For Quantum-mechanics on a Hilbert-space over the complex numbers, the usual scalar product of two states $\langle \phi | \psi \rangle$ and gives the transition amplitude between the two states. The absolute square of this quantity then gives the probability that a particular value associated with $|\phi \rangle$ can be measured when the system is in state $| \psi \rangle$.
However, when one constructs states over super-numbers (for example fermionic coherent states), those states do have supernumbers as coefficients, and thus the scalar product yields a super-number as well.
Can this super-numbers still be used as a transition-amplitude?
For example, in a 2 state-system: $$ |\theta \rangle = | 0 \rangle - \theta | 1 \rangle \\ $$ then $$ \langle 0 |\theta \rangle = 1 \\ \langle 1 | \theta \rangle = - \theta. $$ How would we proceed from here?
- The absolute square would be $ \bar{\theta} \theta $, which is grassmann even - or would it be $\theta \theta = 0$?
- If the square is zero, does that mean that fermionic coherent states essentially are overlapping with the vacuum state?
- Is the concept of transition probabilities simply not defined for states over super numbers?
- If so, could it in principle be defined in a consistent way?
