Prof. Sachindeo Vaidya (CHEP, IISc) | PDF
Previous: Lecture 2
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\[
\left[a_{k\lambda},\,a^{\dagger}_{k'\lambda'}\right] \;=\; i\,\delta_{kk'}\,\delta_{\lambda\lambda'} \qquad (\hbar=1)
\]
Propagator :
\[
\mathcal{D}_{ij}(x,y) \;=\; \langle 0|\,T\left(A_i^{T}(x)A_j^{T}(y)\right)|0\rangle
\]
\[
\mathcal{D}_{ij}(x,y) \;=\; \int\frac{d^4k}{(2\pi)^4}\left(\delta_{ij}-\frac{k_ik_j}{\vec{k}^2}\right)\frac{e^{-ik(x-y)}}{k^2+i\varepsilon}
\]
covariant form
\[
=\; \eta_{\mu\nu}\int\frac{d^4k}{(2\pi)^4}\ \frac{e^{-ik\cdot(x-y)}}{k^2+i\varepsilon}
\]
\[
\mathcal{F} \;=\; \left(e^{-\frac{1}{2}\int_x\int_y \mathcal{D}_{\mu\nu}(x,y)\frac{\delta}{\delta A_\mu(x)}\frac{\delta}{\delta A_\nu(y)}}\right) e^{iA_\mu J^\mu}
\]
\[
\left\{\left.\mathcal{F}\right|_{A=0} = Z_0(J)\right\}
\]
\[
\mathcal{L} \;=\; \left(\mathcal{D}_\mu\varphi\right)\left(\mathcal{D}^\mu\varphi\right)^{*} - m^2\varphi\varphi^{*} + \mathcal{L}_{EM}
\]
\[
\mathcal{D}_\mu \;=\; \partial_\mu - iQ A_\mu
\]
\[
\mathcal{L}_{\text{int}} \;=\; -iQ\,\underbrace{A^\mu\left(\varphi^{*}\partial_\mu\varphi - \varphi\partial_\mu\varphi^{*}\right)}_{J^\mu} \;+\; Q^2 A^\mu A_\mu \varphi^{*}\varphi
\]
The S-matrix Functional:
\[
\mathcal{F} \;=\; \underbrace{e^{\left(\int\!\!\int G(x_1,x_2)\frac{\delta}{\delta\varphi(x_1)}\frac{\delta}{\delta\varphi^{*}(x_2)}\right)}}_{(1)}\ e^{\left(-\frac{1}{2}\int\!\!\int \mathcal{D}_{\mu\nu}(x,y)\frac{\delta}{\delta A_\mu(x)}\frac{\delta}{\delta A_\nu(y)}\right)}\ e^{iS_{\text{int}}}
\]
\[
S_{\text{int}} \;=\; \int d^4x\ \mathcal{L}_{\text{int}}
\]
(1) & \(\mathcal{L}_{\text{int}}\) is not gauge invariant but whole expression is gauge invariant.
\[
\mathcal{L} \;=\; \overline{\Psi}\left(i\!\not\partial-m\right)\Psi + \mathcal{L}_{EM}
\]
\[
=\; \overline{\Psi}\left(i\!\not\partial-m\right)\Psi + e\,\overline{\Psi}\gamma^\mu A_\mu \Psi + \mathcal{L}_{EM}
\]
\[
\mathcal{F} \;=\; \exp\left(-\tfrac{1}{2}\int\!\!\int \mathcal{D}_{\mu\nu}(x,y)\frac{\delta}{\delta A_\mu(x)}\frac{\delta}{\delta A_\nu(y)}\right)\cdot \exp\left(-\int\!\!\int \frac{\delta}{\delta\overline{\Psi}_r(x)}\,S_{r,s}(x,y)\,\frac{\delta}{\delta\Psi_s(y)}\right) e^{ie\int A_\mu \overline{\Psi}\gamma^\mu\Psi}
\]
Basis of \(\varphi\) \(\left\{\text{set of all f}^{n}\text{'s in which we expand }\varphi\right\}\)
Consider \(\varphi(x)\) and \(\varphi(x)+\delta\varphi(x)\)
\[
ds^2 \;\overset{\text{def}^{n}}{\equiv}\; \|\delta\varphi\|^2 \;=\; \int_{\Sigma} d^4x\left(\delta\varphi\right)^2 \;=\; \sum_n \left(\delta c_n\right)^2
\]
\(S\) : action
say \(ds^2\) is metric, where metric depends upon \(\varphi\).
say \(\varphi:\Sigma \longrightarrow S^2\) (\(\Sigma \downarrow\) space-time region ; \(S^2 \rightsquigarrow\) sphere)
if \((\theta,\alpha)\in S^2\) then, \(\varphi\) can be thought of --- functions \(\theta(x)\) & \(\alpha(x)\), \((x\in\Sigma)\)
\[
\varphi+\delta\varphi \;\sim\; \left(\theta+\delta\theta(x),\ \alpha(x)+\delta\alpha(x)\right)
\]
\[
ds^2\left(\varphi,\varphi+\delta\varphi\right) \;=\; \int_{\Sigma} d^4x\left[\left(\delta\theta\right)^2 + \left(\sin^2\theta\right)\left(\delta\alpha\right)^2\right]
\]
The set \(\{c_n\}\equiv\) local coordinates on \(\mathcal{F}\) (set of basis functions).
First consider only \(n\)-modes :
\[
dV^{(N)} \;=\; \sqrt{\left|\det\left(g^{(N)}\right)\right|}\ dc_1\,dc_2\cdots dc_N
\]
\(\downarrow\) \(N\times N\) matrix
\[
I \;=\; \int [d\varphi]\ e^{-\frac{1}{2}\int_x\int_y \varphi(x)M(x,y)\varphi(y)} \qquad \text{where}\quad \varphi(x)=\sum c_n f_n(x)
\]
exponent \(=\sum_{mn} c_m M_{mn} c_n\)
\[
M_{mn} \;=\; \int_{\Sigma}\int_{\Sigma} d^4x\,d^4y\ f_m(x)\,M(x,y)\,f_n(y)
\]
We need \(M_{mn}\) to be diagonalizable, \(\mathrm{Re}(\text{eigen values}) > 0\).
Consider a function \(u_n(x)\) defined by
\[
\int d^4y\ M(x,y)\,u_n(y) \;=\; \lambda_n\,u_n(x)
\]
then write \(\varphi\sim\sum a_n u_n\), then,
\[
\|\delta\varphi\|^2 \;=\; \sum_n \left(\delta a_n\right)^2 \ , \qquad dV \;=\; \prod_n^{N} da_n
\]
\[
I \;=\; \lim_{N\to\infty}\int\!\!\int\cdots\int \prod_n^{N} da_n\ e^{-\frac{1}{2}\sum \lambda_n a_n^2} \qquad \left(\mathrm{Re}(\lambda)>0\right)
\]
\[
=\; \lim_{N\to\infty}\prod_{n=1}^{N}\left(\frac{2\pi}{\lambda_n}\right)^{1/2} \;\equiv\; \left(\det\frac{M}{2\pi}\right)^{-1/2}
\]
Truncating to finite \(N\), then taking \(N\to\infty\) is called a Normalisation procedure.
Similarly consider,
\[
I[J] \;=\; \int [d\varphi]\ e^{-\frac{1}{2}\int_\Sigma\int_\Sigma \varphi(x)M(x,y)\varphi(y) + \int_\Sigma J(x)\varphi(x)}
\]
Completing the square
\[
I \;=\; \int [d\varphi]\ e^{-\frac{1}{2}\int\!\!\int\left(\varphi-JM^{-1}\right)M\left(\varphi-M^{-1}J\right)}\ e^{\frac{1}{2}\int JM^{-1}J}
\]
\[
=\; \left|\frac{M}{2\pi}\right|^{1/2}\ e^{\frac{1}{2}\int\!\!\int J(x)M^{-1}(x,y)J(y)}
\]
If \(\varphi\) is complex, then
\[
I[J,\bar{J}] \;=\; \left(\det\frac{M}{2\pi}\right)^{-1}\ e^{\int\!\!\int \bar{J}M^{-1}J}
\]
Grassmann variables (or numbers) are anti-commuting object on which complex conjugate like operation is also defined.
If \(\alpha,\beta\) are grassmann numbers then \(\alpha\beta\) & \(\beta\alpha\) are also grassmann numbers.
\[
\alpha\beta = -\beta\alpha \qquad\Rightarrow\qquad \left\{\alpha,\beta\right\} = 0
\]
\[
\alpha\beta+\beta\alpha = 0 \qquad (\text{Sum of two grassman numbers is zero.})
\]
For single grassmann variable \(\eta\)
\[
\eta^2 = 0 \qquad \text{i.e.} \qquad \eta\eta+\eta\eta = 0 \;\Rightarrow\; 2\eta^2 = 0 \;\Rightarrow\; \eta^2 = 0
\]
functions may be expanded as
\[
f(\eta) \;=\; f_0\,\eta^0 + f_1\,\eta + \underset{\to\,0}{f_2\,\eta^2} + \cdots
\]
if a function depends upon ordinary variable \(x\) & grassmann variable \(\eta\).
\[
f(x,\eta) \;=\; f_0(x)\,\eta^0 + f_1(x)\,\eta + \underset{\to\,0}{f_2(x)\,\eta^2} + \cdots
\]
If a f\(^{n}\) depends upon two grassmann numbers & an ordinary variable. \((\eta_1,\eta_2)\).
\[
\eta_1^2 = 0 = \eta_2^2\ ;\qquad \left(\eta_1\eta_2 = -\eta_2\eta_1\right)
\]
\[
f(x,\eta_1,\eta_2) \;=\; f_0(x) + f_1(x)\eta_1 + f_2(x)\eta_2 + f_{12}(x)\,\eta_1\eta_2 + 0
\]
\[
\bar{\eta}A\eta \qquad \left\{\ \bar{\eta}=(\bar{\eta}_1,\bar{\eta}_2),\quad \eta=\begin{pmatrix}\eta_1\\ \eta_2\end{pmatrix}\right\}
\]
\(A \longrightarrow\) \(2\times2\) matrix with \(\mathbb{R}\) (or \(\mathbb{C}\)) entries
These pages continue in the notebook after a blank leaf; they form one rough working block on discretising the Klein--Gordon operator on a lattice and are transcribed in the order written.
\[
\delta^2\phi \;=\; \frac{1}{a}\left(\frac{\phi(n+\hat{\mu})-\phi(n)}{a} - \frac{\left(\phi(n)-\phi(n-\hat{\mu})\right)}{a}\right)
\]
\[
=\; \frac{1}{a^2}\left(\phi(n+\hat{\mu}) - \phi(n) - \phi(n) + \phi(n-\hat{\mu})\right)
\]
\[
\delta^2\phi \;=\; \frac{1}{a^2}\left(\phi(n+\hat{\mu}) + \phi(n-\hat{\mu}) - 2\phi(n)\right)
\]
with \(\phi = e^{ik\cdot x}\), \(\phi(n)=e^{ik\cdot n}\) :
\[
=\; e^{ip\cdot(n+\hat{\mu})} + e^{ip\cdot(n-\hat{\mu})} - 2e^{ip\cdot n}
\]
\[
=\; e^{ip\cdot n}\left(e^{ip\cdot\hat{\mu}} + e^{-ip\cdot\hat{\mu}} - 2\right) \;=\; e^{ip\cdot n}\left(e^{ip_\mu} + e^{-ip_\mu} - 2\right)
\]
\[
=\; e^{ip\cdot n}\left(2\cos p_\mu - 2\right) \;=\; -2e^{ip\cdot n}\left(1-\cos p_\mu\right)
\]
\[
=\; \left(4e^{ip\cdot n}\sin^2\frac{p_\mu}{2} + m^2\right)^{-1}
\]
\[
S \;=\; \int \overline{\Psi}\,M\,\Psi\ d^4x
\]
(in the margin :)
\[
\left(\Box_x+m^2\right)G(x,y) = \delta(x-y)\ ;\qquad \left(\partial_x^2+m^2\right)G(x,y) = \delta(x-y)
\]
at \(p_\mu = 0\),
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\[
\frac{1}{\displaystyle\sum_{\mu=1}^{4}\sin^2\frac{p_\mu}{2} + m^2}
\]
\[
\frac{2\pi}{L}\left(n_1,n_2,n_3\right) \qquad\qquad \frac{2\pi}{L}\,n_1
\]
H.W : No doubling in case of scalars, but we see doubling in case of fermions for massless case.
\[
\left(\partial^2+m^2\right)\phi(x) \ , \qquad \phi(x)=\langle x|\phi\rangle
\]
insert \(\mathbb{I}\) :
\[
\langle x|\int dp\ |p\rangle\langle p|\phi\rangle
\]
\[
\left(\partial^2+m^2\right)\int dp\ \langle x|p\rangle\,\hat{\phi}(p)
\]
\[
\delta^2 \;=\; \frac{1}{a^2}\left(\phi(n+\hat{\mu}) + \phi(n-\hat{\mu}) - 2\phi(n)\right)
\]
\[
G(x,y) \;=\; G(|x-y|) \;=\; \int\frac{d^4k}{(2\pi)^4}\ e^{ik\cdot(x-y)}\ \widetilde{G}(k)
\]
\(\langle k|x-y\rangle\)
\[
\left(\Box_x+m^2\right)G(x,y) \;=\; \int\left(-k^2+m^2\right)\frac{d^4k}{(2\pi)^4}\ e^{ik\cdot(x-y)}\ \widehat{G}(k)
\]
\[
=\; \delta^{(4)}(x-y) \;=\; \int\frac{d^4k}{(2\pi)^4}\ e^{ik\cdot(x-y)}
\]
\[
\left(-k^2+m^2\right)\widehat{G}(k) = 1 \qquad\Rightarrow\qquad \widehat{G}(k) = \frac{1}{-k^2+m^2}
\]
\[
e^{ik\cdot(x-y)} \qquad \text{with}\quad x = na,\quad y = ma \qquad \longrightarrow\quad e^{ip\cdot(n-m)}
\]
\[
\phi(x) \;=\; \langle x|\phi\rangle \;=\; \int\frac{d^4p}{(2\pi)^4}\ \hat{\phi}(p)\,e^{ip\cdot x}
\]
\[
\phi(n) \;=\; \int\frac{d p}{(2\pi)^4}\ \hat{\phi}(p)\,e^{ip\cdot n}
\]
\[
\phi(n+\hat{\mu}) \;=\; \int dp\ \hat{\phi}(p)\ \boxed{e^{ip\cdot(n+\hat{\mu})}}
\]
\[
S\left[\phi,\partial_\mu\phi\right] \;=\; \int d^4x\ \phi\left(\int\frac{d^4p}{(2\pi)^4}\ \hat{\phi}(p)\left[e^{ip\cdot(n+\hat{\mu})} + e^{ip\cdot(n-\hat{\mu})} - 2e^{ip\cdot n}\right]\right)
\]
\(\hookrightarrow\) Inverse of propagator
\[
\mathcal{D}_F(x-y) \;=\; \int\frac{d^4k}{(2\pi)^4}\ \frac{e^{ik\cdot(x-y)}}{k^2-m^2}
\]