Supersymmetry of Relativistic Hamiltonians for Arbitrary Spin

Hamiltonians describing the relativistic quantum dynamics of a particle with an arbitrary spin are shown to exhibit a supersymmetric structure when the even and odd elements of the Hamiltonian commute. For such supersymmetric Hamiltonians an exact Foldy-Wouthuysen transformation exits which brings it into a block-diagonal form separating the positive and negative energy subspaces. Here the supercharges transform between energy eigenstates of positive and negative energy. The relativistic dynamics of a charged particle in a magnetic field is considered for the case of a scalar (spin-zero) boson obeying the Klein-Gordan equation, a Dirac (spin one-half) fermion and a vector (spin-one) boson characterised by the Proca equation.

Hamiltonian can indeed be reduced to that of a non-relativistic one. Section 5 discusses the resolvent of 48 supersymmetric relativistic Hamiltonians and again shows that for the three cases under consideration 49 the Green's function in essence may be reduce to that of the associated non-relativistic Hamiltonian. 50 Finally in section 6 we present a short conclusion and an outlook for possible further investigations and 51 in the appendix we collect some useful relations for the spin s = 1 case which are not that commonly 52 known.  54 In the Hamiltonian form of relativistic quantum mechanics one puts the wave equation into a Schrödinger-like form ih∂ t Ψ = HΨ .

Relativistic Hamiltonians for arbitrary spin
(1) The Hamiltonian in above equation in general is of the form where β 2 = 1 acts as a grading operator and m standard for the particle's mass. In addition to the mass term βm the operator E represents the remaining even part of the Hamiltonian, that is, it commutes with the grading operator, [β, E ] = 0. The operator O denotes the odd part of H and obeys the anticommutation relation {β, O} = 0. For a particle with spin s, s = 0, 1 2 , 1, 3 2 , . . ., the Hilbert space H on which H acts is given by that is, the wave function Ψ in (1) is a spinor with 2(2s + 1) components [20]. Let us note that we can decompose the Hilbert space H into a direct sum of the two eigenspaces of the grading operator β with eigenvalue +1 and −1, respectively, Obviously, H ± are simultaneously the subspaces where eigenvalues of H are positive and negative, 55 respectively.
For simplicity let us put the relativistic Hamiltonian (2) into the form is, H = βH † β. 60 Choosing a representation where β takes the diagonal form where M ± : H ± → H ± with M † ± = M ± is an operator mapping positive and negative energy states into positive and negative energy states, respectively. Whereas A : H − → H + maps a negative energy state into a positive energy state and A † : H + → H − vice versa. With above representation the general relativistic spin-s Hamiltonian then takes the form In the following section we will show that under the condition that the even and odd parts commute,  Let us now assume that the even mass operator M and the odd operator O commute, that is, [M, O] = 0. This condition implies that As a consequence of this the squared Hamiltonian (8) becomes block diagonal and the complex supercharges Here m > 0 is an arbitrary mass-like parameter, representing, for example, the mass of the relativistic particle in (2). It is obvious that these operators generate a transformation between positive and negative energy states. A straightforward calculation shows that these operators together with the Witten parity operator W := β form a N = 2 SUSY system, that is, Let us note that M under condition (9) commutes with all operators of above algebra and thus constitutes 65 a center of the SUSY algebra (13). Hence, a relativistic arbitrary-spin Hamiltonian (8) obeying the 66 condition (9) may be called a supersymmetric relativistic arbitrary-spin Hamiltonian.

67
Let us also note that for a supersymmetric relativistic arbitrary-spin Hamiltonian there exists an exact Foldy-Wouthuysen transformation U which brings (8) into the block-diagonal form [20] where the partner Hamiltonians H ± are defined by As a side remark let us mention that the four projections operators projecting onto the subspaces of positive/negative Witten parity and positive/negative eigenvalues of H, respectively, are related to each other via the same unitary transformation as H and H FW That is, the positive and negative energy eigenspaces are transformed via U into spaces of positive and negative Witten parity. In fact, one may verify that U may be represented in terms of these projection operators as follows The non-negative partner Hamiltonians H ± ≥ 0 are essential isospectral which means that their strictly positive eigenvalues are identical. The corresponding eigenstates are related to each other via a SUSY transformation. To be more explicit let us assume these are given by then the SUSY transformation reads [23] Note that the energy eigenvalue ε may be degenerate and above relations are valid for each of these energy eigenstates. We omit an additional index in φ ± ε enumerating such a degeneracy. In addition, both partner Hamiltonians H ± may have a non-trivial kernel, that is, there may exist one or several eigenstates with In this case SUSY is said to be unbroken [23]. For these ground states, again we omit the index for a possible degeneracy, there exists no SUSY transformation relating φ + 0 and φ − 0 . The breaking of SUSY can be studied via the so-called Witten index ∆ [23], which in the current context is identical to the Fredholm index of A, if it is a Fredholm operator, that is, Obviously a non-vanishing Witten index indicates that SUSY is unbroken. In connection with [22] dim ker (Q the kernels of H ± , that is, the number of zero-energy states of H ± are known. In general, however, the 68 operator A is not Fredholm and hence some regularized indices are studied [22,23].

69
Due to the SUSY condition (9) the mass operators commutes with the associated partner Hamiltonians, [M ± , H ± ] = 0, and therefore have an identical set of eigenstates. Let us denote the corresponding eigenvalues of M ± by m ± c 2 , that is then obviously the eigenvalues and eigenstates of (14), are given by Here let us note that the mass eigenvalues may depend on the energy eigenvalues, m ± = m ± (ε). Ψ ± ε = U † ψ ± ε having the same eigenvalues E ± given above. Hence, the eigenvalue problem of a It will turn out in the examples to be discussed below that the partner Hamiltonians H ± and the mass operators M ± are in essence represented by a non-relativistic Schrödinger-like Hamiltonian H NR and/or some constant operator. To be more precise we will show for all three case s = 0, 1 2 and 1 discussed below that the FW-transformed relativistic Hamiltonian takes the form with H NR representing the associate non-relativistic Hamiltonian as

79
In below subsections we will consider a relativistic charged particle with charge e and mass m in an 80 external magnetic field B := ∇ × A characterised by a vector potential A. The symbol π stands for the 81 kinetic momentum, that is, π := p − e A/c, where c denotes the speed of light. We will consider the case 82 of a scalar particle with spin s = 0, a Dirac particle with s = 1 2 and a vector boson having spin s = 1.
Comparing this with the general form (8) we may identify the operators M ± and A as follows.
Obviously these operators are identical up to an additional constant M ± = A + mc 2 and hence the condition (9), that is [M ± , A] = 0 is trivially fulfilled. In other words the Klein-Gordon Hamiltonian (30) for a charged scalar particle in the presence of an arbitrary magnetic field represents a supersymmetric relativistic spin-zero Hamiltonian. Let us note that the two operators (31) in essence are given by the Landau Hamiltonian H L := ( p − e A/c) 2 /2m = π 2 /2m of a non-relativistic spinless charged particle of mass m in a magnetic field, that is, M ± = H L + mc 2 and A = H L . As a consequence the eigenvalue problem for (30) is reduced to that of H L . The FW transformed Hamiltonian explicitly reads Let us denote the eigenvalues of H L by ǫ then the eigenvalues of M ± and H ± = H 2 L /2mc 2 read m ± c 2 = mc 2 + ǫ , ε = ǫ 2 2mc 2 (33) and via relation (27) we find the eigenvalues of the Klein-Gordon Hamilonian As ǫ ≥hω c /2 > 0 so is ε > 0 and hence dim ker H − = dim ker H − = 0. In other words, the Witten index  The Dirac equation representing the relativistic dynamics of spin-1 2 fermions has intensively been studied since its introduction. See for example the excellent book by Thaller [22]. For a charged particle in an arbitrary external magnetic field the Dirac Hamiltonian reads where σ = (σ 1 , σ 2 , σ 3 ) T stands for a three-dimensional vector who's components are given by the Pauli matrices acting on C 2 , thus representing the spin-1 2 degree of freedom. Comparing this with the general form (8) we may identify the operators M ± = mc 2 and A = cσ · π = A † and note that condition (9) is trivially fulfilled. Hence, the Dirac Hamiltonian (36) is indeed supersymmetric and its FW transformed form is know [25] to be expressible in terms of the non-relativistic Pauli Hamiltonian for a spin-1 2 particle with Landé g-factor g = 2.
Obviously the partner Hamiltonians can be identified with the Pauli Hamiltonian H ± = A 2 /2mc 2 = H P and we find for the FW transformed Dirac Hamiltonian for this magnetic field provides an additional SUSY structure within both subspaces H ± .
Finally, for an electron (e < 0) in a constant magnetic field the eigenvalues of H P are well-known E ± = ± m 2 c 2 +h 2 c 2 k 2 z + 2mc 2h ω c (n + 1/2 + s z ) .
The degeneracy for each set of quantum numbers (n, s z , k z ) is given by the largest integer which is stricly 93 less than |F|/Φ 0 and is only finite in case the magnetic field has a compact support.  The Hamiltonian of a charged spin-one particle with a priori arbitrary g-factor is given by, see for example [30,35,36], where S = (S 1 , S 2 , S 3 ) T is a vector who's components are 3 × 3 matrices acting on C 3 and obeying the SO(3) algebra [S i , S j ] = iε ijk S k representing the spin-one-degree of freedom of the particle. Again we may identify the operators From now on let us assume that the magnetic field B is constant, i.e. A = 1 2 B × r. Under this condition one may verify that Hence, the condition (9) is fulfilled if g = 2. In other words the relativistic spin-one Hamiltonian (41) is a supersymmetric Hamiltonian if the gyromagnetic factor is given by g = 2. For a detail discussion we refer to the recent paper [37]. Here we remark that the "Vector Boson" Hamiltonian represents the non-relativistic Hamiltonian of a charged spin-one particle in a magnetic field with gyromagnetic factor g = 2. Note that (44) The eigenvalues of (44) are given by (we assume e < 0) Hence the spectrum of the partner Hamiltonians H ± is given by ε = ǫ 2 /2mc 2 and SUSY is unbroken as ǫ = 0 for n = 0, s z = −1 and k z = ±1/λ L with λ L := √h /mω c = hc/|eB| being the Lamor length [35].
That is, SUSY is unbroken for a spin-1 particle in a homogeneous magnetic field but the Witten index remains zero as H + = H − and therefore dim ker H + = dim ker H − . The corresponding eigenvalues of (41) are given by which is identical in form to the Dirac case (40) but s z now taking the integer values as given in (46). In 103 fact, for k z = 0, n = 0 and s z = −1 above eigenvalues would become complex if |B| > m 2 c 3 /|e|h. This 104 limit would imply λ L < λ C :=h/mc, that is, the Lamor wavelength being smaller than the Compton 105 wavelength of the vector boson. Note that confining a quantum particle to a region of the order of its 106 Compton wavelength ∆x ∼ λ C implies by the uncertainty relation a momentum fluctuation ∆p ∼ mc 107 and thus a single particle description is no longer appropriate. In other words for such large magnetic 108 fields a description via quantum field theory must be applied.

110
In this section we want to study the resolvent or Green's function of supersymmetric relativistic arbitrary-spin Hamiltonians defined as For this is it convenient to first look at the iterated resolvent which is given by and is related with (48) via the obvious relation As H 2 is block-diagonal so is g and hence it can be put into the form As a result the resolvent (48) can be expressed in terms of (51) as follows.
In the following subsections we will now explicitly consider the three cases discussed in the previous section. It will turn out that for these three cases the diagonal elements g ± of the iterated Green's function can be expressed in terms of the Green's function of the corresponding non-relativistic Hamiltonian H NR , that is, Note that the relation ξ = z 2 /2mc 2 − mc 2 /2, which can be put into the form z = ±mc 2 1 + 2ξ/mc 2 , in 111 essence reflects the relation (28). 112

The resolvent of the Klein-Gordon Hamiltonian with magnetic field 113
Following the discussion of the first example of section 4 we may express all relevant operators in terms of the Landau Hamiltonian H L = π 2 /2m. Explicitly we have which results in the iterated resolvents where G L stands for the Green function of the Landau Hamiltonian in terms of which the Klein-Gordan Hamiltonian reads The Green's function then reads in terms of the Landau Hamiltonian (57)

The resolvent of the Dirac particle in a magnetic field
114 As in the above discussion let us first recall the observations made in section 4.2, that is, which provide us with the components of the iterated kernel where G P (ǫ) := (H P − ǫ) −1 is the resolvent of the non-relativistic Pauli Hamiltonian. In terms of this Pauli Green's function and the spin projection operator A the Dirac Green's function can be put into the form Some explicit examples have been worked out in ref. [25].
where the vector Hamilton H V is given in eq. (44). Recalling that A 2 = H 2 V we find for the iterated Green's functions The relativistic spin-one Hamiltonian explicitly reads and leads us to the Green's function (64)

117
In this work we have considered relativistic one-particle Hamiltonians for an arbitrary but fix spin s and have shown that under the condition, that its even part commutes with its odd part, a SUSY structure can be established. Here the SUSY transformations map states of negative energy to those of positive energy and vice versa. This is different to the usual SUSY concepts in quantum field theory where those charges transform bosonic into fermionic states and vice versa. As examples we have chosen the physically most relevant cases of a massive charged particle in a magnetic field for the cases of a scalar particle (s = 0), a Dirac fermion (s = 1/2) and a vector boson (s = 1). In the case of a constant magnetic field SUSY is broken for s = 0 but remains unbroken for s = 1/2 and s = 1. The Witten index is only non-zero in the Dirac case but vanishes for the bosonic cases discussed. However, all three cases have resulted in the notable observation (28) that the FW-transformed Hamiltonian H FW is entirely expressible in terms of a corresponding non-relativistic Hamiltonian H NR . As H 2 FW = H 2 the relativistic energy-momentum relation can be put into the form which allows us to related H NR with the SUSY Hamiltonian (11).

118
There naturally arises the desire to also study the higher-spin cases s ≥ 3/2. Hamiltonians are local operators.

122
Another route for further investigation would be to consider more exotic magnetic fields. For 123 example, choosing an imaginary vector potential such that the kinetic momentum takes the form 124 π = p + imω r in essence leads for s = 1/2 to the so-called Dirac oscillator, which is know to exhibit 125 such a SUSY structure [25]. To the best of our knowledge the corresponding Klein-Gordon and vector In this appendix we present a few relations which provide some additional steps used in section 4.3. For an arbitrary magnetic field let us recall that the components of the kinetic momentum given by π j = p j − (e/c)A j obey the commutation relation [π k , π l ] = (ihe/c) ε klm B m (A1) where we use Einstein's summation convention for repeated indices. From this relation one may derive the commutator [π k , S · π] = (ihe/c) ε klm S l B m which in turn leads us to π 2 , S · π = (eh/c)[ S · B, S · π] + (eh/c)S k S l (π l B k − B l π k ) .
For an arbitrary magnetic field the components of the kinetic momentum do not commute with the components of the magnetic field. However, if we now assume that the magnetic field is constant one may commute in the last term these components. That is, under the assumption that B = const. we arrive at π 2 , S · π = (2eh/c)[ S · B, S · π] , (A3) which in turn results in π 2 , ( S · π) 2 = 2eh c ( S · B), ( S · π) 2 . (A4) Note that relation (A3) was already given in eq. (3.17) of ref. [35]. With the help of (A4) it is easy to calculate the commutator Noting that we have derived this under the assumption of a constant magnetic field the last commutator 134 in above expression vanishes and hence we arrive at eq. (43). Note that [S k , S l ] = iε klm S m and therefore 135 the first term on the right-hand-side above even for a constant magnetic field only vanishes when g = 2.

136
With the assumption that the magnetic field is constant and utilising below properties of the spin-one matrices S i S j S k + S k S j S i = δ ij S k + δ jk S i , ε ijk S i S j B k = i S · B ( S · π) 4 = π 2 − 2eh c ( S · B) ( S · π) 2 + eh c ( B · π)( S · π) , ( S · B), ( S · π) 2 = π 2 − eh c ( S · B) ( S · B) + ( B · π)( S · π) . (A7) Noting that for g = 2 we have and with above relations (A7) immediately follows that A 2 = H 2 V as claimed in the main text. Finally, let 137 us mention the explicit form of the energy eigenfunctions can also be found in ref. [35].