4 Asymptotic multi-particle states
The goal of this section is to show how one can construct genuine multi-particle scattering (or asymptotic) states in a generic interacting QFT in terms of a suitably defined limiting procedure. First, we will present the general idea in a rather sloppy way, focusing on the case of two particles. Then, we will state the precise results of the Haag-Ruelle scattering theory.
4.1 Heuristic construction
Let us consider two momentum-space wavefunctions \(\hat{f}^\alpha(\vec{p})\) for \(\alpha = 1,2\). The goal of Haag-Ruelle scattering theory is to construct a state of two non-interacting particles, which we denote as \(| f^1 , f^2 ; \text{out} \rangle\) (“out” stands for “asymptotic outgoing state”). For the moment, we assume that such a state exists. The game is to guess what kind of properties it should have and how to construct it in a generic interacting QFT.
Under time evolution, the two particles described by the state \(| f^1 , f^2 ; \text{out} \rangle\) should transform independently of one another. Since, for one-particle states, \(e^{-iHt} | f^\alpha \rangle = | f_t^\alpha \rangle\) (see discussion around Equation 3.5), then it is reasonable to expect that the two-particle state transforms as \[ \begin{gather*} e^{-iHt} | f^1 , f^2 ; \text{out} \rangle = | f^1_t , f^2_t ; \text{out} \rangle \ . \end{gather*} \tag{4.1}\]
Let \(\bar{\vec{v}}_\alpha\) and \(\Delta v_\alpha\) be the average velocity and the velocity indetermination, respectively, of the two particles defined as in Equation 3.10. If we choose the two momentum-space wavefunctions \(\hat{f}^\alpha(\vec{p})\) such that average velocities are different and widths of the velocity distributions are small with respect to the velocity difference, i.e. \(\Delta v_1 + \Delta v_2 \ll | \bar{\vec{v}}_1 - \bar{\vec{v}}_2 |\), then the two position-space wavefunctions \(f_t^\alpha(\vec{x})\), constructed as in Equation 3.7, will be significantly different from zero only in two well separated regions of space for large \(t\). In fact, for \(t \to +\infty\), Equation 3.7 yields: \[ \begin{gather*} | \bar{\vec{x}}_1(t) - \bar{\vec{x}}_2(t) | \simeq t \, | \bar{\vec{v}}_1 - \bar{\vec{v}}_2 | \gg t ( \Delta v_1 + \Delta v_2 ) \simeq \Delta x_1(t) + \Delta x_2(t) \ , \end{gather*} \tag{4.2}\] i.e. the distance between the average positions grows linearly with \(t\) and is much larger than the width of the position distribution of each particle. In other words, the two particles become well separated in space for large \(t\).
Let us consider the state \[ \begin{gather*} a(f^1_t)^\dag a(f^2_t)^\dag | \Omega \rangle = \int d^3 x_1 \, f^1_t(\vec{x}_1) \, \phi^\chi(0, \vec{x}_1) \int d^3 x_2 \, f^2_t(\vec{x}_2) \, \phi^\chi(0, \vec{x}_2) | \Omega \rangle \ . \end{gather*} \tag{4.3}\] For a generic value of \(t\), in an interacting QFT, we do not expect this state to represent two non-interacting particles. However, as \(t\) becomes large, the two integrals get contributions only from regions that are very well separated in space and the effect of the interaction between the two particles should become negligible. In this limit, we expect the state \(a(f^1_t)^\dag a(f^2_t)^\dag | \Omega \rangle\) to approximate the state of two non-interacting outgoing particles with wavefunctions \(f^\alpha_t\), i.e. \[ \begin{gather*} a(f^1_t)^\dag a(f^2_t)^\dag | \Omega \rangle \overset{t \to +\infty}{\simeq} | f^1_t , f^2_t ; \text{out} \rangle = e^{-iHt} | f^1 , f^2 ; \text{out} \rangle \ , \end{gather*} \tag{4.4}\] where we have used Equation 4.1. The above equation can be rewritten solving for the asymptotic state at time zero: \[ \begin{gather*} | f^1 , f^2 ; \text{out} \rangle = \lim_{t \to +\infty} e^{iHt} a(f^1_t)^\dag a(f^2_t)^\dag | \Omega \rangle \ . \end{gather*} \tag{4.5}\] One can see this equation as the definition of the asymptotic state of two non-interacting particles. The goal of Haag-Ruelle scattering theory is to prove that the above limit indeed exists and that the asymptotic states satisfy the properties that one would expect from a state of non-interacting particles, like for example Equation 4.1.
In analogy to asymptotic outgoing state, one can also define asymptotic incoming states: \[ \begin{gather*} | f^1 , f^2 ; \text{in} \rangle = \lim_{t \to -\infty} e^{iHt} a(f^1_t)^\dag a(f^2_t)^\dag | \Omega \rangle \ . \end{gather*} \tag{4.6}\] Since the treatment of outgoing and incoming states is very similar, we will focus only on the outgoing states in the following.
4.2 Main results
Let \(\hat{f}(\vec{p})\) be a smooth functions with compact support. We denote by \(\text{csupp}(\hat{f})\) the closed support of \(\hat{f}(\vec{p})\), i.e. the closure of the set of momenta \(\vec{p}\) such that \(\hat{f}(\vec{p}) \neq 0\). The set of velocities associated to \(\hat{f}(\vec{p})\) is defined as: \[ \begin{gather*} V(f) = \left\{ \frac{\vec{p}}{E(\vec{p})} \text{ s.t. } \vec{p} \in \text{csupp}(\hat{f}) \right\} \ . \end{gather*} \tag{4.7}\] The set \(V(f)\) is a compact subset of the unit ball in \(\mathbb{R}^3\).
Let \(\mathcal{H}_{\text{out}}\) be the Hilbert space generated by the vacuum \(| \Omega \rangle\) and all states of the form \(| f^1 , \dots , f^N ; \text{out} \rangle\) for any \(N \ge 1\) and all possible choices of \(\hat{f}^\alpha(\vec{p})\) with non-overlapping velocities. The Hilbert space \(\mathcal{H}_{\text{out}}\) is clearly a subspace of the Hilbert space of the theory.
Notice that the vacuum state \(| \Omega \rangle\) and the Hamiltonian \(H\) which appear in the above theorems are the vacuum and the Hamiltonian of the interacting QFT. Unlike presentations based on perturbation theory, one never needs to decompose the Hamiltonian in a free part and an interaction part.
4.3 Scattering amplitude
This section still needs some improvements.
Consider a scattering process of \(M\) incoming particles with wavefunctions \(\hat{f}^\alpha(\vec{p})\) for \(\alpha = 1, \dots, M\) and \(N\) outgoing particles with wavefunctions \(\hat{g}^\beta(\vec{p})\) for \(\beta = 1, \dots, N\). The corresponding scattering amplitude is given by \[ \begin{gather*} S_{N \leftarrow M}(\hat{g} ; \hat{f}) = \langle g^N , \dots , g^1 ; \text{out} | f^M , \dots , f^1 ; \text{in} \rangle \ . \end{gather*} \tag{4.13}\] Theorem Theorem 4.3 allows to write asymtotic states as: \[ \begin{align*} | f^M , \dots , f^1 ; \text{in} \rangle = & \lim_{t \to -\infty} e^{i H t} a(f^M_t)^\dag \cdots a(f^1_t)^\dag | \Omega \rangle \\ = & \lim_{t \to -\infty} \int \left[ \prod_{\beta=1}^M \frac{d^4 p_\beta}{(2\pi)^4} \hat{f}^\beta(\vec{p}_\beta) \right] \, K_t^M(p) \, \tilde{\phi}(p_M)^\dag \cdots \tilde{\phi}(p_1)^\dag | \Omega \rangle \ , \end{align*} \tag{4.14}\] where we have used the definition of the creation operators, Equation 3.18, the properties of the field in momentum space discussed in Problem 2.2, and the defintion: \[ \begin{gather*} K_t^M(p) = \prod_{\beta=1}^M e^{i [p^0_\beta - E(\vec{p}_\beta)] t} \zeta(p^0_\beta - E(\vec{p}_\beta)) \ . \end{gather*} \tag{4.15}\] For outgoing states, one obtains similar a similar expression: \[ \begin{align*} | g^N , \dots , g^1 ; \text{out} \rangle = & \lim_{t \to +\infty} e^{i H t} a(g^N_t)^\dag \cdots a(g^1_t)^\dag | \Omega \rangle \\ = & \lim_{t \to +\infty} \int \left[ \prod_{\alpha=1}^N \frac{d^4 q_\alpha}{(2\pi)^4} \hat{g}^\alpha(\vec{q}_\alpha) \right] \, K_t^N(q) \, \tilde{\phi}(q_N)^\dag \cdots \tilde{\phi}(q_1)^\dag | \Omega \rangle \ . \end{align*} \tag{4.16}\]
Then the scattering emplitude can be represented as \[ \begin{align*} S_{N \leftarrow M}(\hat{g} ; \hat{f}) = \lim_{t \to +\infty} \int & \left[ \prod_{\alpha=1}^N \frac{d^4 q_\alpha}{(2\pi)^4} \hat{g}^\alpha(\vec{q}^\alpha) \right]^* \left[ \prod_{\beta=1}^M \frac{d^4 p_\beta}{(2\pi)^4} \hat{f}^\beta(\vec{p}_\beta) \right] K_t^N(q)^* K_{-t}^M(p) \\ & \times \langle \Omega | \tilde{\phi}(q_1) \cdots \tilde{\phi}(q_N) \tilde{\phi}(p_M)^\dag \cdots \tilde{\phi}(p_1)^\dag | \Omega \rangle \ . \end{align*} \tag{4.17}\]
One can also extract the wavefunctions and write a formula for the scattering amplitude between plane-wave states: \[ \begin{align*} S_{N \leftarrow M}(\vec{q} ; \vec{p}) = \lim_{t \to +\infty} \int & \left[ \prod_{\alpha=1}^N \frac{d q^0_\alpha}{2\pi} \right] \left[ \prod_{\beta=1}^M \frac{d p^0_\beta}{2\pi} \right] K_t^N(q)^* K_{-t}^M(p) \\ & \times \langle \Omega | \tilde{\phi}(q_1) \cdots \tilde{\phi}(q_N) \tilde{\phi}(p_M)^\dag \cdots \tilde{\phi}(p_1)^\dag | \Omega \rangle \ . \end{align*} \tag{4.18}\] However, this equation needs to be understood in the sense of distributions and, in particular, the limit has to be understood in the weak sense.
The vacuum expectation value in the above equation is the Fourier transform of a Wightman function, which one can rewritten in terms of a generalized spectral function. To understand this, let us consider the \(M=N=2\) case: \[ \begin{align*} & \langle \Omega | \tilde{\phi}(q_1) \tilde{\phi}(q_2) \tilde{\phi}(p_2)^\dag \tilde{\phi}(p_1)^\dag | \Omega \rangle \\ & \quad = \int d^4 y_1 d^4 y_2 d^4 x_1 d^4 x_2 \, e^{ i q_1 y_1 + i q_2 y_2 - i p_2 x_2 - i p_1 x_1 } \langle \Omega | \phi(y_1) \phi(y_2) \phi(x_2)^\dag \phi(x_1)^\dag | \Omega \rangle \\ & \quad = \int d^4 y_1 d^4 y_2 d^4 x_1 d^4 x_2 \, e^{ i q_1 y_1 + i q_2 y_2 - i p_2 x_2 - i p_1 x_1 } \\ & \quad \hspace{2cm} \times \langle \Omega | \phi(0) e^{-i P (y_1-y_2)} \phi(0) e^{-i P (y_2-x_2)} \phi(0)^\dag e^{-i P (x_2-x_1)} \phi(0)^\dag | \Omega \rangle \ , \end{align*} \tag{4.19}\] where we have used the definition of the field in momentum space, Equation 3.19, the transformation rule of the field in position space, Equation 2.13 and the invariance of the vacuum under translations. The Fourier trasforms can be performed in sequence: first the \(y_1\) and \(x_1\) integrals, then the \(y_2\) and \(x_2\) integrals. At every step, one obtains delta functions: \[ \begin{align*} & \langle \Omega | \tilde{\phi}(q_1) \tilde{\phi}(q_2) \tilde{\phi}(p_2)^\dag \tilde{\phi}(p_1)^\dag | \Omega \rangle \\ & \quad = \langle \Omega | \phi(0) (2\pi)^4 \delta^4(P-q_1) \phi(0) (2\pi)^4 \delta^4(P-q_1-q_2) \\ & \phantom{\quad = \langle \Omega |} (2\pi)^4 \delta^4(P-p_1-p_2)\phi(0)^\dag (2\pi)^4 \delta^4(P-p_1) \phi(0)^\dag | \Omega \rangle \\ & \quad = (2\pi)^4 \delta^4(p_1+p_2-q_1-q_2) \\ & \phantom{\quad = } \times \langle \Omega | \phi(0) (2\pi)^4 \delta^4(P-q_1) \phi(0) (2\pi)^4 \delta^4(P-p_1-p_2)\phi(0)^\dag (2\pi)^4 \delta^4(P-p_1) \phi(0)^\dag | \Omega \rangle \ . \end{align*} \tag{4.20}\] We vacuum expectation value in the last line is a generalized spectral function. The above calculation can be easily generalized to the case of \(N\) incoming and \(M\) outgoing particles, yielding: \[ \begin{align*} \langle \Omega | \tilde{\phi}(q_1) \cdots \tilde{\phi}(q_N) \tilde{\phi}(p_M)^\dag \cdots \tilde{\phi}(p_1)^\dag | \Omega \rangle = (2\pi)^4 \delta^4\left( \sum_{\alpha=1}^N q_\alpha - \sum_{\beta=1}^M p_\beta \right) \, \rho_{N,M}(q,p) \ . \end{align*} \tag{4.21}\] Notice that this formula extracts the energy-momentum conservation delta function. The generalized spectral function is defined as follows: \[ \begin{align*} \rho_{N,M}(q,p) = & \langle \Omega | \phi(0) (2\pi)^4 \delta^4(P-q_1) \phi(0) (2\pi)^4 \delta^4\left( P - \sum_{\beta=1}^{2} q_\beta \right) \phi(0) \cdots \phi(0) (2\pi)^4 \delta^4 \left( P - \sum_{\beta=1}^{N-1} q_\beta \right) \phi(0) \\ & \phantom{\langle \Omega | } \times (2\pi)^4 \delta^4 \left( P - \sum_{\alpha=1}^M p_\alpha \right) \phi(0)^\dag \cdots (2\pi)^4 \delta^4\left( P - \sum_{\alpha=1}^2 p_\alpha \right) \phi(0)^\dag (2\pi)^4 \delta^4(P-p_1) \phi(0)^\dag | \Omega \rangle \ . \end{align*} \tag{4.22}\] In terms of this quantity, the scattering amplitude can be rewritten as: \[ \begin{align*} S_{N \leftarrow M}(\vec{q} ; \vec{p}) = & (2\pi)^3 \delta^3\left( \sum_{\alpha=1}^N \vec{q}_\alpha - \sum_{\beta=1}^M \vec{p}_\beta \right) \ \lim_{t \to +\infty} \int \left[ \prod_{\alpha=1}^N \frac{d q^0_\alpha}{2\pi} \right] \left[ \prod_{\beta=1}^M \frac{d p^0_\beta}{2\pi} \right] \\ & \times K_t^N(q)^* K_{-t}^M(p) (2\pi) \delta\left( \sum_{\alpha=1}^N q_\alpha^0 - \sum_{\beta=1}^M p_\beta^0 \right) \, \rho_{N,M}(q,p) \ . \end{align*} \tag{4.23}\] This formula, which can rewritten in several equivalent ways, is the analog of the LSZ reduction formula, as obteined from the Haag-Ruelle scattering theory.
4.4 Proof of Theorem 4.3
The goal of this section is the prove theorems Theorem 4.3 in the case of two-particles, which allows us to discuss the main ideas behind the proof, while avoiding some complications mostly of algebraic nature. We will comment on the generalization to a larger number of particles.
Since we need to deal a lot with smooth functions of rapid decay, it is useful to comment on different equivalent ways to characterize those functions which will be useful later on.
Given a smooth function \(g\) from \([0,+\infty)\) to some normed space (in practice this will be \(\mathbb{R}\), \(\mathbb{C}\) or the Hilbert space of the theory), the following conditions are equivalent:
\(\lim_{t \to +\infty} t^r g(t) = 0\) for any \(r > 0\);
for any \(r>0\), a constant \(C_r\) exists such that \(\|g(t)\| < C_r t^{-r}\) for all \(t > 0\);
for any \(r>0\), a constant \(C_r\) exists such that \(\|g(t)\| < C'_r (1+t)^{-r}\) for all \(t \ge 0\).
Smooth functions which satisfy any (hence, all) of the above conditions are said to decay rapidly or vanish rapidly as \(t \to +\infty\) or, also, to decay/vanish faster than any inverse power of \(t\) as \(t \to +\infty\). The equivalence of the above conditions is proved with elementary methods of calculus (one needs to use the definition of limit and the fact that continuous functions of a single variable are bounded on compact sets. We will leave the proof to the reader.
We want to show that the \(t \to \infty\) limit of the following state exists in the strong sense: \[ \begin{gather*} | \Psi(t) \rangle = e^{iHt} a(f^1_t)^\dag a(f^2_t)^\dag | \Omega \rangle \ . \end{gather*} \tag{4.24}\] One can show (we will not do it here) that this state is infinitely differentiable in \(t\). We will prove that the time derivative of the state \(| \Psi(t) \rangle\) vanishes rapidly in the limit \(t \to +\infty\). Therefore for any \(r > 0\) a constant \(C'_r\) exists such that \[ \begin{gather*} \left\| \frac{d}{dt} | \Psi(t) \rangle \right\| \le C'_r (1 + t^2)^{-r} \qquad \forall t > 0 \ . \end{gather*} \tag{4.25}\] Then we can define \[ \begin{gather*} | \Psi(+\infty) \rangle \equiv | \Psi(0) \rangle + \int_0^\infty ds \, \frac{d}{ds} | \Psi(s) \rangle \ . \end{gather*} \tag{4.26}\] The integral is convergent thanks to Equation 4.25, in fact: \[ \begin{gather*} \left\| \int_0^\infty ds \, \frac{d}{ds} | \Psi(s) \rangle \right\| \le \int_0^\infty ds \, \left\| \frac{d}{ds} | \Psi(s) \rangle \right\| \le C'_1 \int_0^\infty ds \, (1 + s^2)^{-1} < +\infty \ . \end{gather*} \tag{4.27}\] At this stage, we do not know yet that \(| \Psi(+\infty) \rangle\) is indeed the limit of \(| \Psi(t) \rangle\) as \(t \to +\infty\) in the strong sense. In order to prove this, we bound the difference between \(| \Psi(t) \rangle\) and \(| \Psi(+\infty) \rangle\) as follows: \[ \begin{align*} \left\| | \Psi(+\infty) \rangle - | \Psi(t) \rangle \right\| = & \left\| \int_0^\infty ds \, \frac{d}{ds} | \Psi(s) \rangle - \int_0^t ds \, \frac{d}{ds} | \Psi(s) \rangle \right\| \\ = & \left\| \int_t^\infty ds \, \frac{d}{ds} | \Psi(s) \rangle \right\| \\ \le & \int_t^\infty ds \, \left\| \frac{d}{ds} | \Psi(s) \rangle \right\| \\ \le & C'_{\frac{r+1}{2}} \int_t^\infty ds \, (1 + s^2)^{-\frac{r+1}{2}} \\ = & C'_{\frac{r+1}{2}} \int_t^\infty ds \, s^{-r-1} = C'_{\frac{r+1}{2}} r^{-1} t^{-r} \ . \end{align*} \tag{4.28}\] The above bound is valid for any \(r > 0\) and shows that the difference between \(| \Psi(t) \rangle\) and \(| \Psi(+\infty) \rangle\) vanishes rapidly as \(t \to +\infty\). Therefore, the following equation holds: \[ \begin{gather*} \lim_{t \to +\infty} | \Psi(t) \rangle = | \Psi(+\infty) \rangle \ , \end{gather*} \tag{4.29}\] where the limit needs to be interpreted in the strong sense, which proves Equation 4.8 for \(N=2\). The bound Equation 4.28 provides a proof of Equation 4.9 for \(N=2\).
We are left with the task of proving the bound on the derivative given in Equation 4.25. We will do this in several steps in the following subsections.
The strategy presented in this subsection works identically for \(N>2\), with the definition of the state: \[ \begin{gather*} | \Psi(t) \rangle = e^{iHt} a(f^1_t)^\dag \cdots a(f^N_t)^\dag | \Omega \rangle \ . \end{gather*} \]
4.4.1 Step 1: the operators \(A_t\)
The state \(| \Psi(t) \rangle\) can be rewritten as \[ \begin{gather*} | \Psi(t) \rangle = A_t(f^1)^\dag A_t(f^2)^\dag | \Omega \rangle \ , \end{gather*} \tag{4.30}\] where we have defined the operators \[ \begin{gather*} A_t(f)^\dag = e^{iHt} a(f_t)^\dag e^{-iHt} \ . \end{gather*} \tag{4.31}\] The derivative of the state \(| \Psi(t) \rangle\) with respect to \(t\) is readily calculated: \[ \begin{gather*} \frac{d}{dt} | \Psi(t) \rangle = \dot{A}_t(f^1)^\dag A_t(f^2)^\dag | \Omega \rangle + A_t(f^1)^\dag \dot{A}_t(f^2)^\dag | \Omega \rangle \ , \end{gather*} \tag{4.32}\] where we use the dot to denote the time-derivative, i.e. \(\dot{A}_t(f) = \frac{d}{dt} A_t(f)\). We notice that the state \(A_t(f)^\dag | \Omega \rangle\) does not depend on \(t\), since: \[ \begin{gather*} A_t(f)^\dag | \Omega \rangle = e^{iHt} a(f_t)^\dag | \Omega \rangle = e^{iHt} | f_t \rangle = | f \rangle \ , \end{gather*} \tag{4.33}\] where we have used the definition of \(A_t(f)\), Equation 4.31, and Equation 3.5. Therefore the time-derivative of \(A_t(f)^\dag\) annihilates the vacuum, i.e. \[ \begin{gather*} \dot{A}_t(f)^\dag | \Omega \rangle = 0 \ . \end{gather*} \tag{4.34}\] Using this property, one can reduce the expression for the time-derivative of \(| \Psi(t) \rangle\) to \[ \begin{gather*} \frac{d}{dt} | \Psi(t) \rangle = \dot{A}_t(f^1)^\dag A_t(f^2)^\dag | \Omega \rangle \ . \end{gather*} \tag{4.35}\] Because of the following identity: \[ \begin{gather*} \left\| \frac{d}{dt} | \Psi(t) \rangle \right\|^2 = \left\| \dot{A}_t(f^1)^\dag A_t(f^2)^\dag | \Omega \rangle \right\|^2 = \langle \Omega | A_t(f^2) \dot{A}_t(f^1) \dot{A}_t(f^1)^\dag A_t(f^2)^\dag | \Omega \rangle \ , \end{gather*} \tag{4.36}\] the statement that the time-derivative of the state \(| \Psi(t) \rangle\) vanishes rapidly as \(t \to +\infty\) is equivalent to the statement that the 4-point function \(\langle \Omega | A_t(f^2) \dot{A}_t(f^1) \dot{A}_t(f^1)^\dag A_t(f^2)^\dag | \Omega \rangle\) vanishes rapidly as \(t \to +\infty\). The goal of the next steps is to prove this statement.
In the general case, the r.h.s. of Equation 4.35 is replaced by a sum of terms generated by the Leibniz rule. The norm squared of the time derivative of the state \(| \Psi(t) \rangle\) can be expanded as a sum of \((2N)\)-point functions. Each of these \((2N)\)-point functions needs to be shown to vanish rapidly as \(t \to +\infty\).
4.4.2 Step 2: cluster decomposition
We want to decompose the 4-point function in Equation 4.36 in terms of connected vacuum expectation values. We give here the definition of connected vacuum expectation values only for \(n\)-point functions with \(n \le 4\). Given a list of operators \(Q_1\), \(Q_2\) and so on, connected vacuum expectation values of products of these operators are denoted by \(\langle Q_1 Q_2 \cdots Q_N \rangle_c\) and defined as by solving recursively the following equations: \[ \begin{align*} \langle \Omega | Q_1 | \Omega \rangle = & \langle Q_1 \rangle_c \ , \\ \langle \Omega | Q_1 Q_2 | \Omega \rangle = & \langle Q_1 Q_2 \rangle_c + \langle Q_1 \rangle_c \langle Q_2 \rangle_c \ , \\ \langle \Omega | Q_1 Q_2 Q_3 | \Omega \rangle = & \langle Q_1 Q_2 Q_3 \rangle_c \\ & + \langle Q_1 Q_2 \rangle_c \langle Q_3 \rangle_c + \langle Q_1 Q_3 \rangle_c \langle Q_2 \rangle_c + \langle Q_2 Q_3 \rangle_c \langle Q_1 \rangle_c \\ & + \langle Q_1 \rangle_c \langle Q_2 \rangle_c \langle Q_3 \rangle_c \ , \\ \langle \Omega | Q_1 Q_2 Q_3 Q_4 | \Omega \rangle = & \langle Q_1 Q_2 Q_3 Q_4 \rangle_c \\ & + \langle Q_1 Q_2 \rangle_c \langle Q_3 Q_4 \rangle_c + \langle Q_1 Q_3 \rangle_c \langle Q_2 Q_4 \rangle_c + \langle Q_1 Q_4 \rangle_c \langle Q_2 Q_3 \rangle_c \\ & + \langle Q_1 \rangle_c \langle Q_2 Q_3 Q_4 \rangle_c + \langle Q_2 \rangle_c \langle Q_1 Q_3 Q_4 \rangle_c \\ & + \langle Q_3 \rangle_c \langle Q_1 Q_2 Q_4 \rangle_c + \langle Q_4 \rangle_c \langle Q_1 Q_2 Q_3 \rangle_c \\ & + \langle Q_1 \rangle_c \langle Q_2 \rangle_c \langle Q_3 Q_4 \rangle_c + \langle Q_1 \rangle_c \langle Q_3 \rangle_c \langle Q_2 Q_4 \rangle_c + \langle Q_1 \rangle_c \langle Q_4 \rangle_c \langle Q_2 Q_3 \rangle_c \\ & + \langle Q_2 \rangle_c \langle Q_3 \rangle_c \langle Q_1 Q_4 \rangle_c + \langle Q_2 \rangle_c \langle Q_4 \rangle_c \langle Q_1 Q_3 \rangle_c + \langle Q_3 \rangle_c \langle Q_4 \rangle_c \langle Q_1 Q_2 \rangle_c \\ & + \langle Q_1 \rangle_c \langle Q_2 \rangle_c \langle Q_3 \rangle_c \langle Q_4 \rangle_c \ . \end{align*} \tag{4.37}\] The decomposition of n-point functions in terms of connected n-point functions is usually referred to as cluster decomposition. We need to apply these formulae to the operators \(A_t(f^1)\) and \(A_t(f^2)\) and their hermitian conjugates. A number of simplifications occur in this case because of the properties of the operators \(A_t(f)\).
The 1-point functions vanish. This is easily seen using Equation 4.33, which implies that \[ \begin{align*} & \langle A_t(f)^\dag \rangle_c = \langle \Omega | A_t(f)^\dag | \Omega \rangle = \langle \Omega | f \rangle = 0 \ , \\ & \langle A_t(f) \rangle_c = \langle \Omega | A_t(f) | \Omega \rangle = \langle f | \Omega \rangle = 0 \ , \end{align*} \tag{4.38}\] since one-particle states are orthogonal to the vacuum.
There is a total of six 2-point functions that one needs to consider. Three of those 2-point functions vanish because, as we have already discussed, the operator \(\dot{A}_t(f)^\dag\) annihilates the vacuum, therefore: \[ \begin{align*} & \langle A_t(f^2) \dot{A}_t(f^1)^\dag \rangle_c = \langle \Omega | A_t(f^2) \dot{A}_t(f^1)^\dag | \Omega \rangle = 0 \ , \\ & \langle \dot{A}_t(f^1) \dot{A}_t(f^1)^\dag \rangle_c = \langle \Omega | \dot{A}_t(f^1) \dot{A}_t(f^1)^\dag | \Omega \rangle = 0 \ , \\ & \langle \dot{A}_t(f^1) A_t(f^2)^\dag \rangle_c = \langle \Omega | \dot{A}_t(f^1) A_t(f^2)^\dag | \Omega \rangle = \langle \Omega | A_t(f^2) \dot{A}_t(f^1)^\dag | \Omega \rangle^* = 0 \ . \end{align*} \tag{4.39}\]
The operator \(A_t(f)\) annihilates the vacuum. This is a consequence of the fact that \(a(f_t)\) annihilates the vacuum (see Problem 3.4): \[ \begin{gather*} A_t(f) | \Omega \rangle = e^{iHt} a(f_t) | \Omega \rangle = 0 \ . \end{gather*} \tag{4.40}\] Since the above equation holds for any \(t\), it follows that also the operator \(\dot{A}_t(f)\) annihilates the vacuum. Therefore, the following 2-point functions vanish: \[ \begin{align*} & \langle A_t(f^2) \dot{A}_t(f^1) \rangle_c = \langle \Omega | A_t(f^2) \dot{A}_t(f^1) | \Omega \rangle = 0 \ , \\ & \langle \dot{A}_t(f^1)^\dag A_t(f^2)^\dag \rangle_c = \langle \Omega | \dot{A}_t(f^1)^\dag A_t(f^2)^\dag | \Omega \rangle = \langle \Omega | A_t(f^2) \dot{A}_t(f^1) | \Omega \rangle^* = 0 \ . \end{align*} \tag{4.41}\]
Using the above properties, one gets the incredibly simple relation for the 4-point function:
\[ \begin{gather*} \langle \Omega | A_t(f^2) \dot{A}_t(f^1) \dot{A}_t(f^1)^\dag A_t(f^2)^\dag | \Omega \rangle = \langle A_t(f^2) \dot{A}_t(f^1) \dot{A}_t(f^1)^\dag A_t(f^2)^\dag \rangle_c \end{gather*} \tag{4.42}\]
In order to write a general formula for the decomposition of an \(N\)-point function in terms of connected vacuum expectation values, we need some definitions. Given a finite set \(E\), a partition of \(E\) is a set of non-empty disjoint subsets of \(E\) such that the union of all subsets in the partition is equal to \(E\). Let \(\mathcal{P}(E)\) the set of all partitions of \(E\). If \(E \subset \mathbb{N}\) is a set of indices, then we can write the general formula \[ \begin{gather*} \langle \Omega | \rprod{\alpha \in E}{} Q_{\alpha} | \Omega \rangle = \sum_{P \in \mathcal{P}(E)} \ \prod_{J \in P} \ \langle \rprod{\alpha \in J}{} Q_{\alpha} \rangle_c \ , \end{gather*} \] where the superimposed arrow indicates that the factors are ordered in the product such that the index \(\alpha\) increases from left to right. In the decomposition of the relevant \((2N)\)-point functions, several terms survive, and one needs to keep track of all of them. However it is still true that 1-point functions vanish, and that contributions involving only products of 2-point functions vanish as well. The only non-vanishing contributions must contain at least one \(n\)-point function with \(n \ge 3\).
At this point, we want to express the 4-point function in terms of the smeared field. Equation 4.31 with the expression of the creation operator in terms of a position-space integral given in Equation 3.33 yields \[ \begin{gather*} A_t(f)^\dag = \frac{1}{Z^{1/2}} \int d^3x \, f_t(\vec{x}) \, \phi^\chi(t,\vec{x})^\dag \ , \end{gather*} \tag{4.43}\] In order to write \(\dot{A}_t\), we observe that the time derivative of the smeared field is just a smeared field with a different smearing function: \[ \begin{gather*} \dot{\phi}^\chi(x) = \frac{d}{dx_0} \int d^4 y \, \chi(x-y) \, \phi(y) = \int d^4 y \, \dot{\chi}(x-y) \, \phi(y) = \phi^{\dot{\chi}}(x) \ . \end{gather*} \tag{4.44}\] Moreover, the time derivative of the coordinate-space wavefunction \(f_t(\vec{x})\) is just the coordinate-space wavefunction associated to a different momentum-space wavefunction: \[ \begin{align*} \dot{f}_t(\vec{x}) & = \frac{d}{dt} \int \frac{d^3 p}{(2\pi)^3 \sqrt{ 2 E(\vec{p}) }} \, e^{-iE(\vec{p})t + i \vec{p} \vec{x}} \, \hat{f}(\vec{p}) \\ & = \int \frac{d^3 p}{(2\pi)^3 \sqrt{ 2 E(\vec{p}) }} \, e^{-iE(\vec{p})t + i \vec{p} \vec{x}} \, \hat{g}(\vec{p}) = g_t(\vec{x}) \ , \end{align*} \tag{4.45}\] with the definition \[ \begin{gather*} \hat{g}(\vec{p}) = -i E(\vec{p}) \hat{f}(\vec{p}) \ . \end{gather*} \tag{4.46}\] Using these results, we can rewrite the time-derivative of the operator \(A_t(f)\) as \[ \begin{gather*} \dot{A}_t(f)^\dag = \frac{1}{Z^{1/2}} \int d^3x \, g_t(\vec{x}) \, \phi^\chi(t,\vec{x})^\dag + \frac{1}{Z^{1/2}} \int d^3x \, f_t(\vec{x}) \, \phi^{\dot{\chi}}(t,\vec{x})^\dag \ . \end{gather*} \tag{4.47}\] The 4-point function is expanded as follows: \[ \begin{align*} & \langle \Omega | A_t(f^2) \dot{A}_t(f^1) \dot{A}_t(f^1)^\dag A_t(f^2)^\dag | \Omega \rangle = \frac{1}{Z^2} \int d^3x_1 \, d^3x_2 \, d^3x_3 \, d^3x_4 \\ & \qquad \times \Big\{ f^2_t(\vec{x}_4)^* \, g^1_t(\vec{x}_3)^* \, g^1_t(\vec{x}_2) \, f^2_t(\vec{x}_1) \langle \phi^\chi(t,\vec{x}_4) \phi^{\chi}(t,\vec{x}_3) \phi^{\chi}(t,\vec{x}_2)^\dag \phi^\chi(t,\vec{x}_1)^\dag \rangle_c \\ & \qquad \phantom{\times \Big\{} + f^2_t(\vec{x}_4)^* \, g^1_t(\vec{x}_3)^* \, f^1_t(\vec{x}_2) \, f^2_t(\vec{x}_1) \langle \phi^\chi(t,\vec{x}_4) \phi^{\chi}(t,\vec{x}_3) \phi^{\dot{\chi}}(t,\vec{x}_2)^\dag \phi^\chi(t,\vec{x}_1)^\dag \rangle_c \\ & \qquad \phantom{\times \Big\{} + f^2_t(\vec{x}_4)^* \, f^1_t(\vec{x}_3)^* \, g^1_t(\vec{x}_2) \, f^2_t(\vec{x}_1) \langle \phi^\chi(t,\vec{x}_4) \phi^{\chi}(t,\vec{x}_3) \phi^{\dot{\chi}}(t,\vec{x}_2)^\dag \phi^\chi(t,\vec{x}_1)^\dag \rangle_c \\ & \qquad \phantom{\times \Big\{} + f^2_t(\vec{x}_4)^* \, f^1_t(\vec{x}_3)^* \, f^1_t(\vec{x}_2) \, f^2_t(\vec{x}_1) \langle \phi^\chi(t,\vec{x}_4) \phi^{\dot{\chi}}(t,\vec{x}_3) \phi^{\dot{\chi}}(t,\vec{x}_2)^\dag \phi^\chi(t,\vec{x}_1)^\dag \rangle_c \Big\} \end{align*} \tag{4.48}\] Recall that we want to prove that the 4-point function vanishes rapidly as \(t \to +\infty\). In order to do this, it is enough to prove that each of the four terms in the r.h.s. of the above equation vanishes rapidly as \(t \to +\infty\). Notice that the four terms have similar structures. It is more efficient to consider the more general integral \[ \begin{align*} I(t) = \int d^3x_1 \, d^3x_2 \, d^3x_3 \, d^3x_4 \, & h^1_t(\vec{x}_4)^* \, h^2_t(\vec{x}_3)^* \, h^2_t(\vec{x}_2) \, h^1_t(\vec{x}_1) \\ & \times \langle \phi^{\chi_2}(t,\vec{x}_4) \phi^{\chi_1}(t,\vec{x}_3) \phi^{\chi_1}(t,\vec{x}_2)^\dag \phi^{\chi_2}(t,\vec{x}_1)^\dag \rangle_c \ , \end{align*} \tag{4.49}\] which reduces the the four terms in Equation 4.48 for appropriate choices of the wavefunctions \(h^1_t(\vec{x})\) and \(h^2_t(\vec{x})\), and the smearing functions \(\chi_1\) and \(\chi_2\). In all cases, the wavefunctions \(\hat{h}^1(\vec{p})\) and \(\hat{h}^2(\vec{p})\) are smooth functions with compact support and have non-overlapping velocities. In order to prove that the 4-point function vanishes rapidly as \(t \to +\infty\), it is enough to prove that the more general integral \(I(t)\) vanishes rapidly as \(t \to +\infty\). This is the goal of the next steps.
4.4.3 Step 3: Ruelle’s cluster theorem
The importance of the connected n-point functions lies in a fundamental property captured by Ruelle’s cluster theorem: connected n-point functions of smeared fields at equal times vanish rapidly when the distance between any two points goes to infinity. More precisely:
Before jumping into a rigorous discussion, let us discuss why Ruelle’s cluster theorem is relevant to our problem. When \(t\) becomes large, the wavefunctions \(h^2_t(\vec{x}_2)\) and \(h^1_t(\vec{x}_1)\) become localized in regions of space that are far apart from each other. Therefore, the integral \(I(t)\) in Equation 4.49 gets contributions only from regions in which \(|\vec{x}_1-\vec{x}_2|\) and, hence \(R(\vec{x})\) is large. However in this region, the connected n-point function is very small, thanks to Ruelle’s cluster theorem. Which means that the integral \(I(t)\) will become very small when \(t\) becomes large.
Let us go back to the task of designing a bound for \(I(t)\). It is convenient to introduce the variables \[ \begin{gather*} \vec{z}_\alpha = \vec{x}_\alpha - \vec{x}_1 \ , \end{gather*} \tag{4.50}\] and we use translational invariance in the definition \[ \begin{align*} w(\vec{z}_2, \vec{z}_3, \vec{z}_4) \equiv & \langle \phi^{\chi_2}(0,\vec{z}_4) \phi^{\chi_1}(0,\vec{z}_3) \phi^{\chi_1}(0,\vec{z}_2)^\dag \phi^{\chi_2}(0)^\dag \rangle_c \\ = & \langle \phi^{\chi_2}(t,\vec{x}_4) \phi^{\chi_1}(t,\vec{x}_3) \phi^{\chi_1}(t,\vec{x}_2)^\dag \phi^{\chi_2}(t,\vec{x}_1)^\dag \rangle_c \ . \end{align*} \tag{4.51}\]
Following an inspiring chain of inequalities, Ruelle’s cluster theorem implies that, for any \(r > 0\), a constant \(B_r\) exists such that \[ \begin{gather*} \left| w(\vec{z}_2, \vec{z}_3, \vec{z}_4) \right| \le \frac{ B_r }{ ( 1 + |\vec{z}_2| )^{r} \, ( 1 + |\vec{z}_3| )^{r} \, ( 1 + |\vec{z}_4| )^{r} } \ , \qquad \forall \vec{z}_2, \vec{z}_3, \vec{z}_4 \ . \end{gather*} \tag{4.52}\]
The maximal distance between any two points of the 4-point function satisfies \[ \begin{align*} R(\vec{x}) = & \max\{ |\vec{x}_2-\vec{x}_1| \, , \, |\vec{x}_3-\vec{x}_1| \, , \, |\vec{x}_4-\vec{x}_1| \, , \, |\vec{x}_3-\vec{x}_2| \, , \, |\vec{x}_4-\vec{x}_2| \, , \, |\vec{x}_4-\vec{x}_3| \, , \, \} \\ = & \max\{ |\vec{z}_2| \, , \, |\vec{z}_3| \, , \, |\vec{z}_4| \, , \, |\vec{z}_3-\vec{z}_2| \, , \, |\vec{z}_4-\vec{z}_2| \, , \, |\vec{z}_4-\vec{z}_3| \} \\ \ge & \max\{ |\vec{z}_2| \, , \, |\vec{z}_3| \, , \, |\vec{z}_4| \} = \max_\alpha |\vec{z}_\alpha| \ . \end{align*} \] Ruelle’s cluster theorem, together with the above inequality, yields \[ \begin{gather*} \left| w(\vec{z}_2, \vec{z}_3, \vec{z}_4) \right| \le A_r R(\vec{x})^{-r} \le A_r \left\{ \max_\alpha |\vec{z}_\alpha| \right\}^{-r} \ , \end{gather*} \] valid for any \(r>0\) and \(\vec{z}_\alpha\). The function \(w(\vec{z}_2, \vec{z}_3, \vec{z}_4)\) is smooth and, thanks to the above bound, it goes to zero at infinity. Hence, it is bounded. In this case, a constant \(B_r\) exists such that \[ \begin{gather*} \left| w(\vec{z}_2, \vec{z}_3, \vec{z}_4) \right| \le B'_r \left\{ 1 + \max_\alpha |\vec{z}_\alpha| \right\}^{-r} \ . \end{gather*} \] In fact, one can easily prove with elementary methods that \(B'_r = 2^r \max\{ A_r, \|w\|_\infty \}\) is a valid choice. Using the trivial inequality \(1 + \max_\alpha |\vec{z}_\alpha| \ge 1 + |\vec{z}_\beta|\), valid for any \(\beta\), we write \[ \begin{gather*} \left| w(\vec{z}_2, \vec{z}_3, \vec{z}_4) \right| \le B'_r ( 1 + |\vec{z}_\beta| )^{-r} \ . \end{gather*} \] Multipliying the three inequalities for \(\beta = 2,3,4\), we get \[ \begin{gather*} \left| w(\vec{z}_2, \vec{z}_3, \vec{z}_4) \right|^3 \le (B'_r)^3 \prod_\beta ( 1 + |\vec{z}_\beta| )^{-r} \ , \end{gather*} \] which is Equation 4.52 with \(r\) replaced by \(r/3\).
We use this bound to estimate the integral \(I(t)\) in Equation 4.49, after using the change of variables in Equation 4.50: \[ \begin{align*} | I(t) | \le & B_{3r} \int d^3x_1 \, d^3z_2 \, d^3z_3 \, d^3z_4 \, \frac{ \left| h^1_t(\vec{x}_1+\vec{z}_4) \, h^2_t(\vec{x}_1+\vec{z}_3) \, h^2_t(\vec{x}_1+\vec{z}_2) \, h^1_t(\vec{x}_1) \right| }{ ( 1 + |\vec{z}_2| )^{r} \, ( 1 + |\vec{z}_3| )^{r} \, ( 1 + |\vec{z}_4| )^{r} } \\ \le & B_{3r} \, \| h^1_t \|_\infty \, \| h^2_t \|_\infty \, \int \frac{ d^3z_3 }{ ( 1 + |\vec{z}_3| )^{r} } \int \frac{ d^3z_4 }{ ( 1 + |\vec{z}_4| )^{r} } \int d^3x_1 \, d^3z_2 \, \frac{ \left| h^2_t(\vec{x}_1+\vec{z}_2) \, h^1_t(\vec{x}_1) \right| }{ ( 1 + |\vec{z}_2| )^{r} } \ . \end{align*} \tag{4.53}\] Using their definition given in Equation 3.7, the position-space wavefunctions \(h^\alpha_t(\vec{x})\) can be seen to be bounded by a \(t\)-independent constant: \[ \begin{gather*} \| h^\alpha_t \|_\infty = \max_{\vec{x}} | h^\alpha_t(\vec{x}) | \le \int \frac{d^3 p}{(2\pi)^3 \sqrt{ 2 E(\vec{p}) }} \, \left| \hat{h}^\alpha(\vec{p}) \right| \ , \end{gather*} \tag{4.54}\] Defining the \(t\)-independent constant \[ \begin{gather*} C_r = B_{3r} \, \left\{ \int \frac{ dz }{ ( 1 + |\vec{z}| )^{r} } \right\}^2 \prod_{\alpha=1}^2 \int \frac{d^3 p}{(2\pi)^3 \sqrt{ 2 E(\vec{p}) }} \, \left| \hat{h}^\alpha(\vec{p}) \right| \ , \end{gather*} \tag{4.55}\] which is finite for \(r > 3\), we can write our bound more compactly as \[ \begin{gather*} | I(t) | \le C_r \, \int d^3x_1 \, d^3x_2 \, \frac{ | h^2_t(\vec{x}_2) \, h^1_t(\vec{x}_1) | }{ ( 1 + |\vec{x}_2-\vec{x}_1| )^{r} } \equiv J(t) \ . \end{gather*} \tag{4.56}\] Recall that we want to prove that the integral \(I(t)\) vanishes rapidly as \(t \to +\infty\). In order to do this, it is enough to prove that the integral \(J(t)\) vanishes rapidly as \(t \to +\infty\).
4.4.4 Step 4: asymptotic localization of the wavefunctions
Given a momentum-space wavefunction \(\hat{f}(\vec{p})\) with compact support, we have already discussed in Section 3.1.1 how the corresponding position-space wavefunction \(f_t(\vec{x})\) is localized in space for any time \(t\) only in a weak sense: \(f_t(\vec{x})\) decays rapidly at spatial infinity, but it cannot have compact support. However, in the \(t \to +\infty\) limit, a stronger localization occurs, which is best understood in terms of the velocity \(\vec{v} = \vec{x}/t\).
We introduce the velocity-space wavefunction, for \(t>0\), as \[ \begin{gather*} \mathring{f}_t(\vec{v}) = \begin{cases} t^{3/2} e^{i m \gamma^{-1} t} f_t(t \vec{v}) \quad & \text{if } |\vec{v}| < 1 \\ t^{3/2} f_t(t \vec{v}) \quad & \text{if } |\vec{v}| \ge 1 \end{cases} \ , \end{gather*} \tag{4.57}\] where \(\gamma = ( 1 - \vec{v}^2 )^{-1/2}\) is the Lorentz factor associated to the velocity \(\vec{v}\). Notice that the velocity-space wavefunction \(\mathring{f}_t(\vec{v})\) is discontinuous at \(|\vec{v}| = 1\), but it is smooth everywhere else. Since \(|\mathring{f}_t(\vec{v})| = t^{3/2} | f_t(t \vec{v}) |\), from the position-space wavefunction it inherits the following properties:
\(\mathring{f}_t(\vec{v})\) is bounded;
\(\mathring{f}_t(\vec{v})\) decays rapidly for \(|\vec{v}| \to +\infty\);
\(\mathring{f}_t(\vec{v})\) does not have compact support for any time \(t>0\).
The strange prefactors in Equation 4.57 are chosen so that, \(\mathring{f}_t(\vec{v})\) has a well-defined limit as \(t \to +\infty\): \[ \begin{gather*} \mathring{f}_\infty(\vec{v}) \equiv \lim_{t \to +\infty} \mathring{f}_t(\vec{v}) = \begin{cases} \left( \frac{m \gamma}{2 \pi i} \right)^{3/2} \sqrt{ \frac{\gamma}{2m} } \, \hat{f}(m \gamma \vec{v}) \quad & \text{if } |\vec{v}| < 1 \\ 0 \quad & \text{if } |\vec{v}| \ge 1 \end{cases} \end{gather*} \tag{4.58}\]
In the \(t \to +\infty\) limit, the integral defining the position-space wavefunction expressed in terms of the velocity, i.e. \[ \begin{gather*} f_t(t \vec{v}) = \int \frac{d^3 p}{(2\pi)^3 \sqrt{ 2 E(\vec{p}) }} \, e^{-i t [ E(\vec{p}) - \vec{p} \vec{v}]} \, \hat{f}(\vec{p}) \end{gather*} \] localizes around points of stationary phase, i.e. values of \(\vec{p}\) which satisfy the condition \[ \begin{align*} 0 = \nabla_{\vec{p}} [ E(\vec{p}) - \vec{p} \vec{v}] = \frac{\vec{p}}{E(\vec{p})} - \vec{v} \quad \Leftrightarrow \quad \vec{p} = \frac{ m \vec{v} }{ \sqrt{ 1 - v^2 } } \equiv m \gamma \vec{v} \ , \end{align*} \] where one should notice that a solution for \(\vec{p}\) exists only if \(|\vec{v}| < 1\). Changing variable from \(\vec{p}\) to \(\vec{k}\) according to \[ \begin{gather*} \vec{p} = m \gamma \vec{v} + t^{-1/2} \vec{k} \ , \end{gather*} \] in the \(t \to +\infty\) limit, the phase in the complex exponential and the momentum-space wavefunction weighted with \(1/\sqrt{ 2 E(\vec{p}) }\) factor can be expanded as \[ \begin{align*} & t [ E(\vec{p}) - \vec{p} \vec{v}] = m \gamma^{-1} t + \frac{\vec{k}^2 - (\vec{v}\vec{k})^2}{2 m \gamma} + O(t^{-1/2} \xi) \ , \\ & \frac{ \hat{f}(\vec{p}) }{\sqrt{ 2 E(\vec{p}) }} = \frac{ \hat{f}(m \gamma \vec{v}) }{ \sqrt{ 2 m \gamma } } + O(t^{-1/2} \xi) \ . \end{align*} \] Plugging these expansions into the integral defining \(f_t(t \vec{v})\) in Equation 4.57, we get \[ \begin{gather*} f_t(t \vec{v}) = t^{-3/2} e^{- i m \gamma^{-1} t} \frac{ \hat{f}(m \gamma \vec{v}) }{ \sqrt{ 2 m \gamma } } \int \frac{d^3 k}{(2\pi)^3} \, e^{-i \frac{\vec{k}^2 - (\vec{v}\vec{k})^2}{2 m \gamma} } \, \left\{ 1 + O(t^{-1}) \right\} \end{gather*} \] At this order, the integral over \(\vec{k}\) is an improper Gaussian integral and can be computed explicitly: \[ \begin{gather*} \int \frac{d^3 k}{(2\pi)^3} \, e^{-i \frac{\vec{k}^2 - (\vec{v}\vec{k})^2}{2 m \gamma} } = \gamma \left( \frac{m \gamma}{2 \pi i} \right)^{3/2} \ . \end{gather*} \] Equation 4.58 follows easily by combining the previous two equations.
The crucial observation is that the velocity-space wavefunction \(\mathring{f}_\infty(\vec{v})\) at infinite time has compact support, which is exactly the strong localization property we mentioned at the beginning of the section. We want to characterize the support of \(\mathring{f}_\infty(\vec{v})\): \(\vec{v}\) belongs to the closed support of \(\mathring{f}_\infty(\vec{v})\) if and only if \(\vec{p} = m \gamma \vec{v}\) belongs to the closed support of \(\hat{f}(\vec{p})\). Notice that the relation between \(\vec{p}\) and \(\vec{v}\) can be inverted, yielding \(\vec{v} = \vec{p}/E(\vec{p})\). Recalling the definition of the set of velocities in Equation 4.7, it is clear that the closed support of \(\mathring{f}_\infty(\vec{v})\) coincides exactly with the set of velocities \(V(f)\) associated to the momentum-space wavefunction \(\hat{f}(\vec{p})\).
In order to prove that the integral \(J(t)\) defined in Equation 4.56 vanishes rapidly as \(t \to +\infty\), we will need a sharper result: roughly speaking, away from the set of velocities \(V(f)\), the function \(\mathring{f}_t(\vec{v})\) vanishes rapidly both in \(t\) and \(\vec{v}\). More precisely, Ruelle proved the following:
Since the sets of velocities \(V(h^1)\) and \(V(h^2)\) of the momentum-space wavefunctions appearing in the integral \(I(t)\) in Equation 4.56 are compact and disjoint, then it is possible to construct to open sets \(U_1\) and \(U_2\) with the following properties:
\(V(h^\alpha)\) is a subset of \(U_\alpha\);
the distance \(d(U_1,U_2)\) between the two sets \(U_1\) and \(U_2\) is strictly positive.
Let \(d_V\) be the distance between the two sets of velocities \(V(h^1)\) and \(V(h^2)\), i.e. \[ \begin{gather*} d_V = \inf_{ \crampedsubstack{ \vec{v}^1 \in V(h^1) \\ \vec{v}^2 \in V(h^2) } } |\vec{v}^1 - \vec{v}^2| \ . \end{gather*} \] If \(d_V\) were zero, then a sequence of velocities \(\vec{v}^1_n \in V(h^1)\) and \(\vec{v}^2_n \in V(h^2)\) would exist such that \(|\vec{v}^1_n - \vec{v}^2_n| \to 0\) as \(n \to +\infty\). Since the sets \(V(h^1)\) and \(V(h^2)\) are compact, we can assume that the sequences \(\vec{v}^1_n\) and \(\vec{v}^2_n\) are convergent (if not, it is possible to extract convergent subsequences). Since the distance between the two sequences goes to zero, the two limits must coincide. The common limit must belong to both sets \(V(h^1)\) and \(V(h^2)\), which contradicts the assumption that the two sets are disjoint. Therefore, \(d_V > 0\).
Define \[ \begin{gather*} U_\alpha = \bigcup_{\vec{v} \in V(h^\alpha)} B_{d_V/4}(\vec{v}) \ , \end{gather*} \] where \(B_\delta(\vec{v})\) is the open ball of radius \(\delta\) centered at \(\vec{v}\). The two sets \(U_1\) and \(U_2\) are open, contain the sets of velocities \(V(h^1)\) and \(V(h^2)\), and are disjoint. Given two points \(\vec{u}_1 \in U_1\) and \(\vec{u}_2 \in U_2\), by definition of the sets \(U_1\) and \(U_2\), there exist two points \(\vec{v}_1 \in V(h^1)\) and \(\vec{v}_2 \in V(h^2)\) such that \(|\vec{u}_1 - \vec{v}_1| < d_V/4\) and \(|\vec{u}_2 - \vec{v}_2| < d_V/4\). Therefore, the distance between the two points \(\vec{u}_1\) and \(\vec{u}_2\) is bounded by \[ \begin{gather*} |\vec{u}_1 - \vec{u}_2| \ge |\vec{v}_1 - \vec{v}_2| - |\vec{u}_1 - \vec{v}_1| - |\vec{u}_2 - \vec{v}_2| \ge d_V - d_V/4 - d_V/4 = d_V/2 > 0 \ , \end{gather*} \] which means that the distance between the two sets \(U_1\) and \(U_2\) is at least \(d_V/2 > 0\).
Since we are interested in asymptotic bounds at large \(t\), it is not restrictive to assume \(t>1\). In the integral in the r.h.s. of Equation 4.56, we use the change of variables \(\vec{x}_\alpha = t \vec{v}_\alpha\): \[ \begin{gather*} J(t) = C_r \int d^3v_1 \, d^3v_2 \, \frac{ | \mathring{h}^2_t(\vec{v}_2) \, \mathring{h}^1_t(\vec{v}_1) | }{ ( 1 + t |\vec{v}_2-\vec{v}_1| )^{r} } \ . \end{gather*} \tag{4.60}\] We split the integration domain into four pieces, which yields the decomposition: \[ \begin{gather*} J(t) = J(t; U_1, U_2) + J(t; U_1, U_2^c) + J(t; U_1^c, U_2) + J(t; U_1^c, U_2^c) \ , \end{gather*} \tag{4.61}\] where \(U_\alpha^c\) is the complement of the set \(U_\alpha\) in \(\mathbb{R}^3\), and we define the integrals \[ \begin{gather*} J(t; \Omega_1, \Omega_2) = C_r \int_{\Omega_1} d^3v_1 \int_{\Omega_2} d^3v_2 \, \frac{ | \mathring{h}^2_t(\vec{v}_2) \, \mathring{h}^1_t(\vec{v}_1) | }{ ( 1 + t |\vec{v}_2-\vec{v}_1| )^{r} } \ . \end{gather*} \tag{4.62}\]
For \(\vec{v}_\alpha \in U_\alpha^c\), we apply Theorem 4.4 which implies that a constant \(D_{\alpha,r}\) exists for any \(r>0\) such that \[ \begin{gather*} | \mathring{h}^\alpha_t(\vec{v}_\alpha) | \le D_{\alpha,r} (1 + |\vec{v}|)^{-r} t^{-r} \ . \end{gather*} \tag{4.63}\] For \(\vec{v}_\alpha \in U_\alpha\), we use the bound: \[ \begin{gather*} | \mathring{h}^\alpha_t(\vec{v}_\alpha) | = t^{3/2} | h^\alpha_t(t \vec{v}_\alpha) | \le t^{3/2} \int \frac{d^3 p}{(2\pi)^3 \sqrt{ 2 E(\vec{p}) }} \, \left| \hat{h}^\alpha(\vec{p}) \right| \equiv D'_\alpha t^{3/2} \ . \end{gather*} \tag{4.64}\]
Let us look individually at the four integrals in Equation 4.61.
Consider \(\Omega_1 = U_1^c\) and \(\Omega_2 = U_2^c\). Equation 4.63 with \(\alpha=1,2\) and the trivial bound \(( 1 + t |\vec{v}_2-\vec{v}_1| )^{-r} \le 1\) yield: \[ \begin{gather*} J(t; U_1^c, U_2^c) \le C_r D_{1,r} D_{2,r} \left\{ \int \frac{ d^3v }{ ( 1 + |\vec{v}| )^{r} } \right\}^2 t^{-2r} \ . \end{gather*} \tag{4.65}\]
Consider \(\Omega_1 = U_1^c\) and \(\Omega_2 = U_2\). Equation 4.63 with \(\alpha=1\), Equation 4.64 with \(\alpha=2\) and the trivial bound \(( 1 + t |\vec{v}_2-\vec{v}_1| )^{-r} \le ( 1 + |\vec{v}_2-\vec{v}_1| )^{-r}\) yield: \[ \begin{align*} J(t; U_1^c, U_2) \le & C_r D_{1,r} D'_2 t^{3/2-r} \int \frac{ d^3v_1 \, d^3v_2 }{ ( 1 + |\vec{v}_1| )^{r} ( 1 + |\vec{v}_2-\vec{v}_1| )^{r} } \\ = & D_{1,r} D'_2 \left\{ \int \frac{ d^3v }{ ( 1 + |\vec{v}| )^{r} } \right\}^2 t^{3/2-r} \ . \end{align*} \tag{4.66}\]
Consider \(\Omega_1 = U_1\) and \(\Omega_2 = U_2^c\). Equation 4.64 with \(\alpha=1\), Equation 4.63 with \(\alpha=2\) and the trivial bound \(( 1 + t |\vec{v}_2-\vec{v}_1| )^{-r} \le ( 1 + |\vec{v}_2-\vec{v}_1| )^{-r}\) yield: \[ \begin{align*} J(t; U_1, U_2^c) \le & C_r D'_1 D_{2,r} t^{3/2-r} \int \frac{ d^3v_1 \, d^3v_2 }{ ( 1 + |\vec{v}_2| )^{r} ( 1 + |\vec{v}_2-\vec{v}_1| )^{r} } \\ = & D'_1 D_{2,r} \left\{ \int \frac{ d^3v }{ ( 1 + |\vec{v}| )^{r} } \right\}^2 t^{3/2-r} \ . \end{align*} \tag{4.67}\]
Consider \(\Omega_1 = U_1\) and \(\Omega_2 = U_2\). Equation 4.64 with \(\alpha=1,2\) and the trivial bound \(( 1 + t |\vec{v}_2-\vec{v}_1| )^{-r} \le [ t d(U_1,U_2) ]^{-r}\) yield: \[ \begin{gather*} J(t; U_1, U_2) \le C_r D'_1 D'_2 d(U_1,U_2)^{-r} t^{3-r} \left\{ \int \frac{ d^3v }{ ( 1 + |\vec{v}| )^{r} } \right\}^2 \ . \end{gather*} \tag{4.68}\]
Since in the above bounds \(r\) can be arbitrarily large (we only needed \(r>3\) in order to prove Equation 4.56), then each term in the r.h.s. of Equation 4.61 vanishes rapidly as \(t \to +\infty\). This, in turn, implies that the integral \(J(t)\) in Equation 4.56 vanishes rapidly as \(t \to +\infty\), and so does \(I(t)\) defined in Equation 4.49. Thanks to Equation 4.48, the 4-point function which appears in Equation 4.36 vanishes rapidly as \(t \to +\infty\) as well, and so does the time derivative of \(| \Psi(t) \rangle\). This is concludes the proof of Theorem 4.3.
4.5 Problems
Download Matthew Black’s solutions.
Problem 4.1 Let \(\hat{f}^\alpha(\vec{p})\) be the wavefunctions considered in this chapter. Calculate the following connected 2-point functions \[ \begin{gather*} \langle A_t(f^\alpha) A_t(f^\beta)^\dag \rangle_c \end{gather*} \] for any combination of \(\alpha, \beta = 1,2\), in terms of the momentum-space wavefunctions \(\hat{f}^\alpha(\vec{p})\).
Problem 4.2 Let \(\hat{f}^\alpha(\vec{p})\) be the wavefunctions considered in this chapter. Derive the cluster decomposition of the following 4-point function: \[ \begin{gather*} \langle \Omega | A_t(f^1) A_t(f^2) A_t(f^2)^\dag A_t(f^1)^\dag | \Omega \rangle \ , \end{gather*} \] identifying which terms vanish identically. Use the results of Problem 4.1 to simplify your answer.
Problem 4.3 Let \(\hat{f}^\alpha(\vec{p})\) be the wavefunctions considered in this chapter. Calculate the \(t \to +\infty\) limit of the 4-point function defined in Problem 4.2, and show that the limit is approached with an error that vanishes rapidly.