List of results from CQFT 2
On this page, I will learn about the contents of Brennecke (2026) via the method of lists.
I apologize to Jaques Ellul for performing this technisized extraction of information.
Chapter 1: Classical Mechanics
\[ \frac{dx}{dt} = \nabla_p H, \hspace{0.5cm} \frac{dp}{dt} = - \nabla_x H \]
\[ \nabla \cdot E = \rho, \hspace{0.5cm} \nabla \times E = - \frac{1}{c} \partial_t B, \] \[ \nabla \cdot B = 0, \hspace{0.5cm} \nabla \times B = \frac{1}{c}(\partial_t E + j) \]
\[ t \mapsto \rho(t,x) = e \delta_{x(t)} \in \mathcal{D}'(\mathbb R^3) \] and \[ t \mapsto j(x,t) = e \delta_{x(t)} \frac{dx}{dt}(t) \in \mathcal{D}'(\mathbb R^3, \mathbb R^3) \]
Suppose \(F \in C^{\infty}(\mathcal{P})\) and that \(\{F,H\} = 0\). Then \[ \frac{d}{dt} F \circ (x(t), p(t)) = 0. \]
In this case, \(F\) is called a conserved quantity under the Hamiltonian dynamics.
If \(F \in C^{\infty}(\mathcal{P})\) is invariant under the Hamiltionian flow associated to the Hamiltonian vector field \(X_H\), meaning that \[ \Phi_{X_H}(t,\cdot)^*F = F, \] then \(\{F,H\} = 0\).
Let \(F \in C^{\infty}(\mathcal{P})\) be an observable (possibly the Hamiltonian). Then the flow, \(\Phi_{X_H}(t, \cdot)\), of the Hamiltonian vector field \(X_H = \mathbf{J} \nabla H\) is equal to a canonical transformation \(\phi \in C^{\infty}(\mathcal{P}, \mathbb R^{2n})\) for any time \(t\).
Let \(F \in C^{\infty}(\mathcal{P})\). If the Hamilitionian \(H\) is invariant under the flow generated by \(X_F = \mathbf{J} \nabla F\), i.e., \[ H = \Phi_{X_F}(t, \cdot)^* H, \] then \(\{F,H\} = 0\).
Chapter 2.1: Mathematical Interlude: Locally Convex Topological Vector Spaces
Let \(X = (X,\mathbb K, +, \cdot, \tau)\) be a topological vector space, which is the data of a vector space \((X, \mathbb K, +, \cdot)\) together with a topology \(\tau\) of open subsets of \(X\) such that \(+\) (addition) and \(\cdot\) (scalar multiplication) are continuous (from the product topology in the case of addition).
We don’t need any new data to say that \(X\) is locally convex, but there must be a local base \(\mathcal{B}\) whose elements are convex.
Let \((X,\tau)\) be a topological vector space. A local base at \(0\), call it \(\mathcal{B}\), is the following: \(\mathcal{B} \subset \tau\), \(0 \in B\) for all \(B \in \mathcal{B}\), and for any open subset \(A\) containing \(0\), there exists some \(B \in \mathcal{B}\) such that \(B \subset A\).
If \((X,\mathbb K, \tau)\) is a topological vector space with a local base at \(0\), \(\mathcal{B}\), then \(\tau\) is completely determined by \(\mathcal{B}\).
A semi-norm on a vector space \(X\) is a real-valued function \(p: X \to [0,\infty)\) such that \(p(x+y) \le p(x) + p(y)\) and \(p(\lambda x) = |\lambda|p(x)\) for all \(x,y \in X\) and \(\lambda \in \mathbb K\). In other words, \(p\) would be a norm, other than for the possibility that \(p(x) = 0\) when \(x \ne 0\).
Suppose \(\mathcal{P} = (p_n)_{n \in \mathbb N}\) is a family of seminorms, where \(p_n: X \to [0,\infty)\) is a real-valued function on a \(\mathbb K\)-vector space for all \(n\). Then \(\mathcal{P}\) is said to be separating if for any \(x \ne 0\) in \(X\), there is some \(p \in \mathcal{P}\) such that \(p(x) \ne 0\). In particular, when two elements \(f, g \in X\) are distinct, in that \(f - g \ne 0\), then \(\mathcal{P}\) will know this, in the sense that at least one \(p\) thinks that \(p(f-g) \ne p(0) = 0\).
For example, consider the family of the Schwartz seminorms \(|\cdot|_{\alpha, \beta} = \sup_{x \in \mathbb R^n} |(x^{\alpha} \partial^{\beta} \varphi)(x)|\), indexed by multi indices \(\alpha\), \(\beta\). Some of these seminorms might not be able to tell the difference between two Schwartz functions \(f\) and \(g\) in the sense that \(f - g \ne 0\) and yet \(|f-g|_{\alpha, \beta} = 0\). For instance, polynomials, after taking enough derivatives, evaluate to the zero function, so they would vanish under the Schwartz semi norms for high enough \(|\beta|\). Moreover, the \(L^2\) norm does not separate functions, but it does separate equivalence classes of functions.
The Schwarz functions on \(\mathbb R^n\), denoted \(\mathcal{S}(\mathbb R^n)\), with function addition and scalar multiplication, and with a locally convex topology generated by the separating family of seminorms \(|\cdot|_{\alpha,\beta}\), is a locally convex topological vector space.
The dual space of \(\mathcal{S}(\mathbb R^n)\), denoted \(\mathcal{S}'(\mathbb R^n)\) and called the space of tempered distributions, is the vector space of continuous (in the sense of the convex topology on Schwartz function space) linear functionals \(\Lambda: \mathcal{S}(\mathbb R^n) \to \mathbb C\) under pointwise addition and scalar multiplication. Furthermore, this dual space comes with a topology that turns the space of tempered distributions from merely a vector space into a locally convex topological vector space. It is the topology generated by the family of seminorms \((\text{eval}_f)_{f \in S(\mathbb R^n)}\) where \(\text{eval}_f: \mathcal{S}'(\mathbb R^n) \to [0,\infty)\) is defined via \[ \text{eval}_f(\Lambda) = |\Lambda(f)| \] for all test functions \(f \in \mathcal{S}(\mathbb R^n)\). This is precisely the weak* topology \(\sigma(X^*,X) = \sigma(S'(\mathbb R^n), S(\mathbb R^n))\).
Recall that \(\sigma(X,X^*)\) denotes a topology on the set \(X\) determined by the following condition involving \(X^* = \{\Lambda: X \to \mathbb C, \Lambda \text{ linear and bounded}\}\). It is the weakest topology on \(X\) such that the maps \(\text{eval}_\Lambda: X \to [0,\infty)\) defined by \(\text{eval}_\Lambda(x) = |\Lambda(x)|\) are continuous as \(\Lambda\) ranges in \(X^*\).
Let \(\tau\) denote the convex topology on \(\mathcal{S}(\mathbb R^n)\) generated by the family of seminorms \(|\cdot|_{\alpha,\beta}\). Then \(\tau \ne \sigma(\mathcal{S}(\mathbb R^n), \mathcal{S}'(\mathbb R^n)),\) the weak topology on Schwartz space with respect to the dual space of tempered distributions.
Chapter 2.2: Distributions and Tempered Distributions
Given a function \(f\), we identify \(f\) with \[ \Lambda_f: \text{Test Functions} \to \mathbb K, \hspace{0.5cm} \varphi \mapsto \int f \varphi. \]
The Dirac distribution, centered at a point \(x \in \Omega \subset \mathbb R^n\), is the prime example of an “irregular” distribution, because there is no “nice” function \(f\) such that \(\delta_x = f (= \Lambda_f)\) in the sense of duality/distributions.
Consider the Dirac distribution \(\delta_{x_0}\). Then you can define heuristally \(\delta_{x_0}(x \ne x_0) = 0\) and \(\delta_{x_0}(x_0) = \infty\), and in this case, there is a singularity at \(x_0\).
In short, tempered distributions correspond (via duality) to (weak) derivatives of polynomially bounded continuous functions.
The precise statement is as follows. Let \(\Lambda \in \mathcal{S}'(\mathbb R^n)\). Then there exists a polynomially bounded and continuous function \(f \in C(\mathbb R^n)\) such that \[\partial^{\alpha}f = \Lambda,\] which, means that \[\Lambda(\cdot) = \partial^{\alpha} \Lambda_f(\cdot) := (-1)^{|\alpha|} \Lambda(f \partial^{\alpha}(\cdot)), \] which means the for any Schwartz function \(\phi \in \mathcal{S}(\mathbb R^n)\), \[ \Lambda(\phi) = (-1)^{|\alpha|} \int_{\mathbb R^n} dx f(x) (\partial^{\alpha} \phi)(x) \]
(I got confused by \((\partial^{\alpha} \phi)(x)\), so for that just remember that for real-valued functions, the partial derivative at a point is just a number, not a linear map.)
For example, the Dirac delta function centered at \(0\), as a tempered distribition on \(\mathbb R\), is the 2nd weak/distributional derivative of \(\max{(0,x)}\). This is because \(\delta_0\) is the weak derivative of the Heaviside function \(H(x) = \chi_{[0,\infty]}\), which is the weak derivative of \(\max{(0,x)}\).