Structure of Dirac matrices and invariants for nonlinear Dirac equations Tohru Ozawa and Kazuyuki Yamauchi Department of Mathematics Hokkaido University Sapporo 060-080, Japan Introduction In this paper we present invariants for nonlinear Dirac equations in spacetime R n+, by which we prove that a special choice of the Cauchy data yields free solutions. In [] Chadam and Glassey considered a problem of that kind for Klein-Gordon-Dirac equations with Yukawa coupling in one and three dimensions. Their results as well as proofs, however, depend on particular representations of the Dirac matrices. The invariants are especially described in terms of components in the Dirac spinor field. Although their results hold in other cases by the unitary equivalence of the Dirac matrices, this fact seems to make it difficult to understand relations between the Dirac matrices and the invariants. The purpose in this paper is to give a representation-free understanding of the problem and to generalize their results in several directions. Our assumptions (a) and (a2) below are independent of particular representations of the Dirac matrices. Our argument works for Klein-Gordon-Dirac equations with Yukawa coupling as well. This paper is organized as follows. In Section 2, we summarize basic notation and facts about Dirac matrices. Proposition and 2 shall be used in Section 4. In Section 3, we present invariants for the nonlinear Dirac equation in Lemma and Theorem under the assumptions (a) and (a2). As a corollary, we prove that the corresponding constraint on the Cauchy data is preserved and keeps nonlinear interaction null to yield free solutions.
Up to Section 3, results are independent of space dimensions and of specific representations and degree of the Dirac matrices. In Section 4 we examine sufficient conditions for (a) and (a2). For n =, the situation is very simple as seen in Proposition 4. For n 2, we introduce assumption (S) in Proposition 5. In Proposition 6, we prove that the assumption (S) is always satisfied in any space dimension if the degree of the Dirac matrices is chosen to be double of the usual minimal choice 2 [(n+)/2], where [a] is the integral part of nonnegative real number a. In Theorem 2, we prove that (S) is satisfied under some restrictive assumption on space dimensions if the degree of the Dirac matrices is minimal. In Section 5, we reproduce Chadam-Glassey s results in [] as special cases of our results. In Section 6, in connection with Proposition 2 we give an explicit representation of Dirac matrices with real components for n = 8 and N = 6 just for reference. 2 Dirac matrices In this section we recall some basic facts about Dirac matrices. We refer the reader to [4], [5], [7]. Definition Let α 0, α,..., α n be matrices in M N (C). (α 0, {α j } n j= ) is an (n + )-tuple of Dirac matrices when it satisfies the following: () α j = α j for j = 0,,..., n. (2) (Anticommutation Relations) α j α k + α k α j = 2δ jk I N for j, k = 0,,..., n. Here δ jk is the Kronecker delta and I N is the identity matrix in M N (C). Remark Let (a 0, {a j } n j= ) be an (n + )-tuple of Dirac matrices and let l be a non-negative integer with 2l n. Put { α j = a j for j 2l, i l+ a 0 a 2l a j for j > 2l. Then, (α 0, {α j } n j= ) becomes an (n + )-tuple of Dirac matrices. Proposition There is an (n + )-tuple of Dirac matrices in M N (C) when N = 2 [ n+ 2 ]. 2
In the appendix of [4] an explicit construction of Dirac matrices is described. Along this method, for n = 8l, 8l, 8l + with l N, we have an (n + )- tuple of Dirac matrices (α 0, {α j } n j= ) which satisfies that { aj for j 4l, α j = a j otherwise. Through Remark this yields the following property. Proposition 2 Let N = 2 [ n+ 2 ]. There exists an (n + )-tuple of Dirac matrices in M N (R) when n 0,, 7 mod 8. See Appendix for an explicit representation of Dirac matrices with real components for N = 6 and n = 8 3 Invariants of the nonlinear Dirac equation Let (β, {α j } n j= ) be Dirac matrices and let ψ be a classical solution to the nonlinear Dirac equation: t ψ + α ψ + imβψ = iλ(βψ, ψ)βψ, (E) where ψ : R R n (t, x) ψ(t, x) C N, and m and λ R. We use the notation α = α + + α n n, j = / x j, (φ, ψ) = φ ψ + + φ N ψn, φ 2 = (φ, φ) for φ = (φ,..., φ N ), ψ = (ψ,..., ψ N ) and x = (x,..., x n ). We suppose that ψ has sufficient regularity in space-time and decay at infinity in space to ensure formal calculations such as integration by parts and differentiation under integral sign. Integrals without specific domains denote the Lebesgue integrals over R n. We drop the time notation unless otherwise specified. Regarding the Cauchy problem for (E) we refer the reader to [3],[6] and references therein. 3
Proposition 3 ψ 2 dx = constant. Proof. Let ψ = ψ(t) be a solution to (E). Then, t (ψ, ψ) = 2R(α ψ, ψ) 2R ( i(m(βψ, ψ) λ(βψ, ψ) 2 ) ). Since β is Hermitian, (βψ, ψ) is real. Hence t (ψ, ψ) = (α ψ, ψ) (ψ, α ψ) = ((α k k ψ, ψ) + (ψ, α k k ψ)) k = k = k ((α k k ψ, ψ) + (α k ψ, k ψ)) k (α k ψ, ψ), for α k is Hermitian. By integrating with respect to x, the proof is completed. Assumptions Let A be an N N matrix. We itemize assumptions as follows: (a) t βaβ = A. (a2) t α j Aα j = A or t A for any j =,... n. Remark 2 Let ( β, { α j } n j=) be another (n + )-tuple of Dirac matrices. It is well-known that there exists a unitary matrix such that β = U βu and α j = U α j U. We define à = U AU. Then à satisfies (a) and (a2) with (β, {α j } n j=) replaced by ( β, { α j } n j=), provided that U is a real matrix. In this sense the assumptions above are described in a covariant way under orthogonal equivalence of representations of the Dirac matrices. 4
Lemma Assume (a) and (a2). Then, (Aψ, ψ) dx = constant. Proof. Let ψ be a solution to (E). Then, t (Aψ, ψ) = k k (Aα k k ψ, ψ) i(m λ(βψ, ψ))(aβψ, ψ) (Aψ, α k k ψ) i(m λ(βψ, ψ))(aψ, βψ). By the assumption (a) Therefore, (Aβψ, ψ) = ( t βaψ, ψ) = (Aψ, βψ). t (Aψ, ψ) = k { (Aαk k ψ, ψ) + (Aψ, α k ψ) }. Let S be the set {k N : k n, t α k Aα k = A}. When k S, It follows that (Aψ, α k k ψ) = ( t α k Aψ, k ψ) = (Aα k ψ, k ψ). (Aα k k ψ, ψ) + (Aψ, α k ψ) = k (Aα k ψ, ψ). If k / S, by the assumption (a2), t α k Aα k equals t A, so that Aα k is symmetric. Therefore there is a matrix B M N (C) such that Aα k = t BB. Hence k (Bψ, Bψ) = 2(B k ψ, Bψ) = 2( t BB k ψ, ψ) = 2(Aα k k ψ, ψ). Similarly, since t α k A is symmetric, there is C M N (C) such that t CC = t α k A, which satisfies k (Cψ, Cψ) = 2(Aψ, α k k ψ). 5
Consequently, t (Aψ, ψ) = k { (Aαk k ψ, ψ) + (Aψ, α k ψ) } = k S k / S k (Aα k ψ, ψ) { k (B k ψ, B k ψ) + k (C k ψ, C k ψ) }, 2 where t B k B k = Aα k and t C k C k = t α k A. By integrating with respect to x, the proof is completed. Theorem Let A be a unitary matrix of degree N and let µ C. Assume (a) and (a2) for A. Then, µaψ ψ 2 dx = constant. Proof. Since A is unitary, µa ψ 2 = ( + µ 2 ) ψ 2 + 2R(Aψ, ψ). By virtue of Proposition 3 and Lemma, this completes the proof. Remark 3 Let à and ( β, { α j } n j=) be as in Remark 2 and let U be orthogonal. Then the corresponding spinor field ψ is given by ψ = U ψ. In this setting we have (à ψ, ψ) = (Aψ, ψ) and µã ψ ψ 2 = µaψ ψ 2, so that invariants in Lemma and Theorem are also invariant under orthogonal equivalence of representations of the Dirac matrices. Corollary Assume (a) and (a2) for a unitary matrix A and suppose that Aψ(0) = ψ(0). Then (βψ, ψ) = 0 at any time. Namely, ψ is a free solution to (E). 6
Proof. If Aψ(0) = ψ(0), by Theorem, Aψ = ψ for all time. Hence (βψ, ψ) = (βaψ, ψ) = (Aβψ, ψ) (by the assumption (a)) = (βψ, A ψ) ( ) = (βψ, A Aψ) by ψ = Aψ = (βψ, ψ) = (βψ, ψ). Since β is Hermitian, (βψ, ψ) is real. This yields (βψ, ψ) 0. 4 Structure of Dirac matrices In this section we shall be more specific about the main assumptions (a) and (a2). A simple sufficient condition for (a) and (a2) is given by (S) below. The condition (S) requires that only one of Dirac matrices is pure imaginary and the rest of Dirac matrices are real, which in turn provides a structural restriction on the space of Dirac matrices in connection with its degree N (Proposition 6) or its dimension n (Theorem 2). We start with Propositions 4 and 5. The proofs are simple and omitted. Proposition 4 Let n =. Let (β, α) is a 2-tuple of Dirac matrices and suppose that β is a real matrix. Then, (β, α) satisfies (a) and (a2) with A = α. Proposition 5 Let (β, {α j } n j= ) be Dirac matrices. property: Assume the following (S) β = β and there exists j 0 {,..., n} such that α j0 = α j0, α k = α k for all k with k j 0. Set A = iβα j0. Then, A is a real symmetric and orthogonal matrix which fulfills (a) and (a2). 7
Proposition 6 Let N = 2 [ n+ 2 ]+. Then, for any n, there exists an (n + )- tuple of Dirac matrices in M N (C) which satisfies (S). Remark 4 The assumption (S) is preserved up to orthogonal changes of the Dirac matrices. See also Remarks 2 and 3. ( ) Proof. Let b, {a j } n j= be an (n+)-tuple of Dirac matrices in M N (C) and let T : M N (C) M 2N (R) be a mapping, which is real-linear, defined by T (X) = Define ( RX IX IX RX ) as ( β, {α j } n j= ), where RX = ( ) ( ) X + X, IX = X X. 2 2i β = T (b), { it (iaj ) for j = j α j = 0, T (a j ) otherwise. ( ) Then, β, {α j } n j= is an (n + )-tuple of Dirac matrices which satisfies (S). By virtue of Proposition, this completes the proof. Theorem 2 Let N = 2 [ n+ 2 ]. Suppose that n, 2, 3 mod 8. Then, there exists an (n + )-tuple of Dirac matrices in M N (C) which satisfies (S). Proof. In the case of n =, 2, 3, the standard representation yields Dirac matrices with (S), see, e.g. [4]. In other cases, by Proposition 2, there is an (n )-tuple of Dirac matrices {a j } n 2 j=0 in M N(R) with N = [ ] n 2. Let ( ) IN O α 0 =, O I N ( ) O aj α j =, j =,..., n a j O ( ) O iin α n =. ii N O Then, {α j } n j=0 is an (n + )-tuple of Dirac matrices which satisfies (S) with j 0 = n. 8
5 Special Cases In this section we apply Corollary and Proposition 5 to reproduce Chadam- Glassey s result on particular conditions which ensure free solutions in one and three space dimensions [] (see also [2]) as special cases. n =, N = 2 α = A = ( ) 0, β = 0 ( ) 0. 0 ( 0 0 ), The following examples have the structure (S) with j 0 = 2. n = 2, N = 2 ( ) 0 α =, α 0 2 = ( ) 0 A =. 0 ( 0 i i 0 ) ( 0, β = 0 ), n = 3, N = 4 α = α 3 = A =., α 2 = i i, β =. i i,, Here and hereafter, components in matrices without specific representations are understood to be zero. 9
6 Appendix The following are Dirac matrices in M 6 (R). See Proposition 2. β = α = 0
α 2 = α 3 =
α 4 = α 5 = 2
α 6 = α 7 = 3
α 8 = Acknowledgment We are grateful to Hideo Kubo for his interest he has shown in this work and enlightening discussions. References [] John M. Chadam and Robert T. Glassey, On certain global solutions of the Cauchy problem for the (classical) coupled Klein-Gordon-Dirac equations in one and three space dimensions, Archive for Rational Mechanics and Analysis 54 (974), 223 237. [2] John M. Chadam and Robert T. Glassey, On the Maxwell-Dirac equations with zero magnetic field and their solution in two space dimensions, Journal of Mathematical Analysis and Applications 53 (976), no. 3, 495 507. [3] Miguel Escobedo and Luis Vega, A semilinear Dirac equation in H s (R 3 ) for s >, SIAM Journal on Mathematical Analysis 28(997), no.2, 338 362. 4
[4] Hubert Kalf and Osanobu Yamada, Essential self-adjointness of n-dimensional Dirac operators with a variable mass term, Journal of Mathematical Physics 42 (200), no. 6, 2667 2676. [5] H. Blaine Lawson, Jr. and Marie-Louise Michelsohn, Spin Geometry, Princeton Mathematical Series 38, Princeton University Press, Princeton, New Jersey (989). [6] Shuji Machihara, Kenji Nakanishi and Tohru Ozawa, Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation, Revista Mathemática Iberoamericana 9 (2003), no., 79 94. [7] Joachim Weidmann, Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics 258, Springer-Verlag, Berlin (987). 5