Supersymmetric quantum mechanics requires g=2 for vector bosons

There is some common agreement that the gyromagnetic ratio g of charged elementary particles when coupled to an electromagnetic field is g = 2. There are several reasonable arguments for this. The equation of motion for the spin vector, as shown by Bargmann, Michel and Telegdi [1], takes a very simple form when g = 2. As argued by Weinberg [2] and much later by Ferrara, Porrati and Telegdi [3], the requirement to have a good high-energy behaviour of scattering amplitudes, one must choose g = 2 for any spin. In 1994 Jackiw [4] showed that the value g = 2 also follows from a gauge symmetry in the case of a massless spin-1 field coupled to an electromagnetic field. This is of course in agreement with the standard model. Here the currently known electrically charged elementary particles are either spin-$\frac{1}{2}$ fermions or spin-1 bosons. The standard model indeed requires for all these charged particles a value g = 2 at the tree level. Whereas for the elementary fermions $(s=1/2)$ this assertion is consistent with the Belifante [5] conjecture $g=1/s$, it obviously disagrees with the case of vector bosons where s = 1 and the Belifante conjecture would imply g = 1. In fact, precision measurements at the Tevatron [6] resulted in the bounds $1.944 \leq g \leq 2.080$ at $95\%$ C.L. for the W boson. Hence, the case of spin-1 elementary particles is of particular interest as no massive and charged higher-spin elementary particles are known yet.

In 1992 Ferrara and Porrati [7] utilized an N = 1 supersymmetry algebra to show that "the gyromagnetic ratio of arbitrary-spin supersymmetric particles must be equal to 2". Whereas Ferrara and Porrati considered a supersymmetry (SUSY) where the supercharges transform bosons into fermions and vice versa, we consider in this brief report relativistic Hamiltonians for arbitrary spin, that is, characterising the relativistic dynamics of a standard particle with arbitrary but fixed spin. Under the assumption that the even and odd part of such a Hamiltonian commute, it is shown that one can construct a SUSY structure know from SUSY quantum mechanics. Here the SUSY charge transforms between positive and negative energy eigenstates. For this we first generalise the concept of a supersymmetric Dirac Hamiltonian to relativistic Hamiltonians for arbitrary spin $s=0,\frac{1}{2},1,\frac{3}{2},\ldots$. Then we consider the case of a charged particle with s = 1 interacting with an external constant magnetic field. Here the coupling to the spin-degree of freedom is a priori considered with arbitrary gyromagnetic ration g. It is shown that supersymmetric quantum mechanics requires g = 2.

The Hamiltonian of an arbitrary spin-s Hamiltonian can be put into the form [8]

$$\begin{equation} H=\beta {\cal M} + {\cal O}, \end{equation} \tag{ 1 }$$

which acts on the Hilbert space ${\cal H}=L^2(\mathbb{R}^3)\otimes\mathbb{C}^{2(2s+1)}$ whose elements are 2(2s+1)-dimensional spinors. The matrix β obeys the relation $\beta^2 = 1$ and may be represented as a block-diagonal matrix

$$\begin{equation} \beta = \left( \begin{array}{c@{\quad}c} 1 & 0 \\ 0 & -1 \end{array} \right)\!, \end{equation} \tag{ 2 }$$

where the 1 stands for the $2s+1$-dimensional unit matrix. Let us note that the two parts of the Hamiltonian (1) are chosen such that ${\cal M}$ accommodates all the even elements and ${\cal O}$ all the odd elements of H with respect to β. That is, we have the commutation and anti-commutation relations

$$\begin{equation} [\beta, {\cal M}]=0, \quad\{\beta, {\cal O}\}=0. \end{equation} \tag{ 3 }$$

Note that the Hamiltonian (1) is Hermitian, i.e., ${H=H^\dag}$, only for the case of fermions, and hence for half-odd-integer s. For bosons, where s is integer, the Hamiltonian is pseudo-Hermitian, i.e., $H=\beta H^\dag \beta$. Having this in mind it is straightforward to show that in the representation (2) both parts of H are necessarily of the form

$$\begin{equation} {\cal M} = \left(\begin{array}{c@{\quad}c} M_+ & 0 \\ 0 & M_- \end{array}\right),\quad {\cal O} = \left(\begin{array}{c@{\quad}c} 0 & A \\ (-1)^{2s+1}A^\dag & 0 \end{array}\right)\!, \end{equation} \tag{ 4 }$$

where $M_\pm{:}~{\cal H}_\pm\mapsto{\cal H}_\pm$ with $M^\dag_\pm= M_\pm$, $A{:}~{\cal H}_-\mapsto{\cal H}_+$ and $A^\dag{:}~{\cal H}_+\mapsto{\cal H}_-$. Here we have introduced the positive and negative energy subspaces ${\cal H}_+$ and ${\cal H}_-$, which are also eigenspaces of β for eigenvalue $+1$ and –1, respectively. Note that ${\cal H}={\cal H}_+\oplus{\cal H}_-$ and ${\cal H}_\pm=L^2(\mathbb{R}^3)\otimes\mathbb{C}^{2s+1}$. Well-know examples are the Klein-Gordon $(s=0)$ and the Dirac $(s=1/2)$ particle in a magnetic field [9].

Let us now assume that the odd and even parts of H commute, that is, $[{\cal M}, {\cal O}]=0$. This assumption implies

$$\begin{equation} M_+ A = A M_-,\qquad A^\dag M_+ = M_- A^\dag. \end{equation} \tag{ 5 }$$

From this condition it follows that the squared Hamiltonian (1) becomes block diagonal as the off-diagonal blocks vanish due to (5). That is

$$\begin{equation} H^2 = \left(\begin{array}{c@{\quad}c} M^2_+ + (-1)^{2s+1} AA^\dag& 0 \\ 0 & M^2_- +(-1)^{2s+1} A^\dag A \end{array}\right)\!, \end{equation} \tag{ 6 }$$

which allows us to define a SUSY structure in analogy to the Dirac case [10,11]. To be more explicit let us introduce the non-negative SUSY Hamiltonian

$$\begin{eqnarray} H_{\rm SUSY} &:= \displaystyle\frac{(-1)^{2s+1}}{2mc^2}\left( H^2-{\cal M}^2 \right)\nonumber\\ &= \displaystyle\frac{1}{2mc^2} \left(\begin{array}{c@{\quad}c} AA^\dag& 0\\ 0 A^\dag A \end{array}\right)\geq 0 \end{eqnarray} \tag{ 7 }$$

and the corresponding complex SUSY charge

$$\begin{equation} Q:=\frac{1}{\sqrt{2mc^2}} \left(\begin{array}{c@{\quad}c} 0 & A \\ 0 & 0 \end{array}\right)\!,\quad Q^\dag =\frac{1}{\sqrt{2mc^2}} \left(\begin{array}{c@{\quad}c} 0 & 0 \\ A^\dag & 0 \end{array}\right)\!. \end{equation} \tag{ 8 }$$

It is now obvious that these operators together with the Z₂-grading (or Witten) operator $W:=\beta$ obey the SUSY algebra

$$\begin{equation} \hspace{-8pt}\begin{array}{l} H_{\rm SUSY} \!=\!\{Q,Q^\dag\},\quad \{Q,W\}\!=\!0,\quad Q^2=0=(Q^\dag)^2,\\ \left[ W,H_{\rm SUSY} \right] = 0,\quad W^2 =1. \end{array} \end{equation} \tag{ 9 }$$

Hence, an arbitrary spin Hamiltonian of the form (1) obeying the condition (5) may be called a supersymmetric arbitrary-spin Hamiltonian. This is consistent with the usual definition [10,11] of a supersymmetric Dirac Hamiltonian in the case $s=\frac{1}{2}$. Let us note that condition (5) also implies that ${\cal M}$ commutes with all operators of the algebra (9).

It is interesting to note that the condition ${[{\cal M}, {\cal O}]=0}$ in addition implies that there exists an exact Foldy-Wouthuysen transformation [12–14]

$$\begin{equation} U:= \frac{|H|+\beta H}{\sqrt{2H^2+2{\cal M}|H|}}=\frac{1+\beta {\rm sgn} H}{\sqrt{2+\{{\rm sgn}H,\beta\}}}, \end{equation} \tag{ 10 }$$

 ${\rm sgn}H:=H/\sqrt{H^2}$, which brings the Hamiltonian into a block diagonal form, cf. eq. (6),

$$\begin{equation} H_{\rm FW}:= U H U^\dag = \beta \sqrt{H^2}=\beta |H|. \end{equation} \tag{ 11 }$$

As a side remark we mention that the two operators

$$\begin{equation} B_\pm:=\frac{1}{2}\left[1\pm \beta \right],\quad \Lambda_\pm:= \frac{1}{2}\left[1\pm {\rm sgn}H \right] \end{equation} \tag{ 12 }$$

are projection operators onto the subspaces ${\cal H}^\pm$ of positive and negative eigenvalues of β and H, respectively, and they are related to each other via the unitary relation [15]

$$\begin{equation} B_\pm = U \Lambda_\pm U^\dag. \end{equation} \tag{ 13 }$$

That is, the positive and negative energy eigenspaces are transformed via U into eigenspaces of positive and negative eigenvalues of $\beta = W$, cf. eq. (11). Note that we may express U in terms of $B_\pm$ and $\Lambda_\pm$ as follows:

$$\begin{equation} U = \frac{B_+\Lambda_+ +B_-\Lambda_-}{\sqrt{(B_+\Lambda_+ +B_-\Lambda_-)(\Lambda_+B_+ +\Lambda_-B_-)}}. \end{equation} \tag{ 14 }$$

Let us now consider the case of a vector boson with charge e and mass m interacting with a constant external magnetic field $\vec{B}$ characterised via the vector potential $\vec{A}=\frac{1}{2}\vec{B}\times\vec{r}$. The corresponding Hamiltonian is given by

$$\begin{equation} H=\left(\begin{array}{c@{\quad}c} M_+ & A \\ -A & -M_- \end{array}\right)\!, \end{equation} \tag{ 15 }$$

where

$$\begin{equation} \begin{array}{l} \displaystyle M_\pm:= mc^2 + \displaystyle \frac{\vec{\pi}^2}{2m}-\frac{ge\hbar}{2mc}(\vec{S}\cdot\vec{B}),\\ A:= \displaystyle\frac{\vec{\pi}^2}{2m}-\frac{1}{m}(\vec{S}\cdot\vec{\pi})^2+\frac{(g-2)e\hbar}{2mc}(\vec{S}\cdot\vec{B}) =A^\dag \end{array} \end{equation} \tag{ 16 }$$

and $\vec{\pi}:= \vec{p}-e\vec{A}/c$. Here $\vec{S}=(S_1,S_2,S_3)^T$ is a vector whose components are $3\times 3$ matrices obeying the SO(3) algebra $[S_i,S_j]={\rm i} \varepsilon_{ijk}S_k$ representing the spin-one-degree of freedom of the particle, that is $\vec{S}^2=2$ as the spin s = 1 for a vector boson. In the above Hamiltonian we have introduced an arbitrary gyromagnetic factor g which describes the coupling of this spin degree of freedom to the magnetic field $\vec{B}$. Note that the above Hamiltonian was, to the best of our knowledge, first derived in 1940 by Taketani and Sakata [16] with g = 1. At the same time Corben and Schwinger [17] had studied the electromagnetic properties of mesotrons and concluded that g = 2 is required to have a singularity-free theory. For later work using the above Hamiltonian with both g = 1 and g = 2 as well as arbitrary values for g see, for example, refs. [18–23].

Let us now investigate if the above Hamiltonian (15) together with (16) does form a supersymmetric relativistic spin-1 Hamiltonian. For this we recall the relation [22]

$$\begin{equation} [ \vec{\pi}^2,(\vec{S}\cdot\vec{\pi})^2] =\frac{2e\hbar}{c} [(\vec{S}\cdot\vec{B}),(\vec{S}\cdot\vec{\pi})^2], \end{equation} \tag{ 17 }$$

which allows us to explicitly calculate the commutator

$$\begin{equation} [M_\pm,A]=(g-2)\frac{e\hbar}{2m^2c}[(\vec{S}\cdot\vec{B}),(\vec{S}\cdot\vec{\pi})^2]. \end{equation} \tag{ 18 }$$

Obviously for a non-vanishing magnetic field the SUSY condition (5) is only fulfilled when g = 2. In other words, when we require that the relativistic Hamiltonian for a massive charged spin-1 particle in a magnetic field is a supersymmetric Hamiltonian in the above sense, only g = 2 is allowed. This is indeed similar to the argument [11] that the phenomenological non-relativistic Pauli-Hamiltonian for a charged spin-$\frac{1}{2}$ fermion exhibits a SUSY structure only when g = 2.

In a final remark let us note that the above SUSY structure allows to reduce the eigenvalue problem of a supersymmetric arbitrary-spin Hamiltonian to that of a non-relativistic Hamiltonian H_s. As we will show elsewhere [24], for a charged particle in the constant magnetic field $\vec{B}$ the FW-transformed Hamiltonian (11) for the cases s = 0, $s=\frac{1}{2}$ and s = 1 takes the form

$$\begin{equation} H_{\rm FW}=\beta mc^2\sqrt{1+\frac{2H_s}{mc^2}}, \end{equation} \tag{ 19 }$$

where

$$\begin{equation} \begin{array}{rcl} H_0 &:=& \displaystyle \frac{1}{2m}(\vec{p}-e\vec{A}/c)^2,\\ H_\frac{1}{2}&:= &\displaystyle \frac{1}{2m}(\vec{p}-e\vec{A}/c)^2-\frac{e\hbar}{mc}(\vec{\sigma}\cdot\vec{B}),\\ H_1 &:=&\displaystyle \frac{1}{2m}(\vec{p}-e\vec{A}/c)^2-\frac{e\hbar}{mc}(\vec{S}\cdot\vec{B}). \end{array} \end{equation} \tag{ 20 }$$

Obviously, H₀ and $H_\frac{1}{2}$ represent the well-known non-relativistic Landau and Pauli-Hamiltonian, respectively, and H₁ is the correct non-relativistic Hamiltonian for a spin-1 particle in a magnetic field with g = 2. In the above $\vec{\sigma}$ is a vector whose components consist of Pauli's $2\times 2$ matrices representing the spin-$\frac{1}{2}$ degree of freedom.

The purpose of this short note is two-fold. First we generalised the concept of supersymmetric relativistic Hamiltonians known from the Dirac Hamiltonian with $s=\frac{1}{2}$ to the general case of arbitrary s. More explicitly, under the condition (5) it was shown that a SUSY structure, cf. eqs. (7)–(9), can be accommodated. Second, by considering a massive charged vector boson, i.e., s = 1, in the presence of a constant magnetic field, this system resembles a SUSY structure if and only if its gyromagnetic factor g = 2.

I have enjoyed enlightening discussions with Mikhail Plyushchay for which I am very grateful.