[Weinberg QFT] 2.1 Quantum Mechanics
This article is one of the posts in the Textbook Commentary Project.
This book shows that quantum field theory is the only possibility of quantum mechanics, including relativity theory. First, let's look at how the Lorentz transformation, the core of the theory of relativity, works in quantum mechanics.
Let's look at the axioms of quantum mechanics.
1. Physical states are represented by rays in Hilbert Space.
$\bullet$ A Hilbert Space $\mathcal{H}$ is complex vector space:
i) Closed: If $\Phi,\Psi \in \mathcal{H}$, then $\xi \Phi + \eta\Psi \in \mathcal{H}$ for arbitrary complex numbers $\xi, \eta$.
ii) Norm: If $\Phi, \Psi \in \mathcal{H}$, there is a complex number $\left\langle \Phi | \Psi \right\rangle$, such that
$$\left\langle \Phi | \Psi \right\rangle = \left\langle \Psi | \Phi \right\rangle ^*$$
$$\left\langle \Phi | \xi_1 \Psi_1 + \xi_2 \Psi_2 \right\rangle = \xi_1 \left\langle \Phi | \Psi_1 \right\rangle + \xi_2 \left\langle \Phi | \Psi_2 \right\rangle$$
$$\left\langle \eta_1\Phi_1 + \eta_2 \Phi_2 | \Psi \right\rangle = \eta_1^* \left\langle \Phi_1 | \Psi\right\rangle + \eta_2^* \left\langle \Phi_2 | \Psi \right\rangle$$
$$\left\langle \Psi | \Psi \right\rangle \ge 0$, vanish iff $\Psi=0$$ (positivity condition)
(Hidden assumption: limits of vectors within Hilbert space.)
$\bullet$ A ray $\mathcal{R}$ is a set of normalized vectors (i.e., $\left\langle \Psi | \Psi \right\rangle = 1$ ):
$$\Psi,\Psi'\in \mathcal{R}$ if $\Psi'=\xi\Psi$ with $\left| \xi \right| = 1$$.
2. Observables are represented by Hermitian Operators.
$\bullet$ A Hermitian Operators are mappings $\Psi\rightarrow A \Psi$ of Hilbert space into itself:
$$A(\xi \Psi + \eta \Phi) = \xi A \Psi + \eta A \Phi$$ (linear)
$A^\dagger = A$ (reality) ($\left\langle \Phi | A^\dagger \Psi \right\rangle \equiv \left\langle A\Phi | \Psi \right\rangle = \left\langle \Psi | A\Phi \right\rangle^*$)
(Hidden assumption: continuity of $A\Psi$ as a function of $\Psi$.)
$\bullet$ A state represented by a ray $\mathcal{R}$ has a definite value $\alpha$ for the observable represented by an operator $A$ if vectors $\Psi$ belonging to this ray are eigenvectors of $A$ with eigenvalue $\alpha$:
$$A\Psi = \alpha \Psi$ for $\Psi$ in $\mathcal{R}$$.
(Spectral theorem: $A$ Hermitian, then $\alpha$ is real, eigenvectors with different $\alpha$s are orthogonal.)
3. If a system is in a state represented by a ray $\mathcal{R}$, and after the experiment, system become one of the states represented as orthogonal rays $\mathcal{R}_1,\mathcal{R}_2,\cdots$, then the probability of finding system in state represented by $\mathcal{R}_n$ is
$$P(\mathcal{R} \rightarrow \mathcal{R}_n) = \left| \left\langle \Psi | \Psi_n \right\rangle \right|^2$$
where $\Psi$ and $\Psi_n$ are any vectors belonging to rays $\mathcal{R}$ and $\mathcal{R}_n$.
(A pair of rays is orthogonal if the state-vectors from the two rays have vanishing scalar products.)
$\bullet$ Total probability unitary:
$$\sum_n P(\mathcal{R}\rightarrow \mathcal{R}_n) = 1$$
if the state-vectors $\Psi_n$ form a complete set.