From f37472cb1a3ebd1488f9e9d10a926debb07b2a03 Mon Sep 17 00:00:00 2001
From: Trance-0 <60459821+Trance-0@users.noreply.github.com>
Date: Mon, 29 Sep 2025 23:42:28 -0500
Subject: [PATCH] updates

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 .../Math401/Extending_thesis/Math401_S2.md    |   2 +-
 .../Math401/Extending_thesis/Math401_S3.md    |   3 +-
 .../Math401/Extending_thesis/Math401_S4.md    | 139 ++++++++++++++++++
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 create mode 100644 content/Math401/Extending_thesis/Math401_S4.md

diff --git a/content/Math401/Extending_thesis/Math401_S2.md b/content/Math401/Extending_thesis/Math401_S2.md
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--- a/content/Math401/Extending_thesis/Math401_S2.md
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-# Math 401, Fall 2025: Thesis notes, S2, Majorana stellar representation of quantum states.
+# Math 401, Fall 2025: Thesis notes, S2, Majorana stellar representation of quantum states
 
 ## Majorana stellar representation of quantum states
 
diff --git a/content/Math401/Extending_thesis/Math401_S3.md b/content/Math401/Extending_thesis/Math401_S3.md
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-# Math 401, Fall 2025: Thesis notes, S3, Special Barnard space
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+# Math 401, Fall 2025: Thesis notes, S3, Coherent states and POVMs
+
diff --git a/content/Math401/Extending_thesis/Math401_S4.md b/content/Math401/Extending_thesis/Math401_S4.md
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+# Math 401, Fall 2025: Thesis notes, S4, Bargmann space
+
+## Bargmann space (original)
+
+Also known as Segal-Bargmann space or Bargmann-Fock space.
+
+It is the space of [holomorphic functions](../../Math416/Math416_L3#definition-28-holomorphic-functions) that is square-integrable over the complex plane.
+
+> Section belows use [Remarks on a Hilbert Space of Analytic Functions](https://www.jstor.org/stable/71180) as the reference.
+
+A family of Hilbert spaces, $\mathfrak{F}_n(n=1,2,3,\cdots)$, is defined as follows:
+
+The element of $\mathfrak{F}_n$ are [entire](../../Math416/Math416_L13#definition-711) [analytic functions](../../Math416/Math416_L9#definition-analytic) in complex Euclidean space $\mathbb{C}^n$. $f:\mathbb{C}^n\to \mathbb{C}\in \mathfrak{F}_n$
+
+Let $f,g\in \mathfrak{F}_n$. The inner product is defined by
+
+$$
+\langle f,g\rangle=\int_{\mathbb{C}^n} \overline{f(z)}g(z) d\mu_n(z)
+$$
+
+Let $z_k=x_k+iy_k$ be the complex coordinates of $z\in \mathbb{C}^n$.
+
+The measure $\mu_n$ is the defined by
+
+$$
+d\mu_n(z)=\pi^{-n}\exp(-\sum_{i=1}^n |z_i|^2)\prod_{k=1}^n dx_k dy_k
+$$
+
+<details>
+<summary>Example</summary>
+
+For $n=2$,
+
+$$
+\mathfrak{F}_2=\text{ space of entire analytic functions on } \mathbb{C}^2\to \mathbb{C}
+$$
+
+$$
+\langle f,g\rangle=\int_{\mathbb{C}^2} \overline{f(z)}g(z) d\mu(z),z=(z_1,z_2)
+$$
+
+$$
+d\mu_2(z)=\frac{1}{\pi^2}\exp(-|z|^2)dx_1 dy_1 dx_2 dy_2
+$$
+
+</details>
+
+so that $f$ belongs to $\mathfrak{F}_n$ if and only if $\langle f,f\rangle<\infty$.
+
+This is absolutely terrible early texts, we will try to formulate it in a more modern way.
+
+> The section belows are from the lecture notes [Holomorphic method in analysis and mathematical physics](https://arxiv.org/pdf/quant-ph/9912054)
+
+## Complex function spaces
+
+### Holomorphic spaces
+
+Let $U$ be a non-empty open set in $\mathbb{C}^d$. Let $\mathcal{H}(U)$ be the space of holomorphic (or analytic) functions on $U$.
+
+Let $f\in \mathcal{H}(U)$, note that by definition of holomorphic on several complex variables, $f$ is continuous and holomorphic in each variable with the other variables fixed.
+
+Let $\alpha$ be a continuous, strictly positive function on $U$.
+
+$$
+\mathcal{H}L^2(U,\alpha)=\left\{F\in \mathcal{H}(U): \int_U |F(z)|^2 \alpha(z) d\mu(z)<\infty\right\},
+$$
+
+where $\mu$ is the Lebesgue measure on $\mathbb{C}^d=\mathbb{R}^{2d}$.
+
+#### Theorem of holomorphic spaces
+
+1. For all $z\in U$, there exists a constant $c_z$ such that
+    $$
+    |F(z)|^2\le c_z \|F\|^2_{L^2(U,\alpha)}
+    $$
+    for all $F\in \mathcal{H}L^2(U,\alpha)$.
+2. $\mathcal{H}L^2(U,\alpha)$ is a closed subspace of $L^2(U,\alpha)$, and therefore a Hilbert space.
+
+<details>
+<summary>Proof</summary>
+
+First we check part 1.
+
+Let $z=(z_1,z_2,\cdots,z_d)\in U, z_k\in \mathbb{C}$. Let $P_s(z)$ be the "polydisk"of radius $s$ centered at $z$ defined as
+
+$$
+P_s(z)=\{v\in \mathbb{C}^d: |v_k-z_k|<s, k=1,2,\cdots,d\}
+$$
+
+If $z\in U$, we cha choose $s$ small enough such that $P_s(z)\subset U$.
+
+</details>
+
+> [!TIP]
+>
+> [1.] states that point-wise evaluation of $F$ on $U$ is continuous. That is, for each $z\in U$, the map $\varphi: \mathcal{H}L^2(U,\alpha)\to \mathbb{C}$ that takes $F\in \mathcal{H}L^2(U,\alpha)$ to $F(z)$ is a continuous linear functional on $\mathcal{H}L^2(U,\alpha)$. This is false for ordinary non-holomorphic functions, e.g. $L^2$ spaces.
+
+#### Reproducing kernel
+
+Let $\mathcal{H}L^2(U,\alpha)$ be a holomorphic space. The reproducing kernel of $\mathcal{H}L^2(U,\alpha)$ is a function $K:U\times U\to \mathbb{C}$, $K(z,w),z,w\in U$ with the following properties:
+
+1. $K(z,w)$ is holomorphic in $z$ and anti-holomorphic in $w$.
+   $$
+   K(w,z)=\overline{K(z,w)}
+   $$
+
+2. For each fixed $z\in U$, $K(z,w)$ is a square integrable $d\alpha(w)$. For all $F\in \mathcal{H}L^2(U,\alpha)$,
+   $$
+   F(z)=\int_U K(z,w)F(w) \alpha(w) dw
+   $$
+
+3. If $F\in L^2(U,\alpha)$, let $PF$ denote the orthogonal projection of $F$ onto closed subspace $\mathcal{H}L^2(U,\alpha)$. Then
+   $$
+   PF(z)=\int_U K(z,w)F(w) \alpha(w) dw
+   $$
+
+4. For all $z,u\in U$,
+   $$
+   \int_U K(z,w)K(w,u) \alpha(w) dw=K(z,u)
+   $$
+
+5. For all $z\in U$,
+   $$
+   |F(z)|^2\leq K(z,z) \|F\|^2_{L^2(U,\alpha)}
+   $$
+
+### Bargmann space
+
+The Bargmann spaces are the holomorphic spaces 
+
+$$
+\mathcal{H}L^2(\mathbb{C}^d,\mu_t)
+$$
+
+where
+
+$$
+\mu_t(z)=\text{ CONTINUE HERE }
+$$
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