Spin geometry, Dirac operator and Hydrogen atom

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1 Spin geometry, Dirac operator and Hydrogen atom Carsten Balleier February 12, Background from Physics, Notion of Spin 1.1 Stern-Gerlach experiment The Stern-Gerlach experiment was conducted in beam of Ag atoms B In the above setting, the beam of Ag-atoms splits up into two beams. In an inhomogeneous electromagnetic field (as used in the experiment, symbolized by B), the following forces may act: Coulomb force and Lorentz force but they would only act on charged particles, therefore not on the atoms deviation of particles with magnetic moment So the atoms must have a magnetic moment, but where does it come from? Their charge distribution is spherically symmetric and can thus not be the reason! Obviously, the atoms must be spinning, but not in the classical sense since this would yield a continuous distribution on the screen. Are there multiple sorts of silver atoms? Note that silver has electron configuration 3p 1, permitting to attribute this effect to the outer electron. In 1925 Goudsmith and Uhlenbeck propose the notion of spin for an additional, internal property without classical analogue a so-called internal degree of freedom. The spin behaves 1

2 2 Diplomarbeit Carsten Balleier like a fixed angular momentum, knowing only two states (spin-up and spin-down ). In order to find an appropriate mathematical description, recall that the Schrödinger equation works on wave functions ψ L 2 (R 3 ) where ψ 2 = 1 is required. Now, there are two options. Either take ψ L 2 (R 3 Z 2 ), where R 3 Z 2 is the space of all possible states and ψ( r, s) 2 is the probability density to find the particle at the point r R 3 in spin state s Z 2. But people chose to introduce two-component wave functions. Let ψ, ϕ L 2 (R 3 ) be such that ψ 2 + ϕ 2 = 1. Then call the function ( ) ψ : R 3 C 2 ϕ a spinor; later these naïve spinors will be replaced by sections of a certain bundle. A non-relativistic equation to describe particles with spin is the Pauli equation, which was obtained from the Schrödinger equation by adding an interaction between the spin and a magnetic field. Actually, a modified Hamiltonian is used where (Φ, A) describes the electromagnetic field and σ includes the three Pauli matrices: H = 1 ) 2 (ˆ p ea id2 2 +eφ id 2 2 e 2m 2m σ (rot A) 1.2 Relativistic Quantum Mechanics E- p-relation quantization i t, i classical E = p2 2m + V ( q) relativistic E2 = p 2 + m 2 (c = 1), m rest mass i t ψ = 2 2m 3ψ + V ( q)ψ 2 ψ = t 2 3 ψ m 2 ψ Schrödinger eqn. ( 3 : Laplacian on R 3 ) Klein-Gordon eqn. with = 1 The Klein-Gordon equation is suitable for spin-0-particles and can be rewritten as ( g + m 2 )ψ = 0 g = diag(+,,, ). For fermions (i. e., spin- 1 2-particles) one needs to take the square root, roughly stated something like (iγ ν ν + m)(iγ µ µ m)ψ = 0 µ = ( t, 3) It turns out that ψ must be a spinor with four components, hence γ a vector of 4 4-matrices satifying γ µ γ ν + γ ν γ µ = 2g µν id 4 4. This is actually the defining relation for a Clifford algebra. In physicist s notation, we obtain the Dirac operator ( ( ) ) γ µ i µ + m ψ = 0, which will be formalized in the following. Up to now the metric has be semi-riemannian in the sequel, it will be Riemannian for sign convenience.

3 Spin c -Spinorfeldgleichungen 3 2 Spin geometry 2.1 Definitions of the principal objects Definition 2.1. A Clifford algebra Cl(V, q) over a vector space V, equipped with a quadratic form q, is an associative algebra with unit and with a mapping j : V Cl(V, q) satisfying j(v) j(v) = q(v) 1 v V (here: is the algebra multiplication and 1 its unit element). Notice that j is injective, allowing us to identify j(v) with v in what follows. Definition 2.2. The groups Spin(n) and Spin c (n): Spin(n) = {1} {e 1,..., e 2k Cl(R n, x x 2 n k N, e i R n, e i = 1} Spin c (n) = (Spin(n) U(1))/{±1} Observe that U(1), Spin(n) Cl(C n, z z2 n) and U(1) Spin(n) = {±1}. The corresponding Lie algebras are spin(n) = span{e i e j i < j} and spin c (n) = spin(n) ir. Spin representation Note that Cl(C 2k ) = End(C 2k ) holds, which is shown inductively starting at Cl(C) = C C; Cl(C 2 ) = M 2 (C). The isomorphism can be made explicit by (e j C 2k ) ( ) e j Here, we use g 0 = ( ) ( ) ( ) ( ) 0 i and g 1 =. Similarly, Cl(C 2k+1 ) = End(C 2k ) End(C 2k ) is given by ( ) ( ) g (j mod 2). }{{} [ j 1 2 ] times ( ) e j (...,...) (same as above, just twice), e 2k+1 ( i ( ) ( ) ), i.... By these maps, Cl(C n ) acts on C 2k (n = 2k, 2k + 1). Define: called the vector space of Dirac spinors. Σ 2k+1 = Σ 2k = C 2k, Definition 2.3. The spin representation of Cl(C n ) on Σ n (κ : Cl(C n ) End(Σ n )) is given by ( ) for n = 2k and by the projection onto the first component of ( ) for n = 2k + 1.

4 4 Diplomarbeit Carsten Balleier Definition 2.4. Clifford multiplication: This can be extended to the exterior algebra: µ : R Σ n Σ n, (x, ψ) κ(x)ψ =: x ψ µ : k R n Σ n Σ n, (ω, ψ) i 1 <...<i k ω i1...i k κ(e 1 e k )ψ =: ω ψ Coverings The group Spin(n) is constructed such that it double-covers SO(n). The covering is given by a homomorphism λ : Spin(n) SO(n), λ(x) : y x y γ(x), where γ is an antiinvolution γ(x y) = γ(y) γ(x) of the Clifford algebra, and ker λ = {±}. For n 2, Spin(n) is connected, furthermore, for n 3 it is simply connected. The group Spin c (n) is a double-cover by the mapping p([g, z]) = (λ(g), z 2 ). A Spin c representation is obtained from the Spin representation by κ([g, z]) = z κ(g)ψ. From now on, let (M, g) be an orientable, connected, Riemannian man- Spin (c) structure ifold. Definition 2.5. A Spin c structure (P Spin cm, Λ) over the frame bundle P SO M of a manifold M consists of a principle Spin c -bundle and an equivariant map Λ : P Spin cm P SO M such that P Spin cm double-covers P SO U(1) M. A Spin structure is a special case where Spin appears instead of Spin c ; P Spin M double-covers P SO M. Existence: P SO M admits a Spin c structure P U(1) M : w 2 (P SO M) = c 1 (P U(1) M) mod 2, w 2 being the second Stiefel-Whitney class and c 1 being the first Chern class. Definition 2.6. A spinor bundle is an associated bundle ΣM = (P Spin cm Σ n )/Spin c (n), where the Spin c (n) action on the principle bundle is from the right and on Σ n from the left by the spin representation. Sections of this bundle are called spinors, the Clifford multiplication is generalized to µ : (T M) ΣM ΣM, it agrees with the former definition at each point. Connections Let (M n, g) be as before. On P SO M, there is the Levi-Civita connection, given a form Z : T (P SO M) so(n); on P U(1) M, one chooses an arbitrary connection A : T (P U(1) M) u(1) = ir. Therefore, Z A is a connection on P SO U(1) M, which can be lifted: Z A T (P Spin cm) spin(n) ir p π T (P SO U(1) M) Z A so(n) ir

5 Spin c -Spinorfeldgleichungen 5 Using this connection, one has a covariant derivative on ΣM as follows (after an appropriate choice of sections {e i } in P SO M and s in P U(1) M): A Xψ = X(ψ) i<j w ij (X)e i e j ψ + 1 (A s)(x)ψ 2 Definition 2.7. The Dirac operator is the composition D A = µ A of the Clifford multiplication µ and the Spin c connection, where one identifies T M = T M via the metric g. In a local orthonormal basis {e i }, it is given as D A ψ = n e i A e i ψ, i=1 The Laplace operator (on spinors) is given by A ψ = n A e i A e i ψ i=1 2.2 Properties of the Dirac operator ψ Γ(ΣM). n div(e i ) A e i ψ. On a Riemannian manifold (M, g) with scalar curvature R and Spin c structure (U(1)- connection A), there is a well-known relation between the Dirac and the Laplace operator: i=1 D 2 Aψ = A ψ + R 4 ψ da ψ From now on, require additionally (M, g) be complete and compact. Then, one introduces a scalar product (which is well-defined because the spin representation is unitary) by (ψ 1, ψ 2 ) = (ψ 1 (x), ψ 2 (x))dvol g, M the scalar product under the integral sign being the standard hermitian product on the C- vector space Σ n. With respect to this scalar product, D A is essentially self-adjoint in L 2 (ΣM). The last property that will be given here is the following estimate: Theorem 2.8. (M, g) as above, A = 0 (i. e. spin), R > 0, dim M = n. Then: Dψ = λψ λ 1 R0 n 2 n 1, where R 0 is the minimum of R on M. Definition. (new metric covariant derivative) f X ϕ := Xϕ + fx ϕ (f C (M))

6 6 Diplomarbeit Carsten Balleier Then: (D f) 2 = f + 1 4R + (1 n)f 2 Proof. In order to prove the above theorem, integrate the last relation using f = λ n obtain (D λ { n )2 ψ, ψ dvol g = ( 1 M 4 R + 1 n } n 2 λ 2 ) ψ 2 + λ/n ψ 2 dvol g. M Using that ψ is an eigenspinor to the eigenvalue λ, 0 = M { ( 1 4 R + 1 n n λ2 ) ψ 2 + λ/n ψ 2 } dvol g and follows. Assuming λ 2 < 1 4 R 0 n n 1, the r. h. s. is positive unless ψ 0 which is a contradiction. 3 Hydrogen atom Assume: spin- 1 2-particle (electron) in a central field (from static proton). Then by the socalled minimal coupling of field and momentum, physicist s obtain a modified Dirac equation [ ( ) ] γ µ i µ ea m ψ = 0. Here, e < 0 denotes the electron s charge, A = (Φ, A) the electromagnetic field. This form of the Dirac equation corresponds to the Spin c Dirac operator, where P U(1) M encodes the phase information. Unfortunately, up to now no one is able to treat this equation directly; therefore, it is used to derive some improved Pauli equation. Starting with the Schrödinger solutions, one perturbatively finds the energy levels and thus has described the fine structure of the hydrogen atom (i. e. each energy level splits into n components if n is the corresponding main quantum number).