Lecture 10. The Dirac equation. WS2010/11: Introduction to Nuclear and Particle Physics

Transcription

1 Lecture 10 The Dirac equation WS2010/11: Introduction to Nuclear and Particle Physics

2 The Dirac equation The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in The Dirac equation provides a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. The equation demands the existence of antiparticles and actually predated their experimental discovery. This made the discovery of the positron, the antiparticle of the electron, one of the greatest triumphs of modern theoretical physics

3 The Dirac equation Dirac found the equation for free relativistic fermions under the requirement : 1) that the wave-equation as in case of the Schrödinger equation should be of 1st order in/t /x 0 and 2) to allow for a continuity equation with a positive density *. 3) Relativistic covariance (with respect to the Lorentz transformation) then requires that the wave-equation also has to be of 1 st order in the space derivatives /x k (k = 1, 2, 3). Accordingly the equation has to be of the general form (1) with a restmass M and an hermitean Hamiltonian α, β unknown (yet!) coefficients If the equation should describe free particles it s solutions should be plane waves with the proper dispersion relation for particles with restmass M, (2) with (3)

4 The Dirac equation In this case the wavefunction (or each component of) has to be a solution also of the Klein-Gordon-Equation: (4) By comparing the quadratic form of (1), symmetrized in the indices k, l, with the Klein-Gordon-equation (4) (5) one has to require that the unknown coefficients k (k = 1, 2, 3) and must have the properties (6) The conditions (6) can only be fulfilled for matrices. Due to Eq. (2) these matrices have to be hermetian: (7) The eigenvalues then are real and according to (6) can only be + 1.

5 The Dirac equation Furthermore, since k = k due to cyclic invariance of the trace (8) These matrices must be traceless. Accordingly, the dimension of the matrices has to be an even number. In 2 dimensions, however, there are only 3 linearly independent matrices that anticommute, i.e. the Pauli matrices: (9) (10) or unitary transformations of (10). Accordingly, the k, must be at least 4x4 matrices. In Pauli-Dirac standard notation (with 2x2 submatrices) these are given by Unitary equivalent representations are also possible. (11)

6 The Dirac equation The wavefunction (x) in (3) then must have 4 components (12) Its adjoint is and fulfills the equation (13) (14) By multiplication of (1) with from left, multiplication of (14) withfrom right and subtraction we obtain the continuity equation (15) with the positive definite density (16) as desired.

7 -matrices and Dirac-algebra For the analysis of transformation properties it is more convenient to introduce an equivalent set of matrices which is obtained as follows: (17) The Dirac-equation then reads:. (18) with the Pauli-adjoint Spinor (19) For the -matrices Eqs. (6) then become more compact: (20) and Eq. (7) leads to the pseudo-hermeticity (21)

8 -matrices and Dirac-algebra This can be easily proven in the standard representation (EX) (22) In the standard representation 1, 3 are real and 2 is imaginary. (23)

9 -matrices and Dirac-algebra The continuity equation (15) then reads (EX) with (23) and describes the vanishing of the four-divergence of the four-current j (x). With the help of the 4- matrices one can construct a complete set of 4x4 matrices. Since a general 4x4 has 16 independent elements there also have to be 16 basis elements. For convenience one chooses the basis: 1 matrix (4 x4 unit matrix) 4 matrixes 6 matrixes (antisymmetric) (24) 4 matrixes 1 matrix

10 -matrices and Dirac-algebra where 5 is the chirality matrix given by (25) This matrix can also be written as if (µ,ν,ρ,σ) is an even permutation of (0,1,2,3) if (µ,ν,ρ,σ) is an odd permutation of (0,1,2,3) (26) (27) in all other cases with the complete antisymmetric unit tensor of fourth order. The matrix 5 has the properties (EX) (28)

11 -matrices and Dirac-algebra The nonzero elements of the matrices µν are those from nondiagonal spacial components (, ) = (k, l) with k unequal l and (, ) = (0, k). These elements can be expressed as (29) where (30) (31) is the ε -tensor in 3 dimensions. The three 4x4 matrices (29) have the properties of a set of spin-matrices, i.e (EX) (32) which is proven explicitly in the standard representation (33)

12 -matrices and Dirac-algebra Where m are again the Pauli spin-matrices. In this representation we have furthermore and (34) Hermetian properties are fixed by (21), (28) and read (35) The 16 matrices (36) build up (except for Γ 1 ) a traceless orthonormal set of matrices with the scalar product (37) i.e. (38)

13 -matrices and Dirac-algebra In particular, these matrices are linearly independent. Accordingly any 4x4 matrix M can be written as a linear combination in the basis (36): (39) Products of arbitrary matrices thus may also be written in terms of an expansion in the Γ-matrices. For a couple of such products we present the result in the following multiplication table :

14 -matrices and Dirac-algebra (40)

15 -matrices and Dirac-algebra The system of matrices (36) with the multiplication table (40) is denoted as Dirac-Clifford-algebra. An application of (40) is the summation over spin-matrices in transition amplitudes. Further examples are often appearing products such as: if n - odd (41)

16 -matrices and Dirac-algebra Contractions with arbitrary four-vectors a, b v are written in shorthand form e.g. (42) (43) etc. In this context one also has to mention -identities such as: (44) Nowadays the Dirac-Clifford-algebra is encoded in software routines such as Mathemetica, REDUCE, FORM, DERIVE etc.

17 Transformation of Dirac spinors In case of homogenuous Lorentz transformations with (45) the Dirac-equation must be form-invariant according to the principle of special relativity, i.e. read as For(x) we have to allow for a linear mixing of the components of (x), which requires to define a transformation in Hilbert space, i.e. (46) or (47) with a Λ dependent 4 x4 matrix S αβ (Λ). Τhe equations for S are obtained by insertion of (47) in (46); multiplication from left by S 1 gives (48)

18 Transformation of Dirac spinors Comparing (48) with the original Dirac-equation shows that the term [...] ν, i.e. the transformation in Hilbert space must be aquivalent to a Lorentz transformation (49) For Lorentz transformations with det (Λ) = +1 and Λ we employ the Ansatz (50) with an antisymmetric parameter-matrix r v = r v µ. For infinitesimal transformations Λ=1 + r ( r << 1) we get (51) For the matrices µν we obtain according to (49) the condition (52)

19 Transformation of Dirac spinors Using the multiplication table (40) one identifies the unique solution A four-component field function α (x), which transforms under homogeneous Lorentz transformations (47) as is called a Dirac-Spinor. The 4x4 matrices S(Λ) acting in the Hilbert space follow: (53) (54) The group transformations are not unitary, but due to the real and antisymmetric parameter-matrix r v follow: (55) (56)

20 Transformation of Dirac spinors We now consider (54) in particular for rotations and Lorentz boosts for which we obtain: for a rotation with vector for a boost with velocity An expansion of the exp.-function (54) and using the matrix identities (40) gives: (57) 1) For arbitrary complex vectors a,b a pure rotation leads to a unitary transformation (with ) (58)

21 Transformation of Dirac spinors In standard representation this corresponds to a doubling of the Pauli matrices for two-component spinors. It has the characteristic properties for fermions, i.e. for we get: (59) 2) In case of pure Lorentz boosts we obtain the non-unitary transformation: (60) or with the four-unit vector with (61) or

22 Transformation of Dirac spinors For parity transformations we have to allow for a phase = ±1 ( is the eigenvalue of the parity transformation) as in case of the Klein-Gordon field. The condition (49) ( ) leads to: (62) which is solved by (63) Eq. (47) then reads: (64) The Dirac equation is form-invariant under these parity transformations. For the particular matrix 5 this implies: (65) whereas due to the matrices µν - one finds: (66)

23 Transformation of Dirac spinors From the transformation S follows for the Pauli-adjoint spinor: (67) Accordingly a term with arbitrary Dirac spinors is always a Lorentz scalar: (68) while is invariant under ordinary Lorentz transformations but changes sign under parity transformations, i.e. it is a pseudo-scalar: (69)

24 Transformation of Dirac spinors By analyzing the residual matrices of the basis (24) and applying (49) one finds five different bilinear covariants with the particular transformation properties: scalar 4-vector tensor of 2nd order (70) pseudovector pseudoscalar In particular the current (23) is a 4-vector.

25 Solution of the free Dirac equation The solution of the Dirac equation are plane waves (3). In order to find the yet unknown spinors(0) (or a suitable basis) we consider solutions for positive and negative energies, separately: with (71) By insertion in Eq. (1) we obtain: or in -notation: (72) (73) In order to specify the spinors u and υ we consider first the case p = 0 (i.e. we evaluate the spinor in the rest frame of the particle). In this frame (73) reads: (74)

26 Solution of the free Dirac equation Now we can choose for u(0) and υ(0) two orthogonal eigenspinors to 0 eigenvalues +1 and 1, which are denoted by. with Since Tr ( 0 ) = 0 each eigenvalue appears twice. Furthermore, the matrices (29) commute with 0 and we can choose u r (0) and υ r (0) as eigenspinors of with eigenvalues s= +1, where a is an arbitrary unit vector because according to (57) we have : (75) In standard representation one chooses a = (0, 0, 1) which leads to The 4 basis spinors are normalized according to: (76) (77) Accordingly we get the orthogonality relation: (78) since u and υ belong to different eigenvalues of 0.

27 Solution of the free Dirac equation The completeness relation then reads: (79) and any bilinear expansion of 0 wrt. eigenspinors reads as: (80) Consequently, the matrices (81) with (82) have the properties of projection operators on solutions (with p = 0) for positive and negative energies +M.

28 Solution of the free Dirac equation In standard representation these relations become immediately apparent: (83) and the 4x4 matrices read (84) In order to get solutions for we have to apply a Lorentz transformation, i.e. the transformation (61) with the proper Lorentz boost: (85) The appropriate boost velocity is given by: (86)

29 Solution of the free Dirac equation According to (60) one gets with the parameters of (61) : and (87) with (88) The solution is according to (60), (61): (89) Now the spinors for Dirac particles with momentum p read: (90)

30 Solution of the free Dirac equation These are solutions of (73) since (91) Since the lefthand sides of (77) are Lorentz invariants the normalizations read: (92) In order to evaluate the general projectors on solutions with positive and negative energy we use (93) (94)

31 Solution of the free Dirac equation and obtain (EX) (95) The completeness relation (79) then reads: (96) Note: the eigenvalue equation (76) does not hold for finite p, since Σ 3 does not commute with S p. In standard representation the 4 basis spinors are given by: (97) or explicitly:

32 Solution of the free Dirac equation (98) (99) The matrix elements with these u- and υ-spinors are easily calculated to give (EX) (100) such that the matrix elements of the Dirac-equation (73) (after multiplication with M) read: (101) which is equivalent to the free dispersion relation particle of mass M. for a

33 Solution of the free Dirac equation In the representation (100) in the nonrelativistic limit p << M the lower components of u s become smaller by a factor p /(2M) than the upper components, while for υ s the relations are opposite. The relation between u s and υ s may be expressed according to (25) as (EX): i.e. the chirality matrix 5 exchanges u- and υ-spinors. (102) Finally, the explicit form of the Lorentz boost for the transformation (85) reads (using N E = M(E p +M)) (EX): (103)