Quantum states as L^2 functions

[schieghoven]

Hello,

What is a quantum state? Put generalised functions/Schwartz distributions to one side, because a) they're not a Hilbert space, and b) they can't be multiplied, so it's hopeless to even begin to think about Feynman diagrams.

One-particle quantum states seem to be fairly well understood. The state of the system is a function $\psi: \mathbb{R}^3 \rightarrow \mathbb{C}$, and $$|\psi(x)|^2$$ gives the probability density of finding the particle near the space point x. Let's denote by $\Omega_1$ the space of one-particle states. $\Omega_1$ is a Hilbert space with inner product

$$\langle \psi, \phi \rangle = \int d^3x \; \psi^*(x) \phi(x) \qquad \qquad (1)$$​

This Hilbert space is known as $L^2(\mathbb{R}^3)$. The states $$\psi, \phi \in \Omega_1$$ evolve in time according to an equation of motion, and the the inner product (1) is constant in time. Equivalently, the system evolves by a unitary transformation on $\Omega_1$.

Building on this, an n-particle state $$\psi$$ is presumably a function of n space points (x1, x2, ... xn). Assume Bose symmetry, so $$\psi$$ is totally symmetric with respect to x1, x2, ... xn. So in this case $\psi: (\mathbb{R}^3)^n \rightarrow \mathbb{C}$ and $$|\psi(x_1, x_2, \ldots, x_n)|^2$$ gives the probability density of finding the n particles near the space points x1, x2, ..., xn. The set $\Omega_n$ of all n-particle states has a canonical inner product

$$\langle \psi, \phi \rangle = \frac{1}{n!} \int d^3x_1 \ldots d^3x_n \; <br /> \psi^*(x_1, \ldots, x_n) \phi(x_1, \ldots, x_n)$$​

and is a Hilbert space. So far so good. This is not just rigorous - L^2 spaces are stock concepts in pure math - but it's readily understandable as well. States have a direct physical interpretation at all times, not just at asymptotic [itex]t \rightarrow \pm \infty [/tex]. So is it possible to formulate field theory from this standpoint? In field theory, particle number changes with time, so let's suppose the set of all states is<br /> <div style="text-align: center">$$\Omega = \mathbb{C} \oplus \Omega_1 \oplus \Omega_2 \oplus \ldots <br /> \oplus \Omega_n \oplus \ldots$$​</div>(The C is for the vacuum.) Is it possible to define the dynamics of the system in terms of an equation of motion for the n-particle 'wavefunctions'? Would this be another route towards constructing the Feynman series?<br /> <br /> Cheers,<br /> <br /> Dave[/itex]

 

[DarMM]

States have a direct physical interpretation at all times, not just at asymptotic [itex]t \rightarrow \pm \infty [/tex]. So is it possible to formulate field theory from this standpoint? In field theory, particle number changes with time, so let's suppose the set of all states is<br /> <div style="text-align: center">$$\Omega = \mathbb{C} \oplus \Omega_1 \oplus \Omega_2 \oplus \ldots<br /> \oplus \Omega_n \oplus \ldots$$​</div>(The C is for the vacuum.) Is it possible to define the dynamics of the system in terms of an equation of motion for the n-particle 'wavefunctions'? Would this be another route towards constructing the Feynman series?<br /> <br /> Cheers,<br /> <br /> Dave[/itex]

$In general no. The Hilbert space in quantum field theory only has a decomposition as a tower of particle states for the free theory alone.$