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.
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