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Michael Hardy
Michael Hardy
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Consider a system of $N$ fermions in a periodic box $\Lambda \subset \mathbb{R}^{d}$, described by the Hamiltonian $$H_{N} = \sum_{k=1}^{N}(-\Delta_{x_{k}}-\mu) + \lambda \sum_{i< j}V(x_{i}-x_{j}) \tag{1}\label{1}$$$$H_{N} = \sum_{k=1}^N(-\Delta_{x_k}-\mu) + \lambda \sum_{i< j} V(x_i-x_{j}) \tag{1}\label{1}$$ where $\lambda$ is some coupling constant, $\mu \in \mathbb{R}$ is the chemical potential and $V$ is a sufficiently well-behaved interaction function. After second quantization, we consider the fermionic Fock space described by a second quantized Hamiltonian $H$.

For simplicity, I will only consider a fermionic theory here. Suppose we have a theory in finite temperature $\beta = 1/T > 0$. Let $\psi_{x}^{\pm}$$\psi_x^\pm$ be creation and annihilation operators for fermions and, for $\beta \ge t_{i}$$\beta \ge t_i$, $i=1,...,n$$i=1,\ldots,n$, $t_{i}\neq t_{j}$$t_i\neq t_{j}$ for $i \neq j$, define the $n$-point (imaginary time) thermal Schwinger function: $$S_{\sigma_{1},...,\sigma_{n}, \beta}(x_{1},t_{1},...,x_{n},t_{n}) := (-1)^{\pi}\frac{\operatorname{Tr}(e^{-(\beta-t_{\pi(1)})H}\psi_{x_{\pi(1)}}^{\sigma_{\pi(1)}}e^{-(t_{\pi(1)}-t_{\pi(2)})H}\psi_{\pi(2)}^{\sigma_{\pi(2)}}\cdots \psi_{x_{\pi(n)}}^{\sigma_{\pi(n)}}e^{-t_{\pi(n)}H})}{\operatorname{Tr}e^{-\beta H}}. \tag{2}\label{2}$$\begin{align} & S_{\sigma_1,\ldots,\sigma_n, \beta}(x_1,t_1,\ldots,x_n,t_n) \\[8pt] := {} & (-1)^\pi \frac{\operatorname{Tr}(e^{-(\beta-t_{\pi(1)})H} \psi_{x_{\pi(1)}}^{\sigma_{\pi(1)}}e^{-(t_{\pi(1)}-t_{\pi(2)})H} \psi_{\pi(2)}^{\sigma_{\pi(2)}}\cdots \psi_{x_{\pi(n)}}^{\sigma_{\pi(n)}}e^{-t_{\pi(n)}H})}{\operatorname{Tr}e^{-\beta H}}. \tag{2}\label{2} \end{align} Here, $\sigma = \pm 1$, $\pi$ is a permutation of $\{1,...,n\}$$\{1,\ldots,n\}$ such that $t_{\pi(1)} > t_{\pi(2)} > \cdots > t_{\pi(n)}$ and $(-1)^{\pi}$ is the parity of $\pi$.

Consider a system of $N$ fermions in a periodic box $\Lambda \subset \mathbb{R}^{d}$, described by the Hamiltonian $$H_{N} = \sum_{k=1}^{N}(-\Delta_{x_{k}}-\mu) + \lambda \sum_{i< j}V(x_{i}-x_{j}) \tag{1}\label{1}$$ where $\lambda$ is some coupling constant, $\mu \in \mathbb{R}$ is the chemical potential and $V$ is a sufficiently well-behaved interaction function. After second quantization, we consider the fermionic Fock space described by a second quantized Hamiltonian $H$.

For simplicity, I will only consider a fermionic theory here. Suppose we have a theory in finite temperature $\beta = 1/T > 0$. Let $\psi_{x}^{\pm}$ be creation and annihilation operators for fermions and, for $\beta \ge t_{i}$, $i=1,...,n$, $t_{i}\neq t_{j}$ for $i \neq j$, define the $n$-point (imaginary time) thermal Schwinger function: $$S_{\sigma_{1},...,\sigma_{n}, \beta}(x_{1},t_{1},...,x_{n},t_{n}) := (-1)^{\pi}\frac{\operatorname{Tr}(e^{-(\beta-t_{\pi(1)})H}\psi_{x_{\pi(1)}}^{\sigma_{\pi(1)}}e^{-(t_{\pi(1)}-t_{\pi(2)})H}\psi_{\pi(2)}^{\sigma_{\pi(2)}}\cdots \psi_{x_{\pi(n)}}^{\sigma_{\pi(n)}}e^{-t_{\pi(n)}H})}{\operatorname{Tr}e^{-\beta H}}. \tag{2}\label{2}$$ Here, $\sigma = \pm 1$, $\pi$ is a permutation of $\{1,...,n\}$ such that $t_{\pi(1)} > t_{\pi(2)} > \cdots > t_{\pi(n)}$ and $(-1)^{\pi}$ is the parity of $\pi$.

Consider a system of $N$ fermions in a periodic box $\Lambda \subset \mathbb{R}^{d}$, described by the Hamiltonian $$H_{N} = \sum_{k=1}^N(-\Delta_{x_k}-\mu) + \lambda \sum_{i< j} V(x_i-x_{j}) \tag{1}\label{1}$$ where $\lambda$ is some coupling constant, $\mu \in \mathbb{R}$ is the chemical potential and $V$ is a sufficiently well-behaved interaction function. After second quantization, we consider the fermionic Fock space described by a second quantized Hamiltonian $H$.

For simplicity, I will only consider a fermionic theory here. Suppose we have a theory in finite temperature $\beta = 1/T > 0$. Let $\psi_x^\pm$ be creation and annihilation operators for fermions and, for $\beta \ge t_i$, $i=1,\ldots,n$, $t_i\neq t_{j}$ for $i \neq j$, define the $n$-point (imaginary time) thermal Schwinger function: \begin{align} & S_{\sigma_1,\ldots,\sigma_n, \beta}(x_1,t_1,\ldots,x_n,t_n) \\[8pt] := {} & (-1)^\pi \frac{\operatorname{Tr}(e^{-(\beta-t_{\pi(1)})H} \psi_{x_{\pi(1)}}^{\sigma_{\pi(1)}}e^{-(t_{\pi(1)}-t_{\pi(2)})H} \psi_{\pi(2)}^{\sigma_{\pi(2)}}\cdots \psi_{x_{\pi(n)}}^{\sigma_{\pi(n)}}e^{-t_{\pi(n)}H})}{\operatorname{Tr}e^{-\beta H}}. \tag{2}\label{2} \end{align} Here, $\sigma = \pm 1$, $\pi$ is a permutation of $\{1,\ldots,n\}$ such that $t_{\pi(1)} > t_{\pi(2)} > \cdots > t_{\pi(n)}$ and $(-1)^{\pi}$ is the parity of $\pi$.

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MathMath
MathMath
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Why computing $n$-point correlations?

I am trying to find a sufficiently convincing answer to this question, but it has been taking so much of my time and I can't get anywhere. It also follows my previous question on PSE.

In axiomatic QFT, one defines a quantum field theory as a theory that satisfies Wightman's axioms. The famous reconstruction theorem states that the theory can be recovered from the Wightman functions. All of this has imaginary time counterparts; the Osterwalder–Schrader axioms are used to define a Euclidean QFT, and the analytic continuation of the Wightman functions is the so-called Schwinger functions. Schwinger functions are correlations of the underlying Euclidean field, and from the mathematical point of view, one is interested in constructing these Schwinger functions, usually trying to use functional integrals, renormalization group techniques, etc.

Now, suppose I am not interested in QFT (in the sense that I don't want to quantize a classical field) but, instead, I want to study many body quantum mechanics. Then, one goes back to the scenario described in my previous linked question. For completeness, I will recall the definitions here.

Consider a system of $N$ fermions in a periodic box $\Lambda \subset \mathbb{R}^{d}$, described by the Hamiltonian $$H_{N} = \sum_{k=1}^{N}(-\Delta_{x_{k}}-\mu) + \lambda \sum_{i< j}V(x_{i}-x_{j}) \tag{1}\label{1}$$ where $\lambda$ is some coupling constant, $\mu \in \mathbb{R}$ is the chemical potential and $V$ is a sufficiently well-behaved interaction function. After second quantization, we consider the fermionic Fock space described by a second quantized Hamiltonian $H$.

For simplicity, I will only consider a fermionic theory here. Suppose we have a theory in finite temperature $\beta = 1/T > 0$. Let $\psi_{x}^{\pm}$ be creation and annihilation operators for fermions and, for $\beta \ge t_{i}$, $i=1,...,n$, $t_{i}\neq t_{j}$ for $i \neq j$, define the $n$-point (imaginary time) thermal Schwinger function: $$S_{\sigma_{1},...,\sigma_{n}, \beta}(x_{1},t_{1},...,x_{n},t_{n}) := (-1)^{\pi}\frac{\operatorname{Tr}(e^{-(\beta-t_{\pi(1)})H}\psi_{x_{\pi(1)}}^{\sigma_{\pi(1)}}e^{-(t_{\pi(1)}-t_{\pi(2)})H}\psi_{\pi(2)}^{\sigma_{\pi(2)}}\cdots \psi_{x_{\pi(n)}}^{\sigma_{\pi(n)}}e^{-t_{\pi(n)}H})}{\operatorname{Tr}e^{-\beta H}}. \tag{2}\label{2}$$ Here, $\sigma = \pm 1$, $\pi$ is a permutation of $\{1,...,n\}$ such that $t_{\pi(1)} > t_{\pi(2)} > \cdots > t_{\pi(n)}$ and $(-1)^{\pi}$ is the parity of $\pi$.

In the mathematical physics literature, especially in the renormalization group community, these Schwinger functions seem ubiquitous. This is my main problem: from the point of view of many-body quantum mechanics, I don't see what is the real motivation to study these Schwinger functions described by (\ref{2}) (or even their zero-temperature limit). My reasons are the following:

  • The thermodynamic properties of the system are obtained from the partition function or from the free energy. These are "zero" point functions.
  • The two-point Schwinger function (also called the Green function in the physics literature) is, indeed, very important. One can obtain the ground state of the full Hamiltonian and even the spectrum. One can also obtain other interesting physical quantities, such as the occupation number.

But Nowhere I looked mention higher order (i.e. $n \ge 2$) Schwinger (or Green, if you prefer) functions to be fundamentally relevant. In some physics textbooks I checked, I don't even think these are introduced in the first place.

Of course, you can argue that in perturbation theory, one expands the interaction part (via Dyson series, using Trotter or analogous technique) and, if the interaction $V$ is a polynomial in the creation and annihilation operators, one will get the two-point Green function from $n$-point functions. But these $n$-point Green functions are taken with respect to the free theory, not the full interacting theory, so it is not the same as studying (\ref{2}) because the latter takes averages with respect to the interacting theory.

However, even these functions are said to be of great interest in the community. In Gallavotti and Benfatto's book, these Schwinger functions are the primary object of interest in Chapter 3 and they claim that the reason is because (in the $\beta \to \infty$ limit) these define the energy of the ground state. A similar claim is made in this review. However, as discussed in my PSE question, only the two-point Schwinger functions are necessary to obtain the ground state energy and spectrum, and I still don't know the real importance of higher order functions. I am sure there is a reason behind this interest, I just cannot find it explicitly stated.

Once again, I understand that the Schwinger functions are important in the context of axiomatic QFT. But I am taking a different route here: I want to motivate myself that these functions arise naturally in the many-body theory, and their study is interesting not only for mathematical but also by physical reasons.

Could someone clarify the real importance of studying these functions, both in the setting of finite and zero temperature? Thanks in advance!