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On Relativistic Quantum Mechanics of the Majorana Particle On Relativistic Quantum Mechanics of the Majorana Particle H. Arod´z Institute of Theoretical Physics, Jagiellonian University∗ Institute of Nuclear Physics Cracow, May 14th, 2020 PLAN 1. Introduction 2. New observable: the axial momentum 3. The general solution of the Dirac equation 4. The relativistic invariance 5. Summary 1. Introduction 1a. The original motivation: how to construct a quantum theory of non-Grassmannian Majorana field? In particular, what is the Lagrangian for this field? The relativistic quantum mechanics of the single Majorana particle is a step in that direction. It has turned out that it is rather interesting on its own right, because of significant differences with the standard examples of relativistic quantum mechanics. 1b. The Dirac particle: 0 1 1 2 B C α µ (x) = B 3 C ; (x) 2 C; iγ @µ (x) − m (x) = 0: @ A 4 R 3 y Scalar product: h 1j 2i = d x 1(x; t) 2(x; t) with arbitrary t. The Schroedinger form of the Dirac eq.: ^ ^ 0 k 0 y i@t = H ; H = −iγ γ @k + mγ = H : 1. Introduction General complex solution of the Dirac equation: 1 Z (x; t) = d 3p eipx e−iEpt v (+)(p) + eiEpt v (−)(p) ; (1) (2π)3=2 where p denotes eigenvalues of the momentum operator p^ = −ir, and 0 l l 0 (±) (±) p 2 2 (γ γ p + mγ )v (p) = ±Epv (p); Ep = + m + p : (1) is important as the starting point for QFT of the Dirac field. Two views on (1): expansion in the basis of common eigenvectors of commuting observables p^ = −ir and H^ (physics); or merely the Fourier transformation (mathematics). 1c. In the so called Majorana representations, all γµ matrices are purely imaginary. 1. Introduction For example, 0 σ −σ 0 0 σ γ0 = 2 ; γ1 = i 0 ; γ2 = i 1 ; σ2 0 0 σ0 σ1 0 3 0 σ3 0 1 2 3 0 σ0 γ = −i ; and γ5 = iγ γ γ γ = i : σ3 0 −σ0 0 σk – the Pauli matrices, σ0 – the 2 × 2 unit matrix. Charge conjugation C is given by the complex conjugation. The Majorana bispinors are invariant under C – they have real components in the Majorana repr. The Hilbert space of the Majorana bispinors is over R, not C ! The scalar product has the form R 3 T h 1j 2i = d x 1 (x; t) 2(x; t): QM without complex numbers. p^ has the trivial domain! The Dirac bispinor is a composite object, as opposed to the Majorana one: • = 1 + i 2 with real 1;2, C 1 = 1; C(i 2) = −i 2 (similar to the decomposition = R + L into the Weyl bispinors), • two irreducible unitary s = 1=2 reprs of the Poincaré group. 2. New observable: the axial momentum Questions: • Is there ‘eigenvector’ version of expansion (1) for the Majorana bispinors? • What operator could replace the momentum p^ ? 2a. There exists 1:1 mapping M between linear spaces of the Majorana and the right-handed (or left-handed) Weyl bispinors. Take arbitrary right-handed Weyl bispinor φ, γ5φ = φ, and form = φ + φ∗ ≡ M(φ). is a Majorana bispinor. M is invertible: φ = (I + γ5) =2 ≡ R: M preserves linear combinations only if their coefficients are real. The Weyl bispinors are complex, hence the momentum operator p^ = −ir is well-defined for them. It commutes with γ5, therefore also p^φ is right-handed Weyl bispinor. Let us find the Majorana bispinor that corresponds to p^φ: ∗ M(p^φ) = p^φ + (p^φ) = p^( R − L) = −iγ5r ≡ p^5 : Thus the momentum operator from the space of right-handed Weyl bispinors gives rise to p^5 = −iγ5r – the axial momentum operator – in the space of Majorana bispinors. 2. New observable: the axial momentum p^5 commutes with γ5, therefore it can be used also in the space of right-handed Weyl bispinors. It is invariant under the mapping M: if = M(φ) then p^5 = M(p^5φ) because the matrix −iγ5 is real. 2b. Are the two quantum mechanics, Majorana and Weyl, equivalent? No, because the mapping M does not preserve scalar product. Take 1 = M(φ1); 2 = M(φ2), Z Z Z 3 T 3 y y ∗ 3 y d x 1 2 = d x (φ1φ2 + (φ1φ2) ) 6= d x φ1φ2: Moreover, there are differences in evolution equations. In the Weyl case, evolution equation has the form (1) with m = 0; in the Majorana case m 6= 0 is allowed. Using M−1 one can transform Eq. (1) for to the space of right-handed Weyl bispinors: µ ∗ iγ @µφ − mφ = 0: This equation is known as the Majorana equation for φ (φ∗ is charge conjugation of φ). It can not be accepted as the quantum evolution equation for the Weyl φ if m 6= 0, because it is not linear over C. The Hilbert space of the Weyl bispinors is linear over C – it includes all bispinors such that γ5φ = φ. 2. New observable: the axial momentum 2c. The conditions for normalized eigenvectors of the axial momentum: Z 3 T p^5 p(x) = p p(x); d x p (x) q(x) = δ(p − q): The eigenvectors (called the axial plane waves) have the form −3=2 p(x) = (2π) exp(iγ5px) v; v – arbitrary real, constant, normalized (v T v = 1) bispinor, and exp(iγ5px) = cos(px)I + iγ5 sin(px): 2d. Commutator of the axial momentum with position operator x^ j k [x^ ; p^5 ] = iδjk γ5: The implied uncertainty relation is the well-known one 1 h j(∆x^j )2j ih j(∆p^k )2j i ≥ δ ; 5 4 jk j j j k k k where ∆x^ = x^ − h jx^ j i I, ∆p^5 = p^5 − h jp^5 j iI . 2. New observable: the axial momentum 2e. The axial momentum commutes with the Hamiltonian ^ 0 k 0 h = −γ γ @k − imγ only in the massless case: ^ 0 [h; p^5] = −2imγ p^5: The operator r commutes with h^. Therefore, in the Heisenberg picture, p^5(t) = −iγ^5(t)r: It turns out that the operator γ^5(t) has the following form m γ^ (t) = γ + i γ0γ [sin(2!^t) + J^ (1 − cos(2!^t))]: 5 5 !^ 5 p ^ ^ 2 p 2 2 ^2 Here J = h=!;^ !^ = m − ∆ ! Ep = m + ~p : Note that J = −I. Oscillating matrix elements of the axial momentum: Z 2 3 T m T (~)^ ( ) (~) = + (cos( ) − ) ( ) δ(~ − ~) d x p x p5 t q x p 1 2 2Ept 1 v w p q Ep j j m T 0 T γ p −p i sin(2Ept)(v γ w) + (1 − cos(2Ept))(v γ5 w) δ(~p + ~q): Ep Ep 3. The general solution of the evolution equation 3a. The evolution equation for the Majorana bispinor reads ^ ^ 0 k 0 @t = h ; h = −γ γ @k − imγ The Hamiltonian h^ is not Hermitian if m 6= 0! It is anti-symmetric, and the evolution operator exp(h^) is orthogonal. Let us take (x; t) in the form of expansion into the axial plane waves 2 1 X Z (x; t) = d 3p eiγ5px v (+)(p)c (p; t) + v (−)(p)d (p; t) : (2π)3=2 α α α α α=1 (±) The orthonormal basis vα (p) is provided by the eigenvectors of 0 k k γ γ p (this matrix commutes with p5): 1 T v (+)(p) = −p3; p2 − jpj; p1; 0 ; 1 p2jpj(jpj − p2) (+) (+) (−) 0 (+) (−) 0 (+) v2 (p) = iγ5v1 (p); v1 (p) = iγ v1 (p); v2 (p) = −γ5γ v1 (p): This is rather interesting basis: scale invariant, and real. Can be used also in the Dirac quantum mechanics. Another expansion (without p^5): L. Pedro, arXiv:1212.5465 (2012). 3. The general solution of the evolution equation 3b. Time dependence of the real amplitudes cα(p; t); dα(p; t) is found from the evolution equation. To this end, we split the amplitudes into the even and odd parts, 0 00 0 00 cα(p; t) = cα(p; t) + cα (p; t); dα(p; t) = dα(p; t) + dα (p; t); 0 0 00 00 where cα(−p; t) = cα(p; t); cα (−p; t) = −cα (p; t), and same for d 0; d 00. Furthermore, we introduce 0 0 1 0 0 1 0 1 2 3 1 c1 d1 0 −n ±n ±n 00 00 1 3 2 B c1 C ~ B d1 C B n 0 ∓n ±n C ~c(p; t) = B 0 C; d(p; t) = B 0 C; K±(p) = B 2 3 1 C; @ c2 A @ d2 A @ ∓n ±n 0 n A 00 00 3 2 1 c2 d2 ∓n ∓n −n 0 p 2 2 where Ep = p + m , and m p1 jpj m p3 n1 = ; n2 = ; n3 = : p 1 2 3 2 p 1 2 3 2 Ep (p ) + (p ) Ep Ep (p ) + (p ) 3. The general solution of the evolution equation The time dependence of the amplitudes is given by ~ ~ ~c(p; t) = exp(t Ep K+(p)) ~c(p; 0); d(p; t) = exp(t Ep K−(p)) d(p; 0): The matrices K± are antisymmetric, the matrices exp(t Ep K±(p)) belong to the SO(4) group. 3c. This solution can be rewritten in terms of quaternions. The quaternionic units ^i;^j; k^ are introduced as follows: ^ ^ 0 ^ 0 i = iγ5; j = iγ ; k = −γ5γ : They obey the usual conditions ^i2 = ^j2 = k^2 = −I; ^i^j = k^; k^^i = ^j; ^jk^ = ^i: (±) (+) The bispinor basis vα (p) is generated from v1 (p) by acting with ^ ^ ^ 1 ^ 2^ 3^ i; j; k.
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