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I see this statement in many QFT books (e.g. Altland & Simons' Condensed Matter Field Theory) but the author never explains why.

Can you briefly explain why Grassmann fields never have a classical meaning (preferably physical arguments) and possibly point out some good references?

Qmechanic
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Medulla Oblongata
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  • The classical Dirac field is sometimes considered a complex field, sometimes a Grassman field. In the second case, it is simpler to quantize https://doi.org/10.1016/j.shpsb.2019.10.003 – Quillo Feb 14 '24 at 10:39

3 Answers3

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Well, it depends on what is meant by the word classical.

  1. Usually in physics, classical theories mean theories where Planck's constant $\hbar$ is zero. If that's what is meant, then there certainly exist classical Grassmann-odd variables/fermions and a (in that sense) classical notion of supermanifolds. See e.g. this Phys.SE post.

  2. What the authors may refer to is the weird nature of Grassmann-odd variables. Examples:

    • It is not possible to physically measure a Grassmann-odd variable in some detector.

    • There is no non-trivial topology in Grassmann-odd directions.

    • It is not possible to define a definite Berezin integral on a proper subset.

    • See also e.g. this Phys.SE post.

Qmechanic
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Let me put things in broader perspective. Bosonic fields are quantized in terms of commutators with a prefactor $\hbar$. Classical limit leads to commuting variables that may be represented by complex numbers. This is by the way the first step to devise numerical applications of path integrals. Fermionic fields require anticommutators ( consider 4d to focus and avoid special features of low dimension ) and formal $\hbar\to 0$ gives Grassmann algebra, say, $\{\theta_i, \theta_j\}=0$ and these variables cannot be represented as complex numbers ( just take $i=j$) in any simple way.

Matteo Beccaria
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I have proposed a classical interpretation for a pair of Dirac fermions in arxiv:0908.0591. It is a lattice version, works only for a pair, not a single fermion, requires a preferred frame, and the Hamilton operator for the classical $\mathbb{Z}_2$-valued field is only an approximation of the lattice Dirac operator.

The $\mathbb{Z}_2$-valued field can be easily embedded into an $\mathbb{R}$-valued field with degenerate vacuum, which would add a massive scalar field to each pair of Dirac fermions.

Schmelzer
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