A bit of everything

# Path integrals and quantum mechanics

There are many texts in quantum mechanics, and as such many explanations for the motivation of path integrals. We won't try to motivate them here (at least from the start), but start directly from the axiom set and later on show why this will fit quantum mechanics. As a brief reminder, our basic necessity here is that, for a configuration space $C$, we want to get some quantity such that, for $\varphi \in C$,

\begin{equation} \langle F[\varphi] \rangle = \int_{C} \mathcal{D}\varphi F[\varphi] e^{- S[\varphi]} \end{equation}

In physics, this is usually done via some limit procedure

\begin{equation} \int_{C} \mathcal{D}\varphi F[\varphi] e^{- S[\varphi]} = \lim_{\varepsilon \to 0} \int F[\varphi] e^{-S[\varphi]} (\prod_{i = -\infty}^\infty dx^i) \end{equation}

Hopefully, we will get back a similar formula.

## 1. Functional measures

### 1.1 Function spaces

The starting point of path integrals is the definition of a measure on a function space. Function spaces cover quite a wide variety of spaces, but here we'll consider the case of (infinite-dimensional) Banach spaces, ie, a vector space $X$ (over a field $K$, usually $\mathbb{R}$ or $\mathbb{C}$) equipped with a norm $\| \cdot \|_X$ such that, for every Cauchy sequence $\{ x_n \}$, there exists an $x \in X$ such that

\begin{equation} \lim_{n \to \infty} x_n = x \end{equation}

As an example, the set of continuous functions over a finite interval $[a,b]$ of $\mathbb{R}$ to $\mathbb{R}^n$, noted $C([a,b], \mathbb{R}^n)$, forms a Banach space, with the following vector space operations

\begin{eqnarray} (f + g)(x) &=& f(x) + g(x)\\ (\lambda f)(x) &=& \lambda f(x) \end{eqnarray}

with the norm

\begin{eqnarray} \| f \|_C = \sup_{x \in [a,b]} |f(x)| \end{eqnarray}

As a way of linking all these new informations to easier spaces, let's take a moment to remember that, given a set $S$ of finite cardinality $n$, the set of functions $\text{Fun}(S, \mathbb{R})$ is a function space isomorphic to $\mathbb{R}^n$ itself, and each function $f \in \text{Fun}(S, \mathbb{R})$ simply corresponds to a set of real numbers. In which case, the vector space of that function space is isomorphic to the one defined by the canonical $\mathbb{R}^n$ vector space, and our norm is simply

\begin{eqnarray} \| f \|_{\text{Fun}(S, \mathbb{R})} = |f|_{\mathbb{R}^n} \end{eqnarray}

Everything true for Banach spaces in general will be true for $\mathbb{R}^n$ in general, which will help out with understanding the meaning of things.

### 1.2 $\sigma$-algebras and Borel sets

A $\sigma$-algebra $\Sigma$ of a topological space $X$ is a set of sets from $X$, ie, $\Sigma \in \mathcal{P}(X)$, such that

1. $X \in \Sigma$
2. If $A \in \Sigma$, then $X \setminus A \in \Sigma$
3. For a countable set $\{ A_i \}$, $A_i \in \Sigma$, then $\bigcup_i A_i \in \Sigma$

Borel sets are a specific type of $\sigma$-algebra,

What we need therefore is a measure defined on a Banach space, ie, a function $\mu$

\begin{equation} \mu : \Sigma \to \mathbb{R}^* \end{equation}

with $\Sigma$ a $\sigma$-algebra over $X$ and $\mathbb{R}$ the extended real line $\mathbb{R}^* = \mathbb{R} \cup \{ \infty \}$, obeying the usual properties of a measure :

1. For any $E \in \Sigma$, $\mu(E) \geq 0$.
2. $\mu(\varnothing) = 0$

### 1.3. Dual spaces

For every Banach space $X$, it is possible to define a dual space $X'$, such that every $x' \in X'$ is a linear map $x' : X \to \mathbb{R}$. In other words, for every $x' \in X'$, we have

...

In the case of the Banach space $\mathbb{R}^n$, our dual space will be itself isomorphic to $\mathbb{R}^n$, as a set of linear functions of the form

\begin{eqnarray} x' : \mathbb{R}^n &\to& \mathbb{R}\\ (x_1, \ldots, x_n) &\mapsto& x_1'x_1 + \ldots + x_n' x_n \end{eqnarray}

Then every $x' \in X'$ can be expressed as the tuple $(x_1', \ldots, x_n')$, and we can express the action of our dual space by the product

\begin{eqnarray} \langle \cdot, \cdot \rangle : X' \times X \to \mathbb{R}\\ (x', x)&\mapsto \langle x', x \rangle \sum_{i=1}^n x_i' x_i \end{eqnarray}

### Cylinder sets

For a vector space $X$ over a field $K$, the cylinder sets $C$ are functions from $(X')^n$ to $\mathfrak{B}(K^n)$, the Borel sets of the field, of the form

\begin{equation} C_A(f_1, \ldots, f_n) = \left\{ \forall x \in X, (f_1(x), \ldots, f_n(x)) \in A \right\} \end{equation}

for $f_i \in X'$. In particular, if we consider our function space $\text{Fun}(S, \mathbb{R}) \approx \mathbb{R}$, for the Borel set $[0,1]^n \subset \mathbb{R}^n$, the cylinder sets are

\begin{equation} C_{[a,b]}(x_1, \ldots, x_n) = \left\{ \forall x \in \mathbb{R}, (x_1 x, \ldots, x_n x) \in [0,1]^n \right\} \end{equation}

The cylinder set of $(0,\ldots,0)$ will be $\mathbb{R}$, while the cylinder set of $(1, \ldots, 1)$ will be $[0,1]$.

### Fourier transform

For a Banach space $X$ and its dual $X'$, equipped with a measure $\lambda$, we say that the Fourier transform of $\lambda$, $\mathscr{F}\lambda$, is defined by, for $x \in X$, $x' \in X'$,

\begin{equation} \mathscr{F}\lambda(x') = \int_X e^{-i \langle x', x \rangle} d\lambda(x) \end{equation}

## The Gaussian measure

A very common measure used for functional integration is the Gaussian measure. A Gaussian measure $\gamma_C$ must satisfy that, for all $f \in X$,

\begin{equation} S[f] = \int d\gamma_C(\phi) e^{i \phi(f)} = e^{\frac{1}{2} \langle f, Cf \rangle} \end{equation}

It can be seen that

\begin{equation} S = \int_{X'} d\gamma_C(\phi) = 1 \end{equation}

making $d\gamma$ a probability measure.

...

In $\mathbb{R}^n$, the Gaussian measure is defined via

\begin{equation} S[x'] = \int_\mathbb{R} d\gamma_C(x) e^{i x'x} = e^{\frac{1}{2} Cxx'} \end{equation}

## Abstract Wiener space

An abstract Wiener space is a specific case, given a separable Banach space $E$ and a Gaussian measure $\gamma$

### Classic Wiener space

In the case of the classic Wiener space, our Hilbert space is the Hilbert space, $L^{2, 1}_0([t_a, t_b], \mathbb{R}^n, d\mu)$ of square-integrable functions of compact support on $[t_a, t_b]$ to $\mathbb{R}^n$ with the Lebesgue measure, such that their first derivatives are square integrable as well. The Banach space is our space of continuous functions $C([t_a, t_b], \mathbb{R}^n)$.

The Hilbert space is indeed a Hilbert space, as the sum of two such functions still remains of the same compact support on $[t_a, t_b]$, as well as remaining differentiable in the proper way. The inner product for our Hilbert space is

\begin{equation} \langle \psi_1, \psi_2 \rangle = \int_{t_a}^{t_b} \langle \dot{\psi}_1(t), \dot{\psi}_2(t) \rangle_{\mathbb{R}^n} dt \end{equation}

As our derivatives are still square-integrable, this is entirely fine and obeys overall the Hilbert inner product properties.

...

Prove Markov property

### Ornstein–Uhlenbeck measure

Last updated : 2019-08-08 10:39:20
Tags : physics , quantum-mechanics