任意维数下的费米子化
Fermionization in an Arbitrary Number of Dimensions
摘要：
本文的一个目的是表明，至少对于自由无质量粒子，有可能构造一个明确的玻色子理论，它在动量和能量方面与费米子理论完全等价。费米子是以$2^{d/2-1}$族的形式出现的，而与整个费米子系统相对应的玻色子则是一系列完整的卡尔布-雷蒙场，即张量场上每一个指数对应的一组分量。由于Kalb-Ramond场自然地（仅）耦合到扩展的对象或膜，我们怀疑，对于玻色化预备系统来说，不包括膜或类似的东西而将相互作用包含在其中--除了最低维之外--是不可能的。只要使用很久以前由Aratyn和我们中的一个（H.B.F.N.）提出的定理，就很容易看出对家庭的需要，这就是说，为了使费米子系统和玻色子系统的统计力学相匹配，需要使场分量的数量与$\frac{2^{d-1}-1}{2^{d-1}}= \frac{\# bosons}{\# fermions}$成比例，强制要求费米子分量的数量必须是$2^{d-1}$的倍数，其中$d$是时空维度。这种对维数的“解释”对于我们中的一个人（S.N.M.B.）提出的对维数的解释是潜在有用的。因为在自旋-电荷-族理论中很长，并且像后者一样导致典型的（多个）$4$族。这是我们研究任意维数下的费米子化的第二个目的--也就是了解自旋-电荷族理论中包含的族是多么“自然”。
英文摘要：
One purpose of this proceedings-contribution is to show that at least for free massless particles it is possible to construct an explicit boson theory which is exactly equivalent in terms of momenta and energy to a fermion theory. The fermions come as $2^{d/2-1}$ families and the to this whole system of fermions corresponding bosons come as a whole series of the Kalb-Ramond fields, one set of components for each number of indexes on the tensor fields. Since Kalb-Ramond fields naturally (only) couple to the extended objects or branes, we suspect that inclusion of interaction into such for a bosonization prepared system - except for the lowest dimensions - without including branes or something like that is not likely to be possible. The need for the families is easily seen just by using the theorem long ago put forward by Aratyn and one of us (H.B.F.N.), which says that to have the statistical mechanics of the fermion system and the boson system to match one needs to have the number of the field components in the ratio $\frac{2^{d-1}-1}{2^{d-1}}= \frac{\# bosons}{\# fermions}$, enforcing that the number of fermion components must be a multiple of $2^{d-1}$, where $d$ is the space-time dimension. This "explanation" of the number of dimension is potentially useful for the explanation for the number of dimension put forward by one of us (S.N.M.B.) since long in the spin-charge-family theory, and leads like the latter to typically (a multiple of) $4$ families. And this is the second purpose for our work on the fermionization in an arbitrary number of dimensions - namely to learn how "natural" is the inclusion of the families in the way the spin-charge-family theory does.
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