Sarah Alanazi
Jacan Chaplais
April, 2021
In 1927, Dirac proposed that quantum interaction between radiation and matter may be elevated to relativistic theory by quantizing electromagnetic field [1].
However, in 1930 Oppenheimer discovered perturbative approaches led to unphysical results beyond first order calculations, due to self-interaction [2].
In 1947 Bethe suggested a mechanism for handling the infinite terms, called renormalisation [3]. This sparked renewed interest in quantum fields.
Following this, the theory was developed rapidly, with Feynman’s diagrammatic contribution reducing the technical barrier for performing calculations [4].
In this talk we apply the theoretical work that has been done so far. Using these new calculation techniques, we attempt to recover Compton scattering.
This choice is symbolic of movement from semi-classical particle physics, to a full acceptance of the quantum theory [5].
The processes which contribute to this interaction can be written as a scattering amplitude, expressed as the sum of Feynman diagrams.
By applying the Feynman rules to these diagrams and grouping terms, we obtain \[ i \mathcal{M}=i {\color{#b5e48c} e^{2}} {\color{#ffe66d} \epsilon_{\mu \lambda^\prime}^{\ast} \left(k^{\prime}\right)} {\color{#ffe66d} \epsilon_{\nu \lambda}\left(k\right)} {\color{#fe6d73} \bar{u}^{s^{\prime}}\left(p^{\prime}\right) } \left( {\color{#5adbff} \dfrac{% numerator {\color{#b5e48c} \gamma^{\mu}}(\not{p}+k+m) {\color{#b5e48c} \gamma^{\nu}}} {(p+k)^{2}-m^{2}}}% denominator + {\color{#5adbff} \dfrac{% numerator {\color{#b5e48c} \gamma^{\nu}} \left(\not{p}-k^{\prime}+m\right) {\color{#b5e48c} \gamma^{\mu}}} {\left(p-k^{\prime}\right)^{2}-m^{2}}% denominator } \right) {\color{#fe6d73} u^{s}(p)} \]
This unwieldy expression can be reduced a little by expanding the binomials in the denominator, and observing for the numerator \[ \begin{aligned} (\not{p}+m) \gamma^{\nu} u^{s}(p) &= \left(2 p^{\nu} - \gamma^{\nu}\not{p} + \gamma^{\nu} m\right) u^{s}(p) \\ &=2 p^{\nu} u^{s}(p)-\gamma^{\nu}\underbrace{(\not{p}-m) u^{s}(p)}_{ \text{Dirac equation} \implies 0 } \\ &=2_{} p^{\nu} u^{s}(p) \end{aligned} \]
which yields the invariant amplitude in the simpler form \[ i \mathcal{M}= -i e^{2} \epsilon_{\mu \lambda^\prime}^{\ast} \left(k^{\prime}\right) \epsilon_{\nu\lambda}(k) \bar{u}^{s^\prime}\left(p^{\prime}\right) \left( \frac{\gamma^{\mu} \not{k}\gamma^{\nu}+2 \gamma^{\mu} p^{\nu}} {2 p \cdot k} + \frac{-\gamma^{\nu} \not{k}^{\prime} \gamma^{\mu}+2 \gamma^{\nu} p^{\mu}} {-2 p \cdot k^{\prime}}\right) u^{s}(p)\text{.} \]
We consider two intertial frames to obtain our final result, so we write \(\mathcal{M}\) in terms of Mandelstam variables. These are constructed from the incoming and outgoing momenta of the interaction. \[ \begin{array}{lllllllll} s &=& (p+k)^{2} &=& p^{2}+k^{2}+2 p \cdot k &=& m^{2}+2 p \cdot k &=& m^{2}+2 p^{\prime} \cdot k^{\prime} \\ t &=& \left(p^{\prime}-p\right)^{2} &=& p^{\prime 2}+p^{2}-2 p \cdot p^{\prime} &=& 2 m^{2}-2 p \cdot p^{\prime} &=& -2 k \cdot k^{\prime} \\ u &=& \left(k^{\prime}-p\right)^{2} &=& k^{\prime 2}+p^{2}-2 k^{\prime} \cdot p &=& m^{2}-2 k^{\prime} \cdot p &=& m^{2}-2 k \cdot p^{\prime} \end{array} \] Casting transition amplitudes in terms of these manifestly Lorentz invariant quantities allows us to jump between frames with ease.
Inertial frame in which sum of spatial momenta is zero.
Inertial frame in which the electron is at rest. This will also be the assumed rest frame for our particle detectors, hence lab frame.
Cross section, \(\sigma\), used to characterise interaction strength. Analogous to effective cross sectional area of particle through beam, while capturing relativistic and quantum dynamics information.
\[ \begin{aligned} \mathrm{d} \sigma=& \frac{1}{2 E_{\mathcal{A}} 2 E_{\mathcal{B}}\left|v_{\mathcal{A}}-v_{\mathcal{B}}\right|}\left(\prod_{f} \frac{\mathrm{d}^{3} p_{f}}{(2 \pi)^{3}} \frac{1}{2 E_{f}}\right) \\ &\left|\mathcal{M}\left(p_{\mathcal{A}}, p_{\mathcal{B}} \rightarrow\left\{p_{f}\right\}\right)\right|^{2}(2 \pi)^{4} \delta^{(4)}\left(p_{\mathcal{A}}+p_{\mathcal{B}}-\sum p_{f}\right) \end{aligned} \]
Constructed from phase space volume, invariant amplitude \(\mathcal{M}\), and momentum conserving Dirac delta terms.
The phase space volume for a two particle collision may be defined by the integral,
\[ \int \mathrm{d} \Pi_{2}=\int \frac{\mathrm{d}^{3} k^{\prime}} {(2 \pi)^{3} 2 E_{k^{\prime}}} \frac{\mathrm{d}^{3} p^{\prime}}{(2 \pi)^{3} 2 E_{p^{\prime}}}(2 \pi)^{4} \delta^{(4)}\left(p+k-k^{\prime}-p^{\prime}\right) \text{.} \]
By making careful substitutions, and using fundamental theorem of calculus, can extract derivative of cross section wrt specific quantities, eg. \(\mathrm{d}\sigma / \mathrm{d}\cos\theta\).
The phase space of an interaction carries its kinematic information.
In the unpolarised case, we don’t know the spins of particles. The electron orientations are random, so we average over initial particles, and sum over final particles.
\[ \langle |\mathcal{M}|^2 \rangle = \dfrac{1}{4} \sum_\text{spins} | \mathcal{M} |^2 \]
We apply the replacement \(\sum^2_{\lambda=1} \epsilon^\ast_{\mu\lambda} \epsilon_{\nu\lambda} \to - g_{\mu\nu}\), and the property of Dirac spinors, \(\sum_{s=1}^{2} u^{s}(p) \bar{u}^{s}(p) = (\not{p}+m)\), to our spin averaged invariant amplitude, to obtain
\[ \begin{aligned} \frac{1}{4} \sum_{\text {spins}}|\mathcal{M}|^{2}= \dfrac{e^{4}}{4} \bigg[ &\left(\not{p}^{\prime} + m\right) \left( \dfrac{% numerator \gamma^{\mu} \not{k} \gamma^{\nu} + 2 \gamma^{\mu} p^{\nu}} {2 p \cdot k}% denominator + \dfrac{% numerator \gamma^{\nu} \not{k}^{\prime} \gamma^{\mu} -2 \gamma^{\nu} p^{\mu}} {2 p \cdot k^{\prime}}% denominator \right) \\ & \left(\not{p} + m\right) \left( \dfrac{% numerator \gamma_{\nu} \not{k} \gamma_{\mu} + 2 \gamma_{\mu} p_{\nu}} {2 p \cdot k}% denominator + \dfrac{% numerator \gamma_{\mu} \not{k}^{\prime} \gamma_{\nu} - 2 \gamma_{\nu} p_{\mu}} {2 p \cdot k^{\prime}} % denominator \right) \bigg] \end{aligned} \]
Contracting covariant and contravariant indices yields the spin averaged invariant amplitude as a sum of four traces.
\[ \frac{1}{4} \sum_{\text {spins}}|\mathcal{M}|^{2} = \dfrac{e^4}{4} \left[ \dfrac {\operatorname{Tr} A} {\left(2 p \cdot k\right)^2} + \dfrac {\operatorname{Tr} B} {\left(2 p \cdot k\right) \left(2 p \cdot k^{\prime} \right)} + \dfrac {\operatorname{Tr} C} {\left(2 p \cdot k^{\prime} \right)\left(2 p \cdot k\right)} + \dfrac {\operatorname{Tr} D} {\left(2 p \cdot k^{\prime} \right)^2} \right] \]
Clifford algebra provides simplifications, eg. half the traces contain odd number of \(\gamma^\mu\) matrices, so are identically zero. The traces are also related, such that \(\operatorname{Tr} B = \operatorname{Tr} C\), and similarly \(\operatorname{Tr} D=\operatorname{Tr} A \; \left(k \leftrightarrow-k^{\prime}\right)\). Applying this gives
\[ \begin{aligned} \frac{1}{4} \sum_{\text {spins }}|\mathcal{M}|^{2}= 2 e^{4} \left[ \frac{m^2 - u}{s - m^2} + \frac{s - m^2}{m^2 - u}\right. +&\left. 2 m^{2}\left( \frac{2}{s - m^2} - \frac{2}{m^2 - u} \right) \right. \\ +& \left. m^{4}\left( \frac{2}{s - m^2} - \frac{2}{m^2 - u} \right)^{2} \right] \end{aligned} \]
To evaluate the phase space integral introduced earlier \[ \int \mathrm{d} \Pi_{2}= \int \frac{\mathrm{d}^3 k^{\prime}}{(2 \pi)^{3} 2 E_{k^{\prime}}} \frac{\mathrm{d}^3 p^{\prime}}{(2 \pi)^{3} 2 E_{p^{\prime}}} (2 \pi)^{4} \delta^{(4)}\left(p + k - k^{\prime} - p^{\prime}\right) \]
The momentum conserving Dirac delta reduces the calculation to a two dimensional integral, rather than a six dimensional one. We split into spatial and temporal momentum components \[ \delta^{(4)}\left(p + k - k^{\prime} - p^{\prime}\right) = \delta\left( E_\mathbf{p} + \omega - E_\mathbf{p}^\prime - \omega^\prime \right) \delta^{(3)}\left( | \mathbf{p} | + | \mathbf{k} | - | \mathbf{p}^\prime | - | \mathbf{k}^\prime | \right) \]
This may then be evaluated for a specific integration measure, for instance we can substitute the solid angle \[ \mathrm{d}^3 p = | \mathbf{p} |^2 \mathrm{d}p \; \mathrm{d}\Omega \]
In the centre-of-momentum frame, 4-momenta of incoming and outgoing particles are \[ \begin{array}{cccc} p = (E, -\omega\mathbf{\hat{z}}) & k = (\omega, \omega \mathbf{\hat{z}}) & p^\prime = (E, \mathbf{p}^\prime) & k^\prime = (\omega, \omega \sin{\theta}, 0, \omega \cos{\theta}) & \end{array} \] and the phase space integral takes the form \[ \int \mathrm{d} \Pi_{2} = \frac{1}{16 \pi^{2}} \int \mathrm{d} \Omega \frac{\omega}{E_{\text{CoM}}} \]
Whereas in the lab frame, the 4-momenta are given by \[ \begin{array}{cccc} p = (m, 0) & k = (\omega, \omega\mathbf{\hat{z}}) & p^\prime = (E^\prime, \mathbf{p}^\prime) & k^\prime = (\omega^{\prime}, \omega^{\prime} \sin{\theta}, 0, \omega^{\prime} \cos{\theta}) & \end{array} \]
and the phase space integral is obtained as
\[ \int \mathrm{d} \Pi_{2} = \dfrac{1}{16 \pi^2} \int \mathrm{d}\Omega \frac{\omega^{\prime \; 2}}{m \omega} \]
We may now bring our equations for the phase space integral and the invariant amplitude together. Substituting the integration measure, \(\mathrm{d} t = -2\omega^2\mathrm{d}\Omega / 2\pi\) and writing in Mandelstam variables in the centre-of-momentum frame, we obtain \[ \begin{aligned} \dfrac{\mathrm{d}\sigma\left(s, t\right)}{\mathrm{d}t} = \dfrac{2\pi\alpha^2}{\left(s - m^2\right)^2} \left[ \frac{s + t - m^2}{s - m^2} + \frac{s - m^2}{s + t - m^2}\right. +&\left. 2 m^{2}\left( \frac{2}{s - m^2} - \frac{2}{s + t - m^2} \right) \right. \\ +& \left. m^{4}\left( \frac{2}{s - m^2} - \frac{2}{s + t - m^2} \right)^{2} \right] \end{aligned} \]
Expressing in terms of scattering angle, \(t = \frac{ -2\left(\frac{s-m^{2}}{2 m}\right)^{2}(1-\cos \theta) } { 1+\left(\frac{s-m^{2}}{2 m^{2}}\right)(1-\cos \theta) }\), and applying the range of \(\cos\theta\) to this equation yields the domain for \(\mathrm{d}\sigma / \mathrm{d}t\) \[-\dfrac{\left(s - m^2\right)^2}{s} \leq t \leq 0\]
We perform the same procedure in the lab frame, using integration measure \(\mathrm{d}\Omega = -2\pi\mathrm{d}(\cos\theta)\), to obtain the angular dependence.
\[ \begin{aligned} \dfrac{\mathrm{d}\sigma}{\mathrm{d}\cos\theta} = \dfrac{1}{4m\omega} \dfrac{1}{8\pi} \dfrac{\omega^{\prime \; 2}}{\omega m} 2 e^{4} \left[ \frac{m^2 - u}{s - m^2} + \frac{s - m^2}{m^2 - u}\right. +&\left. 2 m^{2}\left( \frac{2}{s - m^2} - \frac{2}{m^2 - u} \right) \right. \\ +& \left. m^{4}\left( \frac{2}{s - m^2} - \frac{2}{m^2 - u} \right)^{2} \right] \end{aligned} \]
Writing in terms of photon momenta obtains the famous Klein-Nishina formula [6]
\[ \dfrac{\mathrm{d}\sigma}{\mathrm{d}\cos\theta} = \frac{\pi \alpha^{2}}{m^{2}} \left(\frac{\omega^{\prime}}{\omega}\right)^{2} \left[\frac{\omega^{\prime}}{\omega}+\frac{\omega}{\omega^{\prime}}-\sin ^{2} \theta\right] \]
In this work, we have used the new Quantum Electrodynamics to recover the known scattering relationship between light and electrons, as written down by Compton.
The processes were interpreted via tree-level Feynman diagrams, at the lowest order.
Future work may involve higher order calculations, and the inclusion of loops, to determine if this formulation makes new predictions about this interaction.
Thank you for your attention, we welcome any questions.
[1] Dirac PAM. Quantum theory of emission and absorption of radiation. Proc Roy Soc Lond A 1927;114:243. https://doi.org/10.1098/rspa.1927.0039.
[2] Oppenheimer JR. Note on the Theory of the Interaction of Field and Matter. Physical Review 1930;35:461–77. https://doi.org/10.1103/PhysRev.35.461.
[3] Bethe HA. The Electromagnetic Shift of Energy Levels. Physical Review 1947;72:339–41. https://doi.org/10.1103/PhysRev.72.339.
[4] Feynman RP. The Theory of Positrons. Physical Review 1949;76:749–59. https://doi.org/10.1103/PhysRev.76.749.
[5] Compton AH. A quantum theory of the scattering of x-rays by light elements. Phys Rev 1923;21:483–502. https://doi.org/10.1103/PhysRev.21.483.
[6] Klein O, Nishina T. Über die Streuung von Strahlung durch freie Elektronen nach der neuen relativistischen Quantendynamik von Dirac. Zeitschrift Fur Physik 1929;52:853–68. https://doi.org/10.1007/BF01366453.
[7] Thomson JJ, Thomson GP. Conduction of electricity through gases. 2nd ed. Cambridge University Press; 1906.