In quantum field theory, the dot product of different multi-particle states is not always zero. It depends on the specific states involved and the properties of the particles being considered. The dot product, or inner product, between two states is determined by the wave functions or field operators associated with those states.
In general, the inner product of two multi-particle states in quantum field theory is given by integrating the product of their complex conjugates over all of spacetime. For example, if we have two states represented by field operators ϕ(x) and ψ(y), the inner product between these states can be written as:
⟨ϕ(x) | ψ(y)⟩ = ∫ d³x d³y ϕ*(x) ψ(y),
where the integral is taken over all spatial coordinates (d³x and d³y) and ϕ*(x) denotes the complex conjugate of ϕ(x).
The value of this inner product is generally not zero unless the wave functions or field operators associated with the states are orthogonal, meaning they have no overlap. In certain cases, such as in the context of symmetries and conservation laws, specific states may have orthogonal properties, and their dot product may evaluate to zero. However, this is not a general property applicable to all multi-particle states in quantum field theory.
It is worth noting that the dot product of different states is crucial in determining transition amplitudes, probabilities, and correlation functions in quantum field theory. The specific values of these dot products play a significant role in understanding particle interactions, scattering processes, and other phenomena in the quantum field theory framework.