First, let me emphasize that position and momentum eigenstates do not belong to $L^2$. Moreover the canonical state for a LPS has not Hilbert space norm.

The fundamental answer to your question is encoded in the underlying phase space structure. In the phase space formulation of quantum mechanics the state of a system is given by a function $F(p,q;t)$ of which only normalization is required; nothing is said about the integral of its square (or any other properly defined scalar product with itself). Normalization can be understood in physical terms (probabilities) or, mathematically, using the unit element relation $\langle 1 \rangle = 1$.

Averages of dynamical quantities are obtained as the product of dynamical phase space functions $b(p,q;t)$ and states $F(p,q;t)$. This in a kind of Banach space with dynamical functions playing the role of 'bras' and states the role of 'kets'. In fact, the phase space average can be rewritten as $\langle \langle b(p,q;t) | F(p,q;t) \rangle \rangle$.

The Hilbert space and the $L^2$ structure can be derived from the underlying phase space by introducing a decomposition of the state $F(p,q;t)$ in complex-valued amplitudes $\Psi(q;t)$. Notice that the phase representation is more general and accounts for mixed quantum states, which are not described by any $\Psi(q;t)$.

First, let me emphasize that position and momentum eigenstates do not belong to $L^2$. Moreover the canonical state for a LPS has not Hilbert space norm.

The fundamental answer to your question is encoded in the underlying phase space structure. In the phase space formulation of quantum mechanics the state of a system is given by a function $F(p,q;t)$ of which only normalization is required; nothing is said about the integral of its square (or any other properly defined scalar product with itself). Normalization can be understood in physical terms (probabilities) or, mathematically, using the unit element relation $\langle 1 \rangle = 1$.

Averages of dynamical quantities are obtained as the product of dynamical phase space functions $b(p,q;t)$ and states $F(p,q;t)$. This in a kind of Banach space with dynamical functions playing the role of 'bras' and states the role of 'kets'. In fact, the phase space average can be rewritten as $\langle \langle b(p,q;t) | F(p,q;t) \rangle \rangle$.

The Hilbert space and the $L^2$ structure can be derived from the underlying phase space by introducing a decomposition of the state $F(p,q;t)$ in complex-valued amplitudes $\Psi(q;t)$.

$$ \langle \langle 1 | F(p,q;t) \rangle \rangle = \langle \Psi(q;t) | \Psi(q;t) \rangle $$

Notice that the phase structure is more general than Hilbert and $L^2$ spaces and accounts for mixed quantum states, which are not described by any $\Psi(q;t)$.

First, let me emphasize that position and momentum eigenstates do not belong to $L^2$. Moreover the canonical state for a LPS has not Hilbert space norm.

The fundamental answer to your question is encoded in the phase space structure. In the phase space formulation the state of a system is given by a function $F(p,q:t)$ of which only normalization is required; nothing is said about the integral of its square (or any other properly defined scalar product with itself). Normalization can be understood in physical terms of probabilities or mathematically from the unit relation $\langle 1 \rangle = 1$.

Averages of dynamical quantities are obtained as the product of dynamical phase space functions $b(p,q:t)$ and states $F(p,q:t)$. This in a kind of Banach space with dynamical functions playing the role of 'bras' and states the role of 'kets'. In fact, the phase space average can be rewritten as $\langle \langle b(p,q:t) | F(p,q:t) \rangle \rangle$.

The Hilbert space and the $L^2$ structure can be derived from the phase space by introducing a decomposition of the state $F(p,q:t)$ in complex-valued amplitudes.

First, let me emphasize that position and momentum eigenstates do not belong to $L^2$. Moreover the canonical state for a LPS has not Hilbert space norm.

The fundamental answer to your question is encoded in the underlying phase space structure. In the phase space formulation of quantum mechanics the state of a system is given by a function $F(p,q;t)$ of which only normalization is required; nothing is said about the integral of its square (or any other properly defined scalar product with itself). Normalization can be understood in physical terms (probabilities) or, mathematically, using the unit element relation $\langle 1 \rangle = 1$.

Averages of dynamical quantities are obtained as the product of dynamical phase space functions $b(p,q;t)$ and states $F(p,q;t)$. This in a kind of Banach space with dynamical functions playing the role of 'bras' and states the role of 'kets'. In fact, the phase space average can be rewritten as $\langle \langle b(p,q;t) | F(p,q;t) \rangle \rangle$.

The Hilbert space and the $L^2$ structure can be derived from the underlying phase space by introducing a decomposition of the state $F(p,q;t)$ in complex-valued amplitudes $\Psi(q;t)$. Notice that the phase representation is more general and accounts for mixed quantum states, which are not described by any $\Psi(q;t)$.