the quantization procedure, we found the physical state conditions given in Eq. (5.4) and saw that they eventually reduce to the algebraic mass-shell condition (5.13) and the simultaneous differential equations (5.14d) and (5.14e) for the twistor function F J . This function seems to correspond to the wave function of a physical state found by Plyushchay [6]. In the twistor formulation, however, we succeed to derive an arbitrary-rank massive spinor field Ψ by means of the Penrose transform of F J [see Eq. (6.1)]. This is an advantage of the twistor formulation developed in this paper. Intriguingly, the spinor field Ψ has extra indices in addition to the usual dotted and undotted spinor indices and satisfies the generalized DFP equations (6.13a) and (6.13b). (The extra indices will be related to the particle-antiparticle degrees of freedom [42].) We also verified that the totally symmetrized spinor field Ψ (S) satisfies the (ordinary) DFP equations (6.18a) and (6.18b). It is worth mentioning that the mass of the spinor fields Ψ and Ψ (S) is determined depending on the spin quantum number J. More precisely, these spinor fields of rank 2J have the mass M J defined in Eq. (5.6).
We proved, in the context of the twistor formulation, that the spin quantum number J of a massive particle with rigidity can take only non-negative integer values. Although the method of proof is completely different from that of Plyushchay [6], both methods led to the same result concerning the allowed values of J. (Hence the statement of Deriglazov and Nersessian [17] is contradicted.) In the twistor formulation, an essential condition for leading to this result is ultimatelyχ (a)− |F = 0, given in Eq. (5.4d). In fact, we derived the allowed values of J by using the conditions (6.5a) and (6.5b), which were found by evaluating the homogeneity degrees of the particular solution F Js * of the differential equation (5.14d) originating fromχ (a)− |F = 0.
Finally we try to extend the twistor model so that J can take positive half-integer values as well as non-negative integer values. For this purpose, we introduce the 1-dimensional U(1) Chern-Simons terms with real constants s i . Since the field a i transforms as Eq. (3.18) under the proper reparametrization, S i is obviously reparametrization invariant. Also, S i remains invariant under the gauge transformation (3.19), provided that θ i satisfies an appropriate boundary condition such as θ i (τ 1 ) = θ i (τ 0 ). Now we consider the extended twistor model governed by the actionS := τ 1 τ 0 dτ L + S 1 + S 2 , with L being the Lagrangian (3.32). In quantizing the extended twistor model, the physical state condition χ (a)− −s − |F = 0 with s − := s 1 −s 2 is imposed on |F , instead ofχ (a)− |F = 0. Then it can be shown that s − takes only integer or half-integer values. In addition, Eq. (6.7a) is modified as N + : = q + p = 2(q 1 + p 2 − s − ) = 2(q 2 + p 1 + s − ) . (7.2) Using Eqs. (7.2) and (6.8), and noting N + ≥ 0, we see that J can take both nonnegative integer and positive half-integer values. In this way, the extended twistor model is established as a model for massive particles with integer or half-integer spin. However, in the case s − = 0, the extended twistor model cannot be regarded as a twistor representation of the massive rigid particle model. In view of this situation, it would be interesting to extend the massive rigid particle model so as to correspond to the extended twistor model.
As can easily be verified,Ŵ αα , and henceŴ 2 , commutes with the following operators given in Eq. (5.4): For this reason, we can put the eigenvalue equationŴ 2 |F = Λ|F , with an eigenvalue Λ, together with the conditions (5.4a)-(5.4f). (These conditions can be regarded as eigenvalue equations with the common eigenvector |F and vanishing eigenvalues.) It is straightforward to see that in the particle's rest frame, the commutation relation (A22) acting on |F reduces to Ŵ r ,Ŵ s = iǫ rstŴt by the use of Eqs. (5.4e) and (5.4f) (see Appendix C). Since the operatorsŴ r fulfill the SU (2) commutation relation, the eigenvalue ofŴ rŴr is found to be J(J + 1) under a proper condition. Here, J denotes the spin quantum number that takes non-negative integer or positive half-integer values. Notinĝ W 2 = −Ŵ rŴr , we thus obtain so that Λ is determined to be −J(J + 1). Now, we can expect that the operator version of Eq. (A7) holds at the quantum mechanical level. Acting this operator on |F , we havê where relevant commutation relations have been used after applying the Weyl ordering rule. With Eqs. (5.4e) and (5.4f), Eq. (A25) becomeŝ (A28) a. Proposition 1 Let G 1 and G 2 be piecewise smooth functions ofπ i α and π iα , let |G 2 be the ket vector such that G 2 (π, π) = π, π|G 2 , and let − → T 2 := − → T r − → T r be the differential operator representation ofT 2 , defined from holds, provided that the integrals on both sides are finite. Here, d 4π := dπ 1 0 ∧dπ 1 1 ∧dπ 2 0 ∧dπ 2 1 and d 4 π := dπ 10 ∧ dπ 11 ∧ dπ 20 ∧ dπ 21 .
Proof : Let us first recall Eq. (A15), or equivalently, By using Eq. (B3) and the traceless property σ rk k = 0, it can be shown that π, π|T r = − → T r π, π|. Applying this twice to the left-hand side of Eq. (B5), we have Carrying out integration by parts twice on the right-hand side of Eq. (B7) leads to the right-hand side of Eq. (B5): Here, in addition to σ rk k = 0, we have used the fact that the boundary terms appearing in the integrations by parts vanish, because the integrals on both sides of Eq. (B5) are assumed to be finite. Combining Eqs. (B7) and (B8) thus gives Eq. (B5). Now we consider the following function: where E is a function of z ααπi α π iα , andP (S) is the operator defined in Eq. (6.28). We here employ z αα rather than x αα , because z αα is useful for defining finite integrals. By using the eigenvalue equations in Eq. (B3a) repeatedly, Eq. (B9) can be written as where P (S) is defined by with P i 1 ...ip α 1 ...αp; j 1 ...jq,α 1 ...αq (π, π) := (−1) p π j 1α1 · · · π jqαqπ i 1 We now apply Proposition 1 to the case of G 1 = P (S) E and |G 2 = |F . Then Eq. (B5) reads whereF (π, π) := π, π|F . The functions E andF are assumed to be chosen in such a way that they are piecewise smooth and the integrals in Eqs. (B10) and (B13) are finite. By means of the fact that − → T r E z ααπi α π iα = 0, Eq. (B13) becomes where N + := p + q.
Since the function E is completely arbitrary, as long as it is piecewise smooth and makes the integral in Eq. (B18) finite, we can conclude that I = N + /2. Combining this with Eq. (A32), we have The proof of Eq. (6.8) is thus complete.

Appendix C: Plyushchay's noncovariant formulation
In this appendix, we extract some part of Plyushchay's noncovariant formulation [3,6] from the current twistor formulation. It is again shown that the spin quantum number J takes only non-negative integer vales.
Here, the approximate expression W(ε ×n) ≃ (ε ×n)W has been used. From (C11), we can see that n|W is the eigenvector ofn corresponding to the eigenvalue n − ε × n.
Hence it follows that n|W = n − ε × n| holds at least in the physical subspace in which |F lives. In this way, we have n|W|F = n − ε × n|F , or equivalently, where F (n) is defined by F (n) := n|F . Expanding both sides of Eq. (C12) with respect to ε and equating terms of order ε yield n|Ŵ |F = LF (n) , L := −in × ∂ n .
The operator L is identified as the orbital angular-momentum operator defined in the internal space parametrized by n. By using the commutation relations p 0 ,Ŵ r = 0, (C9a), (C9b), and (A18), it can be shown thatŴ r |F is also a physical state vector satisfying the conditions (C8a)-(C8d). Then we can use Eq. (C13) twice to obtain In this way, we can reproduce the procedure given by Plyushchay. As we have seen, the existence ofn µ satisfying the commutation relation (C9a) is essential for the argument in this appendix.
Taking the 3-dimensional inner product of Eq. (C13) with n, and using the eigenvalue equation n|n = n n|, we have n|n rŴr |F = 0 .
(C16) This is consistent with the condition (C8c). Equation (C16) actually reduces to Eq. (C8c) under the natural assumption that the set of n|'s constitutes a complete set in the relevant Hilbert space. We thus see that Eq. (C13) can yield Eq. (C8c) and eventually leads to Eq. (5.4d), i.e.χ (a)− |F = 0 via the operator version of Eq. (C2e). Since both of the allowed values of J and the condition (5.4d) can be obtained by exploiting Eq. (C13), we can expect that the condition (5.4d) is related to deriving the allowed values of J. In