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Prof. Sachindeo Vaidya (CHEP, IISc) | PDF

Next: Lecture 2

• QFT 1 recap
  • Scalar & spinor fields
  • \(L\) & \(H\) formulation
  • canonical quantisation
  • symmetries & conservation laws
  • Interaction & \(S\) matrix
  • QED scattering examples
• Functional derivatives / Integrals
• Renormalisation
• Gauge theories (Gauge principle, parallel transport)
• Quantisation
  • Functional Integrals
  • BRST
  • Renormalisation.
• Spontaneous symmetry breaking, Anomalies
• Gauge theories (some non-perturbative questions)
  • Anomalies / Index theories
  • Confinement / Area law
  • t'Hooft & polyakov monopoles

1. Itzykson & Zuber
2. Weinberg V II
3. Srednicki QFT

\(\hookrightarrow\) Choose any one book.

• Peskin & Schroeder
• Schwartz
• Ramond.

• Assignments -- every week or one per 10 days
• Exams -- Mid term, End term \(\Big\}\) 4 hrs or more if required.

\[ ds^2 \;=\; dt^2 - (d\vec{x})^2 \]

Minkowski metric

\[ \eta_{\mu\nu} \;=\; \mathrm{diag}\,(1,-1,-1,-1) \] \[ p\cdot x \;=\; p_\mu x^\mu \;=\; p^\mu x_\mu \;=\; p_0 x_0 - \vec{p}\cdot\vec{x} \] \[ \epsilon^{ijk} \;=\; \epsilon^{123} \;=\; \underline{1} \ , \qquad \epsilon^{\mu\nu\rho\sigma} \;=\; \epsilon^{0123} \;=\; 1 \]

\(\partial\) is sometimes used to denote boundary of special or space-time region. (So, \(\partial V\) or \(\partial\Sigma\) are boundaries of \(V\) & \(\Sigma\) respectively.)

Particle	Field
(1) spin 0 (unchaged, massive)	scalar field
(2) spin 0 charged boson	complex scalar field
(3) photon (spin \(-1\) massless)	\(A_\mu(\vec{x},t)\) real vector field / potential.
(4) spin \(\frac{1}{2}\) fermions \((e^{\pm},\gamma,\text{quarks}\dots)\)	spinor field \(\Psi_r(\vec{x},t)\)

\[ u_k(\vec{x}) \;=\; \frac{e^{-ik\cdot x}}{\sqrt{2\omega_k V}} \ , \qquad \omega_k=\sqrt{k^2+m^2} \ , \qquad (V\to\infty \text{ in end}) \] \[ \int d^3x \left( u_k^{*}\left(-i\partial_0 u_{k'}\right) - \left(i\partial_0 u_k^{*}\right)u_{k'}\right) \;=\; \delta_{k,k'} \]

where

\[ k \equiv (k_1,k_2,k_3) \;=\; \frac{2\pi}{L}\,(n_1,n_2,n_3) \ , \qquad V=L^3 \]

for \(V\to\infty\), \(k\) becomes continuous &

\[ \delta_{k,k'} \;\longrightarrow\; \frac{(2\pi)^3}{V}\,\delta(k-k') \ , \qquad \sum_k \;\longrightarrow\; V\int \frac{d^3k}{(2\pi)^3} \]

these wave fun. obey K.G. eq\(^{n}\)

\[ \left(\partial^2+m^2\right)u_k = 0 \qquad \text{i.e.} \qquad \left(\Box+m^2\right)u_k = 0 \]

K.G. eq\(^{n}\) comes from

\[ \mathcal{L} \;=\; \tfrac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi) - \tfrac{1}{2}m^2\phi^2 \]

obey dirac eq\(^{n}\) --

\[ \left(-i\gamma^\mu\partial_\mu + m\right)\Psi(x) = 0 \qquad (\text{or}) \qquad \left(i\!\not\partial-m\right)\psi = 0 \]

where

\[ \left\{\gamma^\mu,\gamma^\nu\right\} \;=\; 2\eta^{\mu\nu}\,\mathbb{I}_{4\times4} \]

dirac eq\(^{n}\) comes from

\[ \mathcal{L}_{\text{Dirac}} \;=\; \overline{\Psi}\left(i\!\not\partial-m\right)\Psi \]

Spinor field `\(\Psi\)' can be expanded as

\[ \Psi_r(x) \;=\; \sum_{p,r}\sqrt{\frac{m}{E_p V}}\left(a_{p,r}\,u_r(p)\,e^{-ip\cdot x} \;+\; b^{\dagger}_{p,r}\,v_r(p)\,e^{ip\cdot x}\right) \]

where \(u,v\) are sol\(^{n}\) of dirac eq\(^{n}\).

we will study about other fields later on.

we know, Action

\[ (S) \;=\; \int_{\Sigma} d^4x\ \mathcal{L}(\varphi,\partial_\mu\varphi) \]

\(\Sigma \rightsquigarrow\) space time region.

Given a field config\(^{n}\) & its derivative we get a real number. So it is a functional.

\(S\) = Functional of field & its derivatives.

Note -- Thinking of functions as expansion in sine & cosine fun. (Fourier series) is more important in QM & QFT than thinking it in terms of taylor series.

We expand `\(\varphi\)' in terms of basis functions \(\{f_n\}\)

\[ \varphi(x) \;=\; \sum_n c_n f_n(x) \]

specifying \(\varphi\) is same as specifying \(c_n\).

\[ (\varphi+\delta\varphi) \;\sim\; \sum \left(c_n+\delta c_n\right) f_n(x) \]

we are interested in changing \(\varphi\) in bulk / volumetric region of \(\Sigma\) & keeping \(\varphi\) fixed at surface / boundary represented by \(\partial\Sigma\). i.e.

\[ \left.\partial\varphi\right|_{\partial\Sigma} \;=\; 0 \]

\(\partial\Sigma \rightsquigarrow\) represents boundary of region \(\Sigma\).

\[ \delta I \;=\; \int_{\Sigma} d^4x\ \sigma(x)\,\delta\varphi(x) \qquad \left\{\begin{aligned}&\sigma(x)\text{ is arbitrary fun.}\\ &\text{depend upon }\varphi\ \&\ \partial_\mu\varphi.\end{aligned}\right. \]

then functional derivative

\[ \frac{\delta I}{\delta\varphi(x)} \;=\; \sigma(x) \qquad (\text{defination}) \]

ex :

\[ I\left[\varphi,\partial_\mu\varphi\right] \;=\; \int_{\Sigma} d^4x\left(\tfrac{1}{2}(\partial_\mu\varphi)(\partial^\mu\varphi)-\tfrac{1}{2}m^2\varphi^2\right) \]

then

\[ \frac{\delta I}{\delta\varphi(x)} \;=\; -\left(\partial^2+m^2\right)\varphi(x) \]

then we require that,

\[ \int_{\Sigma} d^4x\ \varphi^2(x) \;<\; \infty \qquad \text{and} \qquad \int_{\Sigma} d^4x\ (\partial\varphi)^2(x) \;<\; \infty \]

\(\Big\}\) To make action finite & finite energy \(\int d^3x\,\mathcal{H} \to\) finite.

\[ \mathcal{F} \;=\; \left\{\text{all }\varphi\text{'s such that }\ \varphi:\Sigma\to\mathbb{R}\ \text{ with above cond}^{n}\right\} \]

A Functional (like action \(S/I\)) is a real valued function on \(\mathcal{F}\) (where \(\mathcal{F}\) is infinite dimensional).

Action \(S\) or \(I\) has physical significance.

if \(S \sim \hbar\) use Q.M

if \(\frac{S}{\hbar} \ggg 1\) we can safely use C.M.

we also expand action as perturbation series of \(\hbar\):

\[ I \;=\; I_0 + \hbar I_1 + \hbar^2 I_2 + \cdots \]

\(I_0 \downarrow\) classical action ; \(I_1 \downarrow\) first order quantum correction \(\cdots\)

For

\[ \mathcal{L} \;=\; \tfrac{1}{2}(\partial_\mu\varphi)^2 - \frac{m^2\varphi^2}{2} \]

we know

\[ \Pi(\vec{x},t) \;=\; \partial_0\varphi(\vec{x},t) \] \[ H \;=\; \tfrac{1}{2}\int d^3x\left(\Pi^2 + (\nabla\phi)^2 + m^2\varphi^2\right) \]

Imposing canonical commutation relations --

\[ \left[\varphi(\vec{x},t),\Pi(\vec{x}\,',t)\right] \;=\; i\,\delta^3(\vec{x}-\vec{x}\,') \]

all other equal time commutator vanishes.

\[ \mathcal{L}_0 \;=\; \overline{\Psi}\left(i\!\not\partial-m\right)\Psi \] \[ \Pi \;=\; i\psi^{\dagger}\left(\gamma^0\right)^2 \;=\; i\psi^{\dagger} \qquad \ldots\ \left\{(\gamma^0)^2=+\mathbb{I}\right\} \] \[ H \;=\; \int d^3x\ \psi^{\dagger}\left(i\gamma^0\gamma^i\partial_i + m\gamma^0\right)\psi \]

\(H\) is hermitian (we expect it to be).

\[ H \;=\; \sum_{p,r} E_p\left(a^{\dagger}_{p,r}a_{p,r} + b^{\dagger}_{p,r}b_{p,r}\right) \;-\; \boxed{2\sum_p E_p} \]

\(\swarrow\) Becomes infinity. Just throw it.

We can't impose canonical commutation relations, as \(H\) is unbounded from below. So we use anti-commutation relations. {also \(\Psi(x)\) should be valued in Grassmann no.s}.

Impose anti-commutation relations --

\[ \left\{\Psi_r(\vec{x},t),\ \Pi_s(\vec{x}\,',t)\right\} \;=\; i\,\delta_{r,s}\,\delta(\vec{x}-\vec{x}\,') \]

(Others are zero.)

(No class -- Next week)