Consider a $Riemannian\space n-dimensional\space manifold$ with coordinates {$x^i$} $(i=1,...,n)$. Let the arc length parameter be $t$. So that $\frac{d}{dt}x^i(t)\equiv\dot{x}^i(t)$ is the usual tangent vector along the curve $x^i(t)$. On this manifold, we consider prescribing the following theory that comprises of $n$ bosonic variables and $n$ fermionic variables {$\psi^i$} and their conjugates {$\bar{\psi}^i$}:$$ L=\frac{1}{2}g_{ij}\dot{x}^i\dot{x}^j+\frac{i}{2}g_{ij}\bigl(\bar{\psi}^iD_t\psi^j-D_t\bar{\psi}^i\psi^j\bigr)-\frac{1}{4}R_{ijkl}\psi^i\psi^j\bar{\psi}^k\bar{\psi}^l. \tag{1}$$ The Riemann tensor terms are self explanatory and $D_t$ is the covariant derivative:$$ D_t\psi^i=\partial_t\psi^i+\dot{x}^j\Gamma^i_{jk}\psi^k. \tag{2}$$Query 1: I am having trouble understanding this covariant derivative. I run in to the following paradox. $D_t$ is just the directional derivative along a curve on the manifold. So I can use the chain rule to write:$$ D_t\psi^i=\dot{x}^jD_j\psi^i. \tag{3}$$ Now the Lagrangian is a scalar and hence I can deduce that the fermions with the raised indices must be vectors, for only then does the last term in (1) come out a scalar. So the raised indices on the fermions must be contravariant indices. Then using the rule for their covariant differentiation I have from the above relation:$$D_t\psi^i=\dot{x}^j\bigl(\partial_j\psi^i+\Gamma^i_{jk}\psi^k\bigr)=\dot{x}^j\partial_j\psi^i+\dot{x}^j\Gamma^i_{jk}\psi^k. \tag{4}$$ Apparently, the second last term above is just:$$\dot{x}^j\partial_j\psi^j=\partial_t\psi^i. \tag{5}$$ And so $(3)$ comes out consistent with $(2)$. What bothers me is that (from what I undertstand:) the fermions and bosons are independent entities, and so the bosonic derivative of the fermion must vanish:$$\partial_j\psi^i=0. \tag{6}$$That's the issue, for now $(2)$ and $(3)$ are no longer consistent with each other.
Query 2: Consider computing the partial derivatives of the Lagrangian, that is:$$\frac{\partial L}{\partial x^m},\frac{\partial L}{\partial \dot{x}^m},\frac{\partial L}{\partial \psi ^m},\frac{\partial L}{\partial \dot{\psi}^m}\frac{\partial L}{\partial\bar{\psi}^m},\frac{\partial L}{\partial \dot{\bar{\psi}}^m}. \tag{7}$$ Then one way of looking at this is to use $(2)$ in $(1)$ and proceed in the usual way, taking $x^i$, $\dot{x}^i$, $\psi^i$, $\dot{\psi}^i$, $\bar{\psi}^i$, $\dot{\bar{\psi}}^i$ to be independent entities, that is bosonic positions and velocities, and fermionic positions and velocities. The other is to view the fermionic covariant derivatives in $(1)$ as being entities that are independent of bosonic velocities $\dot{x}^m$ (for instance), so that the derivative operator for bosonic velocity $\frac{\partial}{\partial \dot{x}^m}$ will not enter the expression for $D_t\psi^i$ I mean would the $\dot{x}^m$ dependence in (2) count as an explicit dependence (so that $\frac{\partial}{\partial \dot{x}^m}$ enters) or an implicit dependence (so that $\frac{\partial}{\partial \dot{x}^m}$ does not enter)? Kindly help.


1 Answer 1

  1. The time derivative of the fermion is not a transport term (3), i.e. eq. (3) does not apply.

  2. The non-covariant partial derivatives (7) always work. See also e.g. this related Phys.SE post.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.