phy2403

New aggregate collection of notes for PHY2403, QFT I : up to lecture 21.

December 2, 2018 phy2403 No comments

I’ve now uploaded a new version of my class notes for PHY2403, the UofT Quantum Field Theory I course, taught this year by Prof. Erich Poppitz.

This version includes the following content.  Fermion related content is new since the last upload, as well as some work to make the scattering content (lecture 17,18) somewhat sensible (it was in it’s raw post-class form before, which made little sense)

  • Chapter 1: Fields, units, and scales.
  • 1.1: What is a field?
  • 1.2: Scales.
  • 1.3: Natural units.
  • 1.4: Gravity.
  • 1.5: Cross section.
  • 1.6: Problems.
  • Chapter 2: Lorentz transformations.
  • 2.1: Lorentz transformations.
  • 2.2: Determinant of Lorentz transformations.
  • 2.3: Problems.
  • Chapter 3: Classical field theory.
  • 3.1: Field theory.
  • 3.2: Actions.
  • 3.3: Principles determining the form of the action.
  • 3.4: Principles (cont.)
  • 3.5: Least action principle.
  • 3.6: Problems.
  • Chapter 4: Canonical quantization, Klein-Gordon equation, SHOs, momentum space representation, raising and lowering operators.
  • 4.1: Canonical quantization.
  • 4.2: Canonical quantization (cont.)
  • 4.3: Momentum space representation.
  • 4.4: Quantization of Field Theory.
  • 4.5: Free Hamiltonian.
  • 4.6: QM SHO review.
  • 4.7: Discussion.
  • 4.8: Problems.
  • Chapter 5: Symmetries.
  • 5.1: Switching gears: Symmetries.
  • 5.2: Symmetries.
  • 5.3: Spacetime translation.
  • 5.4: 1st Noether theorem.
  • 5.5: Unitary operators.
  • 5.6: Continuous symmetries.
  • 5.7: Classical scalar theory.
  • 5.8: Last time.
  • 5.9: Examples of symmetries.
  • 5.10: Scale invariance.
  • 5.11: Lorentz invariance.
  • 5.12: Problems.
  • Chapter 6: Lorentz boosts, generators, Lorentz invariance, microcausality.
  • 6.1: Lorentz transform symmetries.
  • 6.2: Transformation of momentum states.
  • 6.3: Relativistic normalization.
  • 6.4: Spacelike surfaces.
  • 6.5: Condition on microcausality.
  • Chapter 7: External sources
  • 7.1: Harmonic oscillator.
  • 7.2: Field theory (where we are going).
  • 7.3: Green’s functions for the forced Klein-Gordon equation.
  • 7.4: Pole shifting.
  • 7.5: Matrix element representation of the Wightman function.
  • 7.6: Retarded Green’s function.
  • 7.7: Review: “particle creation problem”.
  • 7.8: Digression: coherent states.
  • 7.9: Problems.
  • Chapter 8: Perturbation theory.
  • 8.1: Feynman’s Green’s function
  • 8.2: Interacting field theory: perturbation theory in QFT.
  • 8.3: Perturbation theory, interaction representation and Dyson formula
  • 8.4: Next time.
  • 8.5: Review
  • 8.6: Perturbation
  • 8.7: Review
  • 8.8: Unpacking it.
  • 8.9: Calculating perturbation
  • 8.10: Wick contractions
  • 8.11: Simplest Feynman diagrams
  • 8.12: Phi fourth interaction
  • 8.13: Tree level diagrams.
  • 8.14: Problems.
  • Chapter 9: Scattering and decay.
  • 9.1: Definitions and motivation.
  • 9.2: Definitions and motivation (cont.)
  • 9.3: Calculating interactions
  • 9.4: Example diagrams.
  • 9.5: The recipe.
  • 9.6: Back to our scalar theory
  • 9.7: Review: S-matrix
  • 9.8: Scattering in a scalar theory
  • 9.9: Decay rates.
  • 9.10: Cross section.
  • 9.11: More on cross section.
  • 9.12: d(LIPS)_2.
  • 9.13: Problems.
  • Chapter 10: Fermions, and spinors.
  • 10.1: Fermions: R3 rotations.
  • 10.2: Lorentz group
  • 10.3: Weyl spinors.
  • 10.4: Lorentz symmetry.
  • 10.5: Dirac matrices.
  • 10.6: Dirac Lagrangian.
  • 10.7: Review.
  • 10.8: Dirac equation.
  • 10.9: Helicity
  • 10.10: Next time.
  • 10.11: Review.
  • 10.12: Normalization.
  • 10.13: Other solution.
  • 10.14: Lagrangian.
  • 10.15: General solution and Hamiltonian.
  • 10.16: Problems.
  • Appendix A: A Useful formulas and review.
  • A.1: Review of old material.
  • A.2: Useful results from new material.
  • Appendix B: Momentum of scalar field.
  • B.1: Expansion of the field momentum.
  • B.2: Conservation of the field momentum.
  • Appendix C: Reflection using Pauli matrices.

Problem set 1-3 solutions are redacted.  If interested (and not a future student of PHY2403), feel free to contact me for an un-redacted copy.

PHY2403H Quantum Field Theory. Lecture 21, Part II: Dirac Hamiltonian, Hamiltonian eigenvalues, general solution, creation and anhillation operators, Dirac Sea, antielectrons. Taught by Prof. Erich Poppitz

December 1, 2018 phy2403 No comments , , , , ,

This post contains a summary of my lecture notes for the second half of last Wednesday’s QFT-I lecture.
[Click here for a PDF with the full notes for this portion of the lecture.]

DISCLAIMER: Very rough notes from class, with some additional side notes.

These are notes for the UofT course PHY2403H, Quantum Field Theory, taught by Prof. Erich Poppitz, fall 2018.

Summary:

  • We found that the Dirac Hamiltonian is
    \begin{equation*}
    H
    =
    \int d^3 x
    \Psi^\dagger
    \lr{
    – i \gamma^0 \gamma^j \partial_j \Psi + m \gamma^0
    }
    \Psi.
    \end{equation*}
  • We found that our plane wave solutions
    \(\Psi_u = u(p) e^{-i p \cdot x}\), and \( \Psi_v = v(p) e^{i p \cdot x} \), were eigenvectors of the operator portion of the Hamiltonian
    \begin{equation*}
    \begin{aligned}
    -\gamma^0 \lr{ i \gamma^j \partial_j – m } \Psi_u &= p_0 \Psi_u \\
    -\gamma^0 \lr{ i \gamma^j \partial_j – m } \Psi_v &= -p_0 \Psi_v.
    \end{aligned}
    \end{equation*}
  • We formed a linear superposition of our plane wave solutions
    \begin{equation}\label{eqn:qftLecture21b:800}
    \Psi(\Bx, t)
    =
    \sum_{s = 1}^2
    \int \frac{d^3 p}{(2 \pi)^3 \sqrt{ 2 \omega_\Bp } }
    \lr{
    e^{-i p \cdot x} u^s_\Bp a_\Bp^s
    +
    e^{i p \cdot x} v^s_\Bp b_\Bp^s
    }.
    \end{equation}
  • and expressed the Dirac Hamiltonian in terms of creation and anhillation operators
    \begin{equation*}
    H_{\text{Dirac}}
    =
    \sum_{r = 1}^2
    \int \frac{d^3 p }{(2\pi)^3 }
    \omega_\Bp
    \lr{
    a^{r \dagger}_\Bp
    a^r_\Bp

    b^{r \dagger}_{-\Bp}
    b^r_{-\Bp}
    }.
    \end{equation*}
  • Finally, we interpreted this using the Dirac Sea argument

    Dirac Sea

  • It was claimed that the \( a, b\)’s satisfied anticommutator relationships
    \begin{equation}\label{eqn:qftLecture21b:940}
    \begin{aligned}
    \symmetric{a^s_\Bp}{a^{r \dagger}_\Bq} &= \delta^{sr} \delta^{(3)}e(\Bp – \Bq) \\
    \symmetric{b^s_\Bp}{b^{r \dagger}_\Bq} &= \delta^{sr} \delta^{(3)}(\Bp – \Bq),
    \end{aligned}
    \end{equation}
    where all other anticommutators are zero
    \begin{equation}\label{eqn:qftLecture21b:960}
    \symmetric{a^r}{b^s} =
    \symmetric{a^r}{b^{s\dagger}} =
    \symmetric{a^{r\dagger}}{b^s} =
    \symmetric{a^{r\dagger}}{b^{s\dagger}} = 0.
    \end{equation}
    and used these to algebraically remove the negative energy states of the Hamiltonian.

PHY2403H Quantum Field Theory. Lecture 21, Part I: Dirac equation solutions, orthogonality conditions, direct products. Taught by Prof. Erich Poppitz

November 29, 2018 phy2403 No comments , , , , ,

[Click here for a PDF of this notes with full details.]

DISCLAIMER: Rough notes from class, with some additional side notes.

These are notes for the UofT course PHY2403H, Quantum Field Theory, taught by Prof. Erich Poppitz, fall 2018.

Overview.

See the PDF above for full notes for the first part of this particular lecture. We covered

  • Normalization:
    \begin{equation*}
    u^{r \dagger} u^{s}
    = 2 p_0 \delta^{r s}.
    \end{equation*}
  • Products of \( p \cdot \sigma, p \cdot \overline{\sigma} \)
    \begin{equation*}
    (p \cdot \sigma) (p \cdot \overline{\sigma})
    =
    (p \cdot \overline{\sigma}) (p \cdot \sigma)
    = m^2.
    \end{equation*}
  • Adjoint orthogonality conditions for \( u \)
    \begin{equation*}
    \overline{u}^r(\Bp) u^{s}(\Bp) = 2 m \delta^{r s}.
    \end{equation*}
  • Solutions in the \( e^{i p \cdot x} \) “direction”
    \begin{equation}\label{eqn:qftLecture21:99}
    v^s(p)
    =
    \begin{bmatrix}
    \sqrt{p \cdot \sigma} \eta^s \\
    -\sqrt{p \cdot \overline{\sigma}} \eta^s \\
    \end{bmatrix},
    \end{equation}
    where \( \eta^1 = (1,0)^\T, \eta^2 = (0,1)^\T \).
  • \(v\) normalization
    \begin{equation*}
    \begin{aligned}
    \overline{v}^r(p) v^s(p) &= – 2 m \delta^{rs} \\
    v^{r \dagger}(p) v^s(p) &= 2 p^0 \delta^{rs}.
    \end{aligned}
    \end{equation*}
  • Dirac adjoint orthogonality conditions.
    \begin{equation*}
    \begin{aligned}
    \overline{u}^r(p) v^s(p) &= 0 \\
    \overline{v}^r(p) u^s(p) &= 0.
    \end{aligned}
    \end{equation*}
  • Dagger orthogonality conditions.
    \begin{equation*}
    \begin{aligned}
    v^{r \dagger}(-\Bp) u^s(\Bp) &= 0 \\
    u^{r\dagger}(\Bp) v^s(-\Bp) &= 0.
    \end{aligned}
    \end{equation*}
  • Tensor product.

    Given a pair of vectors
    \begin{equation*}
    x =
    \begin{bmatrix}
    x_1 \\
    \vdots \\
    x_n \\
    \end{bmatrix},
    \qquad
    y =
    \begin{bmatrix}
    y_1 \\
    \vdots \\
    y_n \\
    \end{bmatrix},
    \end{equation*}
    the tensor product is the matrix of all elements \( x_i y_j \)

    \begin{equation*}
    x \otimes y^\T =
    \begin{bmatrix}
    x_1 \\
    \vdots \\
    x_n \\
    \end{bmatrix}
    \otimes
    \begin{bmatrix}
    y_1 \cdots y_n
    \end{bmatrix}
    =
    \begin{bmatrix}
    x_1 y_1 & x_1 y_2 & \cdots & x_1 y_n \\
    x_2 y_1 & x_2 y_2 & \cdots & x_2 y_n \\
    x_3 y_1 & \ddots & & \\
    \vdots & & & \\
    x_n y_1 & \cdots & & x_n y_n
    \end{bmatrix}.
    \end{equation*}

  • Direct product relations.
    \begin{equation*}
    \begin{aligned}
    \sum_{s = 1}^2 u^s(p) \otimes \overline{u}^s(p) &= \gamma \cdot p + m \\
    \sum_{s = 1}^2 v^s(p) \otimes \overline{v}^s(p) &= \gamma \cdot p – m \\
    \end{aligned}
    \end{equation*}

PHY2403H Quantum Field Theory. Lecture 20: Dirac Lagrangian, spinor solutions to the KG equation, Dirac matrices, plane wave solution, helicity. Taught by Prof. Erich Poppitz

November 27, 2018 phy2403 No comments , , , , , , , , ,

[Here is another PDF only post, containing my notes for Lecture 20 of the UofT QFT I (quantum field theory) course.]

In this lecture Professor Poppitz derived a rest frame solution of the Dirac equation, then demonstrated that the generalization to non-zero momentum satisfied the equation. We also saw that Dirac spinor solutions of the Dirac equation are KG equation solutions, and touched on the relation of some solutions to the helicity operator.

This post doesn’t have a web version, since my latex -> wordpress-mathjax script doesn’t have support for the theorem/lemma environments that I used for Monday’s notes, and I don’t have time to implement that right now.

PHY2403H Quantum Field Theory. Lecture 19: Pauli matrices, Weyl spinors, SL(2,c), Weyl action, Weyl equation, Dirac matrix, Dirac action, Dirac Lagrangian. Taught by Prof. Erich Poppitz

November 24, 2018 phy2403 No comments , , , , , , ,

[Here are my notes for lecture 19 of the UofT course PHY2403H, Quantum Field Theory, taught by Prof. Erich Poppitz, fall 2018.] For this lecture my notes are pdf only, due to length. While the after-class length was 8 pages, it ended up expanded to 17 pages by the time I finished making sense of the material.

These also include a portion of the notes from Lecture 18 (not yet posted), as it made sense to group all the Pauli matrix related content.  This particular set of notes diverges from the format presented in class, as it made sense to me to group things in this particular lecture in a more structured definition, theorem, proof style.  I’ve added a number of additional details that I found helpful, as well as a couple of extra problems (some set as formal problems at the end, and others set as theorem or lemmas in with the rest.)

Reflection using Pauli matrices.

November 22, 2018 phy2403 No comments , , , , ,

[Click here for a PDF of this post with nicer formatting]

In class yesterday (lecture 19, notes not yet posted) we used \( \Bsigma^\T = -\sigma_2 \Bsigma \sigma_2 \), which implicitly shows that \( (\Bsigma \cdot \Bx)^\T \) is a reflection about the y-axis.
This form of reflection will be familiar to a student of geometric algebra (see [1] — a great book, one copy of which is in the physics library). I can’t recall any mention of the geometrical reflection identity from when I took QM. It’s a fun exercise to demonstrate the reflection identity when constrained to the Pauli matrix notation.

Theorem: Reflection about a normal.

Given a unit vector \( \ncap \in \mathbb{R}^3 \) and a vector \( \Bx \in \mathbb{R}^3 \) the reflection of \( \Bx \) about a plane with normal \( \ncap \) can be represented in Pauli notation as
\begin{equation*}
-\Bsigma \cdot \ncap \Bsigma \cdot \Bx \Bsigma \cdot \ncap.
\end{equation*}

To prove this, first note that in standard vector notation, we can decompose a vector into its projective and rejective components
\begin{equation}\label{eqn:reflection:20}
\Bx = (\Bx \cdot \ncap) \ncap + \lr{ \Bx – (\Bx \cdot \ncap) \ncap }.
\end{equation}
A reflection about the plane normal to \( \ncap \) just flips the component in the direction of \( \ncap \), leaving the rest unchanged. That is
\begin{equation}\label{eqn:reflection:40}
-(\Bx \cdot \ncap) \ncap + \lr{ \Bx – (\Bx \cdot \ncap) \ncap }
=
\Bx – 2 (\Bx \cdot \ncap) \ncap.
\end{equation}
We may write this in \( \Bsigma \) notation as
\begin{equation}\label{eqn:reflection:60}
\Bsigma \cdot \Bx – 2 \Bx \cdot \ncap \Bsigma \cdot \ncap.
\end{equation}
We also know that
\begin{equation}\label{eqn:reflection:80}
\begin{aligned}
\Bsigma \cdot \Ba \Bsigma \cdot \Bb &= a \cdot b + i \Bsigma \cdot (\Ba \cross \Bb) \\
\Bsigma \cdot \Bb \Bsigma \cdot \Ba &= a \cdot b – i \Bsigma \cdot (\Ba \cross \Bb),
\end{aligned}
\end{equation}
or
\begin{equation}\label{eqn:reflection:100}
a \cdot b = \inv{2} \symmetric{\Bsigma \cdot \Ba}{\Bsigma \cdot \Bb},
\end{equation}
where \( \symmetric{\Ba}{\Bb} \) is the anticommutator of \( \Ba, \Bb \).
Inserting \ref{eqn:reflection:100} into \ref{eqn:reflection:60} we find that the reflection is
\begin{equation}\label{eqn:reflection:120}
\begin{aligned}
\Bsigma \cdot \Bx –
\symmetric{\Bsigma \cdot \ncap}{\Bsigma \cdot \Bx}
\Bsigma \cdot \ncap
&=
\Bsigma \cdot \Bx –
{\Bsigma \cdot \ncap}{\Bsigma \cdot \Bx}
\Bsigma \cdot \ncap

{\Bsigma \cdot \Bx}{\Bsigma \cdot \ncap}
\Bsigma \cdot \ncap \\
&=
\Bsigma \cdot \Bx –
{\Bsigma \cdot \ncap}{\Bsigma \cdot \Bx}
\Bsigma \cdot \ncap

{\Bsigma \cdot \Bx} \\
&=

{\Bsigma \cdot \ncap}{\Bsigma \cdot \Bx}
\Bsigma \cdot \ncap,
\end{aligned}
\end{equation}
which completes the proof.

When we expand \( (\Bsigma \cdot \Bx)^\T \) and find
\begin{equation}\label{eqn:reflection:n}
(\Bsigma \cdot \Bx)^\T
=
\sigma^1 x^1 – \sigma^2 x^2 + \sigma^3 x^3,
\end{equation}
it is clear that this coordinate expansion is a reflection about the y-axis. Knowing the reflection formula above provides a rationale for why we might want to write this in the compact form \( -\sigma^2 (\Bsigma \cdot \Bx) \sigma^2 \), which might not be obvious otherwise.

References

[1] C. Doran and A.N. Lasenby. Geometric algebra for physicists. Cambridge University Press New York, Cambridge, UK, 1st edition, 2003.

my course evaluation comments for PHY2403 — Quantum Field Theory I.

November 20, 2018 phy2403 No comments , , ,

Here are my evaluation comments for QFT I. The university provides an anonymous facility to submit course feedback, but since I have no conflicts that require anonymization, I’m posting my commentary (and rationale for some of my list selections) publicly.

Q) Please comment on the overall quality of the instruction in this course.

Professor Poppitz’s knowledge of the subject matter is impressive and thorough. I expect that this is a particularly difficult course to teach and think that he has done an admirable job trying to work through the maximum amount of material in the limited time available in this course.

It is challenging but fun game (albeit a slightly masochistic one) to keep up with Prof Poppitz’s blistering pace through the course material. Poppitz often says “Phew!!” at the end of the race to complete a long derivation in the allotted time, and I’ll be saying the same thing at the end of this course.

The barrage of abstract material covered in a lecture is often sufficient to leave me with a headache, and it takes a few hours to recover from each class. It takes a few more hours after that to digest the material at a human pace.

This course would strongly benefit from video recorded lectures that some of the more educationally progressive academic institutions currently provide (i.e. MIT’s OCW and Yale’s “Open Yale” courses). With the exception of the UofT SciNet group (phy1610 — Scientific computing for Physicists), I’m not aware of any UofT physics courses that provide such recordings. For phy2403, video recordings would be particularly valuable, as it would allow the student to “pause” the Professor and work through the material presented at an individually suitable pace.

Q) Please comment on any assistance that was available to support your learning in the course.

Prof Poppitz was available continually on the course forum, after class briefly, and in weekly office hours. I received a great deal of helpful assistance from him during the course.

Q) Compared to other courses, the workload for this course was…

I picked Heavy (not Very Heavy), but I’m not in a good position to evaluate since I’m only taking one course.

Q) I would recommend this course to other students.

I picked Mostly (not Strongly). I wouldn’t recommend this course to anybody who was not adequately prepared. I’m not sure that I was. It is a very tough course. I was continually impressed with the other students in the class. I’ve worked slowly for years to gradually build up the background required to take this course, and all the rest of these younglings are downing the material with seeming ease. There are a lot of exceptionally smart students enrolled on this course.

New aggregate collection of class notes for phy2403: up to lecture 17.

November 14, 2018 phy2403 No comments

I’ve now uploaded a new version of my class notes for PHY2403, the UofT Quantum Field Theory I course, taught this year by Prof. Erich Poppitz.

This version includes the following chapters, roughly one per lecture, plus some extras

  • Introduction
  • Units, scales, and Lorentz transformations.
  • Lorentz transformations and a scalar action.
  • Scalar action, least action principle, Euler-Lagrange equations for a field, canonical quantization.
  • Klein-Gordon equation, SHOs, momentum space representation, raising and lowering operators.
  • Canonical quantization, Simple Harmonic Oscillators, Symmetries
  • Symmetries, translation currents, energy momentum tensor.
  • 1st Noether theorem, spacetime translation current, energy momentum tensor, dilatation current.
  • Unbroken and spontaneously broken symmetries, Higgs Lagrangian, scale invariance, Lorentz invariance, angular momentum quantization
  • Lorentz boosts, generator of spacetime translation, Lorentz invariant field representation.
  • Microcausality, Lorentz invariant measure, retarded time SHO Green’s function.
  • Klein-Gordon Green’s function, Feynman propagator path deformation, Wightman function, Retarded Green’s function.
  • Forced Klein-Gordon equation, coherent states, number density, time ordered product, perturbation theory, Heisenberg picture, interaction picture, Dyson’s formula
  • Time evolution, Hamiltonian pertubation, ground state
  • Perturbation ground state, time evolution operator, time ordered product, interaction
  • Differential cross section, scattering, pair production, transition amplitude, decay rate, S-matrix, connected and amputated diagrams, vacuum fluctuation, symmetry coefficient
  • Scattering, decay, cross sections in a scalar theory.
  • Problem Set 1.
  • Problem Set 2.
  • Independent study problems
  • Useful formulas and review.
  • Momentum of scalar field.
  • Index
  • Bibliography

Problem set 1-2 solutions are redacted.  If you aren’t a UofT student taking PHY2403, feel free to contact me for an un-redacted copy.

PHY2403H Quantum Field Theory. Lecture 16: Differential cross section, scattering, pair production, transition amplitude, decay rate, S-matrix, connected and amputated diagrams, vacuum fluctuation, symmetry coefficient. Taught by Prof. Erich Poppitz

November 13, 2018 phy2403 No comments , , , , , , , ,

Here are my [lecture notes from last Wednesday’s class], which are posted out of sequence and only in PDF format this time.

PHY2403H Quantum Field Theory. Lecture 17: Scattering, decay, cross sections in a scalar theory. Taught by Prof. Erich Poppitz

November 12, 2018 phy2403 No comments , , ,

[Click here for a PDF of this post with nicer formatting]

DISCLAIMER: Very rough notes from class (today VERY VERY rough).

These are notes for the UofT course PHY2403H, Quantum Field Theory, taught by Prof. Erich Poppitz, fall 2018.

Review: S-matrix

We defined an \( S-\)matrix
\begin{equation}\label{eqn:qftLecture17:20}
\bra{f} S \ket{i} = S_{fi} = \lr{ 2 \pi }^4 \delta^{(4)} \lr{ \sum \lr{p_i – \sum_{p_f} } } i M_{fi},
\end{equation}
where
\begin{equation}\label{eqn:qftLecture17:40}
i M_{fi} = \sum \text{ all connected amputated Feynman diagrams }.
\end{equation}
The matrix element \( \bra{f} S \ket{i} \) is the amplitude of the transition from the initial to the final state. In general this can get very complicated, as the number of terms grows factorially with the order.

We also talked about decays.

Scattering in a scalar theory

Suppose that we have a scalar theory with a light field \( \Phi, M \) and a heavy field \( \varphi, m \), where \( m > 2 M \). Perhaps we have an interaction with a \( z^2 \) symmetry so that the interaction potential is quadratic in \( \Phi \)
\begin{equation}\label{eqn:qftLecture17:60}
V_{\text{int}} = \mu \varphi \Phi \Phi.
\end{equation}
We may have \( \Phi \Phi \rightarrow \Phi \Phi \) scattering.

We will denote diagrams using a double line for \( \phi \) and a single line for \( \Phi \), as sketched in

fig. 1. Particle line convention.

 

There are three possible diagrams:

fig 2. Possible diagrams.

The first we will call the s-channel, which has amplitude

\begin{equation}\label{eqn:qftLecture17:80}
A(\text{s-channel}) \sim \frac{i}{p^2 – m^2 + i \epsilon} =
\frac{i}{s – m^2 + i \epsilon}
\end{equation}

\begin{equation}\label{eqn:qftLecture17:100}
(p_1 + p_2)^2 = s
\end{equation}
In the centre of mass frame
\begin{equation}\label{eqn:qftLecture17:120}
\Bp_1 = – \Bp_2,
\end{equation}
so
\begin{equation}\label{eqn:qftLecture17:140}
s = \lr{ p_1^0 + p_2^0 }^2 = E_{\text{cm}}^2.
\end{equation}

To the next order we have a diagram like fig. 3.

fig. 3. Higher order.

and can have additional virtual particles created, with diagrams like fig. 4.

fig. 4. More virtual particles.

 

We will see (QFT II) that this leads to an addition imaginary \( i \Gamma \) term in the propagator
\begin{equation}\label{eqn:qftLecture17:160}
\frac{i}{s – m^2 + i \epsilon}
\rightarrow
\frac{i}{s – m^2 – i m \Gamma + i \epsilon}.
\end{equation}
If we choose to zoom into the such a figure, as sketched in fig. 5, we find that it contains the interaction of interest for our diagram, so we can (looking forward to currently unknown material) know that our diagram also has such an imaginary \( i \Gamma \) term in its propagator.

fig. 5. Zooming into the diagram for a higher order virtual particle creation event.

Assuming such a term, the squared amplitude becomes
\begin{equation}\label{eqn:qftLecture17:180}
\evalbar{\sigma}{s \text{near} m^2}
\sim
\Abs{A_s}^2 \sim \inv{(s – m)^2 + m^2 \Gamma^2}
\end{equation}

This is called a resonance (name?), and is sketched in fig. 6.

fig. 6. Resonance.

 

Where are the poles of the modified propagator?

\begin{equation}\label{eqn:qftLecture17:220}
\frac{i}{s – m^2 – i m \Gamma + i \epsilon}
=
\frac{i}{p_0^2 – \Bp^2 – m^2 – i m \Gamma + i \epsilon}
\end{equation}

The pole is found, neglecting \( i \epsilon \), is found at
\begin{equation}\label{eqn:qftLecture17:200}
\begin{aligned}
p_0
&= \sqrt{ \omega_\Bp^2 + i m \Gamma } \\
&= \omega_\Bp \sqrt{ 1 + \frac{i m \Gamma }{\omega_\Bp^2} } \\
&\approx \omega_\Bp + \frac{i m \Gamma }{2 \omega_\Bp}
\end{aligned}
\end{equation}

Decay rates.

We have an initial state
\begin{equation}\label{eqn:qftLecture17:240}
\ket{i} = \ket{k},
\end{equation}
and final state
\begin{equation}\label{eqn:qftLecture17:260}
\ket{f} = \ket{p_1^f, p_2^f \cdots p_n^f}.
\end{equation}
We defined decay rate as the ratio of the number of initial particles to the number of final particles.

The probability is
\begin{equation}\label{eqn:qftLecture17:280}
\rho \sim \Abs{\bra{f} S \ket{i}}^2
=
(2 \pi)^4 \delta^{(4)}( p_{\text{in}} – \sum p_f )
(2 \pi)^4 \delta^{(4)}( p_{\text{in}} – \sum p_f )
\times \Abs{ M_{fi} }^2
\end{equation}

Saying that \( \delta(x) f(x) = \delta(x) f(0) \) we can set the argument of one of the delta functions to zero, which gives us a vacuum volume element factor
\begin{equation}\label{eqn:qftLecture17:300}
(2 \pi)^4
\delta^{(4)}( p_{\text{in}} – \sum p_f ) =
(2 \pi)^4
\delta^{(4)}( 0 )
= V_3 T,
\end{equation}
so
\begin{equation}\label{eqn:qftLecture17:320}
\frac{\text{probability for \( i \rightarrow f\)}}{\text{unit time}}
\sim
(2 \pi)^4 \delta^{(4)}( p_{\text{in}} – \sum p_f )
V_3
\times \Abs{ M_{fi} }^2
\end{equation}

\begin{equation}\label{eqn:qftLecture17:340}
\braket{\Bk}{\Bk} = 2 \omega_\Bk V_3
\end{equation}

coming from

\begin{equation}\label{eqn:qftLecture17:360}
\braket{k}{p} = (2 \pi)^3 2 \omega_\Bp \delta^{(3)}(\Bp – \Bk)
\end{equation}
so
\begin{equation}\label{eqn:qftLecture17:380}
\braket{k}{k} = 2 \omega_\Bp V_3
\end{equation}

\begin{equation}\label{eqn:qftLecture17:400}
\frac{\text{probability for \(i \rightarrow f\)}}{\text{unit time}}
\sim
\frac{
(2 \pi)^4 \delta^{(4)}( p_{\text{in}} – \sum p_f )
\Abs{ M_{fi} }^2 V_3
}
{
2 \omega_\Bk V_3
2 \omega_{\Bp_1}
\cdots
2 \omega_{\Bp_n} V_3^n
}
\end{equation}

If we multiply the number of final states with \( p_i^f \in (p_i^f, p_i^f + dp_i^f) \) for a particle in a box
\begin{equation}\label{eqn:qftLecture17:420}
p_x = \frac{ 2 \pi n_x}{L}
\end{equation}

\begin{equation}\label{eqn:qftLecture17:440}
\Delta p_x = \frac{ 2 \pi }{L} \Delta n_x
\end{equation}

\begin{equation}\label{eqn:qftLecture17:460}
\Delta n_x
=
\frac{L}{2 \pi} \Delta p_x
\end{equation}

and

\begin{equation}\label{eqn:qftLecture17:480}
\Delta n_x
\Delta n_y
\Delta n_z
= \frac{V_3}{(2 \pi)^3 }
\Delta p_x
\Delta p_y
\Delta p_z
\end{equation}

\begin{equation}\label{eqn:qftLecture17:500}
\begin{aligned}
\Gamma
&=
\frac{\text{number of events \( i \rightarrow f \)}}{\text{unit time}} \\
&=
\prod_{f} \frac{ d^3 p}{(2 \pi)^3 2 \omega_{\Bp^f} }
\frac{ (2 \pi)^4 \delta^{(4)}( k – \sum_f p^f ) \Abs{M_{fi}}^2 }
{
2 \omega_{\Bk}
}
\end{aligned}
\end{equation}

Note that everything here is Lorentz invariant except for the denominator of the second term ( \(2 \omega_{\Bk}\)). This is a well known result (the decay rate changes in different frames).

Cross section.

For \( 2 \rightarrow \text{many} \) transitions

\begin{equation}\label{eqn:qftLecture17:520}
\frac{\text{probability \( i \rightarrow f \)}}{\text{unit time}}
\times \lr{
\text{ number of final states with \( p_f \in (p_f, p_f + dp_f) \)
}
}
=
\frac{ (2 \pi)^4 \delta^{(4)}( \sum p_i – \sum_f p^f ) \Abs{M_{fi}}^2 {V_3} }
{
2 \omega_{\Bk_1} V_3
2 \omega_{\Bk_2} {V_3 }
}
\prod_{f} \frac{ d^3 p}{(2 \pi)^3 2 \omega_{\Bp^f} }
\end{equation}

We need to divide by the flux.

In the CM frame, as sketched in fig. 7, the current is
\begin{equation}\label{eqn:qftLecture17:540}
\Bj = n \Bv_1 – n \Bv_2,
\end{equation}
so if the density is
\begin{equation}\label{eqn:qftLecture17:560}
n = \inv{V_3},
\end{equation}
(one particle in \(V_3\)), then
\begin{equation}\label{eqn:qftLecture17:580}
\Bj = \frac{\Bv_1 – \Bv_2}{V_3}.
\end{equation}

fig. 7. Centre of mass frame.

 

This is where [1] stops,
\begin{equation}\label{eqn:qftLecture17:640}
\sigma
=
\frac{ (2 \pi)^4 \delta^{(4)}( \sum p_i – \sum_f p^f ) \Abs{M_{fi}}^2 {V_3} }
{
2 \omega_{\Bk_1}
2 \omega_{\Bk_2}
\Abs{\Bv_1 – \Bv_2}
}
\prod_{f} \frac{ d^3 p}{(2 \pi)^3 2 \omega_{\Bp^f} }
\end{equation}

There is, however, a nice Lorentz invariant generalization
\begin{equation}\label{eqn:qftLecture17:600}
j = \inv{ V_3 \omega_{k_A} \omega_{k_B}} \sqrt{ (k_A – k_B)^2 – m_A^2 m_B^2 }
\end{equation}

(Claim: DIY)
\begin{equation}\label{eqn:qftLecture17:620}
\begin{aligned}
\evalbar{j}{CM}
&=
\inv{V_3}
\lr{
\frac{\Abs{\Bk}}{\omega_{k_A}}
+
\frac{\Abs{\Bk}}{\omega_{k_B}}
} \\
&=
\inv{V_3} \lr{ \Abs{\Bv_A} + \Abs{\Bv_B} } \\
&=
\inv{V_3} \Abs{\Bv_1 – \Bv_2 }.
\end{aligned}
\end{equation}

\begin{equation}\label{eqn:qftLecture17:660}
\sigma
=
\frac{ (2 \pi)^4 \delta^{(4)}( \sum p_i – \sum_f p^f ) \Abs{M_{fi}}^2 {V_3} }
{
4 \sqrt{ (k_A – k_B)^2 – m_A^2 m_B^2 }
}
\prod_{f} \frac{ d^3 p}{(2 \pi)^3 2 \omega_{\Bp^f} }.
\end{equation}

References

[1] Michael E Peskin and Daniel V Schroeder. An introduction to Quantum Field Theory. Westview, 1995.