Lecture 3

Chasse_neige

Foresee

How fundamental principles restrain interactions, delve into scattering amplitude (using analyticity to restrain the form of the scattering amplitude).

Scattering Theory

Axioms of Scattering Theory

1. Assume a Hilbert Space $\mathcal{H}$ with a Hamiltonian $H$, there exists a lowest-energy state-vacuum $|\mathrm{\Omega }⟩$, which is normalized. $|\mathrm{\Omega }⟩$ trivially spans a subspace of $\mathcal{H}$: ${\mathcal{H}}^{(0)}$.

2. Assume that there exists a subspace ${\mathcal{H}}^{(1)}\in \mathcal{H}$ - 1PS (one-particle state), the state of all stable particles. 1 PS is specified by $|n,\overrightarrow{p},\sigma ⟩$.

$$H|n,\overrightarrow{p},\sigma ⟩=E_p|n,\overrightarrow{p},\sigma ⟩E_p=\sqrt{m^2+p^2}$$

​ Normalize 1 PS:

$$⟨\overrightarrow{p}|\overrightarrow{q}⟩=2E_p(2\pi {)}^3{\delta }^3(\overrightarrow{p}-\overrightarrow{q})$$

​ The complete Hilbert space

$$\mathcal{H}={\mathcal{H}}^{(0)}\oplus {\mathcal{H}}^{(1)}\oplus {\mathcal{H}}_{\text{rest}}$$

Fock Space

Redifine the Hilbert Space as

$${\mathcal{H}}_0\equiv \underset{p=0}{\overset{\mathrm{\infty }}{⨁}}[{\mathcal{H}}^{(1)}{]}^p$$$$[{\mathcal{H}}^{(1)}{]}^0\equiv {\mathcal{H}}^{(0)}$$

Naturally, we induce the Hamiltonian $H_0$ (‘free Hamiltonian’), and we can interpret $H-H_0$ as interaction.

3. We can choose a complete set of eigenstate of $\mathcal{H}$ called in states$$|{\mathrm{\Psi }}_{+}⟩\equiv |\{n_1,{\overrightarrow{p}}_1,{\sigma }_1,\cdots ,n_N,{\overrightarrow{p}}_N,{\sigma }_N{\}}_{+}⟩N=0,1,2,\cdots$$and $|{\mathrm{\Psi }}_{+}⟩$ asymptotically approach $|{\mathrm{\Psi }}_0⟩\in {\mathcal{H}}_0$ as $t\to \mathrm{\infty }$. (in the wave-packet sense)

$$\underset{t\to -\mathrm{\infty }}{lim}{\int }_{p}g(\{\overrightarrow{p}\})|\{\overrightarrow{p}{\}}_{+}⟩={\int }_{p}g(\{\overrightarrow{p}\})|\{\overrightarrow{p}{\}}_0⟩$$

​ where $g$ is the wave packet. Normalize $|{\mathrm{\Psi }}_{+}⟩$ in the same way as $|{\mathrm{\Psi }}_0⟩$. However, in a system with Lang-range force, $|{\mathrm{\Psi }}_{+}⟩\to |{\mathrm{\Psi }}_0⟩$ not fast enough which will induce IR divergence.

4. Another complete set of the eigenstates of the full Hamiltonian $H$, approaching multiparticle states in the far future. ‘out state’, we call them $|{\mathrm{\Psi }}_{-}⟩$. By construction

$$|{\mathrm{\Psi }}_{-}⟩\{\begin{array}{ll}=|{\mathrm{\Psi }}_{+}⟩& (\text{if}|{\mathrm{\Psi }}_{+}⟩\in {\mathcal{H}}^{(0)}\oplus {\mathcal{H}}^{(1)})\\ \ne |{\mathrm{\Psi }}_{+}⟩& (\text{otherwise})\end{array}$$

5. Amplitude $|{\mathrm{\Psi }}_{+}⟩\to |{\mathrm{\Phi }}_{-}⟩$

   S-matrix $S_{\mathrm{\Phi }\mathrm{\Psi }}=⟨{\mathrm{\Phi }}_{-}|{\mathrm{\Psi }}_{+}⟩$

   S-operator: acting on states in ${\mathcal{H}}_0$

   $$⟨{\mathrm{\Phi }}_0|S|{\mathrm{\Psi }}_0⟩\equiv ⟨{\mathrm{\Phi }}_{-}|{\mathrm{\Psi }}_{+}⟩$$

Properties of S-matrix

Lorentz inv, Locality, Unitarity

Poincare inv.

$$[S,H]=[S,P]=[S,J]=[S,K]=0$$

Consequences: $|{\mathrm{\Psi }}_{+}⟩\equiv |\{{\overrightarrow{p}}_1,\cdots ,{\overrightarrow{p}}_M{\}}_{+}⟩$ and $|{\mathrm{\Phi }}_{-}⟩=|\{{\overrightarrow{q}}_1\cdots ,{\overrightarrow{q}}_N{\}}_{-}⟩$

$$S_{\mathrm{\Phi }\mathrm{\Psi }}=(2\pi {)}^4{\delta }^4(q_1+\cdots +q_N-p_1-\cdots -p_M)A_{\mathrm{\Phi }\mathrm{\Psi }}$$

where $A_{\mathrm{\Phi }\mathrm{\Psi }}$ is a scalar made of $p$ and $q$’s.

Refined version of P-inv.

Globel symmetry act quasi-locally. In particular, we can perform separate LGTs to each particle in $|{\mathrm{\Psi }}_{+}⟩$ and $|{\mathrm{\Phi }}_{-}⟩$. We require $S_{\mathrm{\Phi }\mathrm{\Psi }}$ transform covariantly as UIRs of LGTs.

$$⟨{\mathrm{\Phi }}_{-}|\{\cdots (\overrightarrow{p},\sigma {)}_i\cdots {\}}_{+}⟩\to \sum _{{\sigma }^{\prime }}{W}_{{\sigma }^{\prime }{\sigma }^{\prime }}⟨{\mathrm{\Phi }}_{-}|\{\cdots (\overrightarrow{p},{\sigma }^{\prime }{)}_i\cdots {\}}_{+}⟩$$

$A_{\mathrm{\Phi }\mathrm{\Psi }}$ is a Lorentz scalar constructed by any of $p^{\mu }$ and $q^{\mu }$ s, and $e_i$ for each in-particle, and $e_i^{\ast }$ for each out-particle.

Locality

very important, highly ununique - Analyticity

Cluster decomposition: for an in state $|{\mathrm{\Psi }}_{+}⟩=|\{{\psi }_1,{\psi }_2{\}}_{+}⟩$, where ${\psi }_1$ and ${\psi }_2$ are very far apart.

$$\underset{|{\overrightarrow{x}}_1-{\overrightarrow{x}}_2|\to \mathrm{\infty }}{lim}⟨\{{\varphi }_1,{\varphi }_2{\}}_{-}|\{{\psi }_1,{\psi }_2{\}}_{+}⟩=⟨{\varphi }_{1-}|{\psi }_{1+}⟩⟨{\varphi }_{2-}|{\psi }_{2+}⟩$$

Question: How singular can $A_{\mathrm{\Phi }\mathrm{\Psi }}$ be as a function of p and q?

It can be very singular as ${\delta }^3({\overrightarrow{p}}_1-{\overrightarrow{q}}_1){\delta }^3({\overrightarrow{p}}_2-{\overrightarrow{q}}_2)$

Redefine the interaction matrix as

$$A_{p_1{p}_2\to p_3{p}_4}={\delta }_{13}{\delta }_{24}+(2\pi {)}^4{\delta }^4(p_1+p_2-p_3-p_4)\times i{M}_{34,12}$$

With the property

$$\underset{|{\overrightarrow{x}}_1-{\overrightarrow{x}}_2|\to \mathrm{\infty }}{lim}{M}_{34,12}=0$$

Interpretation

$$\underset{|{\overrightarrow{x}}_1-{\overrightarrow{x}}_2|\to \mathrm{\infty }}{lim}\int \prod _a\left[{{\mathrm{d}}^3\overrightarrow{p}}_a{e}^{i{\overrightarrow{p}}_a\cdot {\overrightarrow{x}}_a}\right]M_{34,12}=0$$

Consider a simple example

$$g(\overrightarrow{x})=\int {\mathrm{d}}^3\overrightarrow{p}f(\overrightarrow{p})e^{i\overrightarrow{p}\cdot \overrightarrow{x}}$$

If $f(\overrightarrow{p})$ is analytic within the strip, then $g(x)\to 0$ exponentially fast as $x\to \mathrm{\infty }$.

We wat a long-range force $g(x)\to \frac{1}{x}$ as $x\to \mathrm{\infty }$, which means $f(p)$ is non-analytic at $p=0$.

Summary

$M$ has no singularity, other than pole/branch cuts.

Unitarity

$S$ is unitary, which means $S^{†}S=1$.

Optical theorem: $2\to 2$ forward scattering

$$S=1+i(2\pi {)}^4{\delta }^4(p)M$$$$\mathrm{\Im }M(p_1{p}_2\to p_1{p}_2)=\sum _{X\in \text{final states}}(2\pi {)}^4{\delta }^4(p_1+p_2-p_X)|M(p_1{p}_2\to X){|}^2$$

Unitarity tells us that the poles in S-matrices cannot be more singular than simple poles.

$s=-(p_1+p_2{)}^2$, if $M$ has a simple pole at $s=m^2-iϵ$

$$\begin{array}{c}M\to \frac{R}{-s+m^2-iϵ}=P\frac{R}{-s+m^2}+i\pi R\delta (s-m^2)\\ \Rightarrow \mathrm{\Im }M=\pi R\delta (s-m^2)\end{array}$$

RHS of Optical theorem

$$\sum _X{\delta }^4(p_{12}-p_X)|M{|}^2{M}_{12\to X}=C+\cdots$$$$\begin{array}{c}\Rightarrow \int \frac{{{\mathrm{d}}^{}\overrightarrow{p}}_X}{(2\pi {)}^3}\frac{1}{2E_{pX}}{\delta }^4(p_X-p_1-p_2)\times |C{|}^2+\cdots \\ \propto \delta (s-m^2)\times |C{|}^2+\cdots \end{array}$$

Remarks: causality [$E_X$]

• P-inv., Locality, Unitarity, Causality $\Rightarrow$ Analyticity of S-matrix, which is a powerful tool to constrain the form of S-matrix.