CFT/TFT Lecture note
[2/26] 0. Whence CFT&TFT?
Conformal field theory is a speical kind of quangum field theory with a scaling symmetry.
\rightarrow The system is invariant under scale transformation \vec{x}\rightarrow \lambda\vec{x}\\ t\rightarrow \lambda t
\bullet QFT = quantum mechanics with "\infty'ly" many degrees of freedom that fills the space
When the size of lattice spacing is very small, we can treat dynamical degrees of freedom at each point to be effectively continuous \phi(n_1\hat{e}_1+n_2\hat{e_2}),\ \frac{1}{a^2}\sum_{n_1,n_2}\rightarrow \phi(x,y),\ \frac{1}{L^2}\int dx\, dy Such a continuum limit is descrived by a QFT.
\bullet In high-energy particle physics, quantum fields describe elementary particles such as quarks, leptons, photons, gauge bosons, ...
\bullet In condensed matter physics, fields can be local spin degrees of freedom of anything that can be associated with space.
\bullet In HEP, quantum fields are conisdered to be elementary.
\bullet In CMP, quantum fields are low-energy effective description. (E\ll \frac{1}{a})
Whence CFT?
Confirmal symmetry \sim "scale invariance"
\rightarrow no characteristic scale
Critical phenomena
eg) Ising model H=\sum_{\langle ij\rangle}\sigma_i\sigma_j \langle\sigma_i\sigma_j\rangle -\langle \sigma_i\rangle\langle\sigma_j\rangle \sim \exp \left( -\frac{|i-j|}{\xi}\right) For generic T, the system have no scale invariance, (a,\xi) introduce scales.
For T=T_c, \xi\rightarrow \infty. (much larger that a).
"critical point" \Rightarrow scale symmetry emergent.
\bullet Ising model is just one among as \infty models that can provide approx. desc. of cpx systems with local interacts "universality class".
\Rightarrow CFT: study of such universality class
Q) Can we calssify "all" such class?
A) Only partically in 1+1d.
Quantum critical system (T=0)
gapless excitation \Rightarrow \langle \sigma\sigma\rangle\sim \frac{1}{|x|^2}$
- edge modes in quantum Hall system
- Knodo effect
Deep inelastic scattering
\bullet scale free natrue of QCD \leftrightarrow "asymptotic freedom"
(scale-invariance at short distance (UV))
String theory
\bullet world-sheet theory has scale invariance
Quantum gravity
Ads/CFT correspondence
[2/28] Whence TFT?
Topological field theory = QFT that does not depend on the choice of spacetime metric g_{\mu\nu}
rightarrow It does depend on topology.
M \rightarrow Z[M,g] = \int \mathcal{D}\phi \, e^{iS[M,g,\phi]}
TQFT does depend on M but does not depend on g. \Rightarrow 'topological' 'topology of M'
Physically, \hat{H}=0, no local excitations.
all local excitations are massive. (m\rightarrow \infty)
\bullet On a space with boundary, gapless edge modes can appear. (QHE)
For any physical system, one would like to know low-E excitations "gapped phase"
\bullet It may look trivial, but on a topologically non-trivial space (locally trivial), non-local observables (eg: line operators) can be non-trivial.
Another case is "gapless phase" which has continuous level of energy.
\bullet trivial means free theory, non-trivial means non-trivial CFT.
1. From particles to fields to particles again
Let particles are on lattice with spring, and their position is x_n=na, and q_n be displacement away from equilibrium. H=\sum_{n=1}^N\left( \frac{p_n^2}{2m}+\frac{1}{2}K(q_n-q_{n-1})^2\right)+\cdots (Boundary condition: q_{n+N}=q_n) Individual fluctuation is linear sum of normal modes.
Translatino invariance: T:x\mapsto x+a \hat{q}_k=\frac{1}{\sqrt{N}}\sum_{n=1}^Ne^{ikx_n}q_n\\ \hat{p}_k=\frac{1}{\sqrt{N}}e^{ikx_n}p_n with boundary condition, q_n=\frac{1}{\sqrt{N}}\sum_k e^{-ikx_n}\tilde{q}_k k be quantized with e^{ikNa}=1, with k=\frac{2\pi m}{Na}, m=a,\cdots, N-1
Brilliomn zone is -\frac{\pi}{a}\le k<\frac{\pi}{a}\quad k\sim k+\frac{2\pi}{a}.
System in a box \Rightarrow k-space discrete (q_{n+N}=q_n) "IR regulator"
System on a lattice \Rightarrow k-space finite (a) "UV regulator"
H=\sum \frac{p_n^2}{2m}+\frac{1}{2}K(q_{n+1}-q_n)^2 put p_n=\frac{1}{?}\sum e^{-ikx}\tilde{p}_k then H=\sum_k\left( \frac{\tilde{p}_k\tilde{p}_{-k}}{2m}+\frac{1}{2}m\omega_k^2\tilde{q}_k\tilde{q}_{-k}\right) with \omega_k\equiv 2\sqrt{\frac{k}{m}}\sin \frac{|k|a}{2}
i\partial_tq_n=[q_n,H]=\frac{p_n}{m} i\partial p_n=[p_n,H]=\sum_n \frac{1}{2}k[p+n,(q_n-q_{n-1})^2=k(q_{n+1}-2q_n-q_{n-1}) \Rightarrow m\ddot{q}_n=-k(2q_n-q_{n+1}-q_{n-1})
then m\ddot{tilde{q}}_k=-k(2-2\cos ka)\tilde{q}_k put \tilde{q}_k=\sum_\omega e^{-i\omega t}q_{k,\omega} then \sum_\omega -m\omega^2 q_{k,\omega}+k(2-2\cos ka)q_{k,\omega}=0 0=(\omega^2-\omega_k^2)q_{k,\omega}\simeq (w-v_s^2k^2)q_{v,\omega}\quad (k\ll \frac{1}{a}) with (\partial_t^2-v_s^2\partial_x^2)q(x,t)=0 \Rightarrow q is a wave (v_q\equiv \left. \frac{\partial \omega}{\partial k}\right|_{k=0}=a\sqrt{\frac{k}{m}}
QM: [q_n,p_{n'}]=i\delta_{nn'} \Rightarrow [\tilde{q}_k,\tilde{p}_{k'}]=\sum_{nn'}u_{kn}u_{k'n'} [q_n,p_n] (u_{k,n}=e^{ikx_n}/\sqrt{N}) =\sum_n u_{kn}u_{k'n'}i=i\delta_{k-k'}
Finally, \hat{q}_k=\sqrt{\frac{\hbar}{2m\omega_k}}(a_k+a_{-k}^\dagger) \hat{p}_k=\frac{1}{i}\sqrt{\frac{\hbar m\omega_k}{2}}(a_k-a_{-k}^\dagger) [a_k,a^\dagger_{k'}]=\delta_{kk'}
\Rightarrow H=\sum_{k\ne 0}\hbar \omega_k \left( a^\dagger_{k}a_k+\frac{1}{2}\right) +\frac{p_0^2}{2m} sum of decoupled SHO's + center of mass motion
The ground state |0\rangle a_k|0\rangle=0 \quad \forall k,\quad p_0|0\rangle=0
First excited state: a^\dagger_k|0\rangle (one phonon with momentum \hbar k: |k\rangle)
Second excited state: a^\dagger_k a^\dagger_{k'}|0\rangle p=\hbar (k+k')\quad E=\hbar (\omega_k+\omega_{k'})\quad N_k=a^\dagger_ka_k
Fork space \mathcal{H}: 1 particle Hilbert space
\mathcal{F}=\otimes \mathcal{H} / symmetrize or anti-symmetrize
In above example, a_ks are automatically symmetry (spin-statistics for boson).
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Chain of oscillators H=\sum_k \hbar \omega_k (a^\dagger_k a_k+\frac{1}{2}+\frac{p_0^2}{2m} with \omega_k=\sqrt{\frac{2k}{m}}\sim \frac{|k|a}{2}.
\mathcal{H}=\mbox{span}\{ a^\dagger_{k_1}a^\dagger_{k_2}\cdots|0\rangle\}
Fork space \sim \otimes \mathcal{H}_{1-ptc}/(symmetrization)
In QFT, we naturally get symmetrization with a_ks.
|k|\ll \frac{1}{a} \Rightarrow \omega_k^2\simeq k^2V_s^2 photon E=pc is same as \omega=kc.
(\partial_t^2-v_s^2\nabla^2)\phi=0 \phi=\sum A_\omega e^{i\omega t} \omega^2=v_s^2k^2
And this can be Lorentz invariant \partial_\mu \partial^\mu \equiv \partial_t^2-\nabla^2
Mossbawer effect
Scalar field theory (in 1+1d)
Path-integral formulation Z=\int dq_1\, dq_2\, \cdots dq_N\, e^{iS[q_1,\cdots,q_N]} when S[q]=\int dt [\frac{1}{2}\sum m\dot{q}_n^2-V(q)] and V(q)=\frac{1}{2}k\sum^N_{n=1}(q_{n+1}-q_n)^2.
Continuum limit or long-wavelength limit a\rightarrow 0, N\rightarrow \infty
then (q_{n+1}-q_n)^2\rightarrow a^2(\partial_x q)^2\quad a\sum_n f(x_n)\rightarrow \int dx\, f(x)
Z=\int [Dq]e^{iS[q(x)]}
S[g]=\int dt\int dx(\frac{1}{2}(\partial_tq)^2-\mu V_s^2(\partial_xq)^2-rq^2-sq^4\cdots)=\int dx\, dt\, \mathcal{L}(q,\partial_tq,\partial_xq) Action functional is spacetime integral of Lagrangian density.
(when k\ll \frac{1}{a}, higher-derivatives neglected)
[x,p]=\hbar became [q,\pi]. momentum conj. for q(x,t) \pi(x,t)=\frac{\partial\mathcal{L}}{\partial \dot{q}}=\mu\dot{q} Hamiltonian became H=\sum_n p_n\dot{q}_n-L=\int dx(\frac{\pi(x)^2}{2\mu}+\mu V_s^2(\partial_xq)^2+\cdots)
In \infty-vol limit, \frac{1}{L^d}\sum_k\rightarrow \int \frac{d^dk}{(2\pi)^d} L^d\delta_{kk'}\rightarrow (2\pi)^d\delta^{(d)}(k-k') Continuum scalar field theory in (d+1)-dim. S[\phi]=\int d^dx\, dt\, (\frac{1}{2}\dot{\phi}^2-\frac{1}{2}v_s^2(\nabla\phi)^2-V(\phi)) EOM comes from "Least(Stationary) Action Principle" 0=S[\phi+\delta\phi]-S[\phi]\Leftrightarrow \frac{\delta S}{\delta \phi}=0 This became \delta S=\int d^dx\, dt\, (-\ddot{\phi}+v_s\nabla^2\phi-V'(\phi))\delta\phi=0 for example, V(\phi)=\frac{1}{2}m^2\phi^2 and V_s=1, this become (-\partial_t+\nabla^2-m^2)\phi=0
Conjugate momentum is \pi(x)=\frac{\partial\mathcal{L}}{\partial \dot{\phi}}=\dot{\phi}\\ \Rightarrow H=\int d^dx(\frac{1}{2}\pi(x)^2+\frac{1}{2}V_s^2(\nabla \phi)^2+\frac{1}{2}m^2\phi^2)\ge 0 \phi(x)=\int \frac{d^dk}{(2\pi)^d}e^{i\vec{k}\cdot\vec{x}}\phi_\vec{k} \pi(x)=\int \frac{d^dk}{(2\pi)^d}e^{-i\vec{k}\cdot\vec{x}}\pi_\vec{k} \Rightarrow H=\int \frac{d^dk}{(2\pi)^d}(\frac{1}{2}\pi_\vec{k}\pi_{-\vec{k}}+\frac{1}{2}(V_s^2\vec{k}^2+m^2)\phi_k\phi_{-k}) Dispersion relation is \omega_\vec{k}=m^2+V_s^2\vec{k}^2 Now we can represent as creation and annihilation operator \phi_k\equiv \frac{1}{\sqrt{2\omega_k}}(a_k+a^\dagger_{-k})\\ \pi_k\equiv \frac{1}{i}\sqrt{\frac{\omega_k}{2}}(a_k-a^\dagger_{-k}) and [\phi(x),\pi(y)]=i\delta^{(d)}(x-y)\Rightarrow [a_k,a^\dagger_{k'}]=(2\pi)^d\delta^{(d)}(k-k') \Rightarrow H=\int \frac{d^dk}{(2\pi)^d}\omega_k(a^\dagger_k a_k+\frac{1}{2}L^d)
\pi(x)=\int \frac{d^dk}{(2\pi)^d}\frac{1}{\sqrt{2\omega_k}}(e^{i\vec{k}\cdot\vec{x}}a_\vec{k}+e^{-i\vec{k}\cdot\vec{x}}a^\dagger_\vec{k}) \pi(x)=\int \frac{d^dk}{(2\pi)^d}\sqrt{\frac{\omega_k}{2}}(e^{i\vec{k}\cdot\vec{x}}a_\vec{k}-e^{-i\vec{k}\cdot\vec{x}}a_{-\vec{k}}^\dagger) For time difference, \phi(x,t)=e^{iHt}\phi(x)e^{-iHt}.
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Symmetries in field theory
\mathcal{L}=\mathcal{L}(\phi,\partial_\mu\phi) (which means locality) S[\phi]=\int d^{d+1}x\, \mathcal{L}(\phi,\partial_\mu\phi)
eg) \mathcal{L}_{KG}=\frac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi)-\frac{1}{2}m^2\phi^2(-V(\phi)) real massive scalar field.
eg) \mathcal{L}_{EM}=-\frac{1}{4e^2}F_{\mu\nu}F^{\mu\nu} where F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu and A_\mu=(\phi,\vec{A})
dimensional analysis
\hbar=c=1 "natural unit" \mbox{mass}\sim\mbox{energy}\sim\mbox{length}^{-1}\sim\mbox{time}^{-1} \Rightarrow [p_\mu]=[m]=[\partial_\mu]=+1 [X^\mu]=-1 [S]=0\quad \Rightarrow [\mathcal{L}]=d+1=D
Equation of motion (\Leftarrow least action principle)
\frac{\delta S}{\delta \phi(x)}=0=\int d^Dx\left[ \frac{\partial\mathcal{L}}{\partial \phi}\delta \phi+\frac{\partial\mathcal{L}}{\partial (\partial_\mu\phi)}\delta(\partial_\mu\phi)\right]=\partial_\mu \left[ \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\delta\phi\right]-\left[ \partial_\mu\left( \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\right)\right]\delta\phi =\int d^Dx\left[ \frac{\partial\mathcal{L}}{\partial\phi}-\partial_\mu\left( \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\right)\right]\delta\phi+\int (\mbox{total derivative})=0
Noether's theorem
Suppose S is invariant under a continuous transformation \phi\rightarrow \phi'(x)+\epsilon\Delta\phi(x) (continuous) symmetry \delta S=S[\phi+\epsilon(x^\mu)\Delta\phi]-S[\phi]=\int d^Dx(\partial_\mu\epsilon)j^\mu=^{IBP}-\int d^Dx\, \epsilon(\partial_\mu j^\mu)=0 by EOM is satisfied. so, \partial_\mu j^\mu=0. Which means conserved currrent.
\partial_0j^0-\vec{\nabla}\cdot\vec{j}=0 Q_R\equiv \int_R d^dx\, j^0 then \partial_t Q_R=\int d^dx\, \partial_0j^0=-\int_Rd^dx\, \nabla\cdot \vec{j}=0 Q$ is conserved.
Noether's method
\epsilon\rightarrow \epsilon(x^\mu)
symmetry: \mathcal{L}(\phi',\partial\phi')=\mathcal{L}(\phi,\partial\phi)+\epsilon(\partial_\mu\Theta^\mu) Taylor expand, \mathcal{L}(\phi',\partial\phi')=\mathcal{L}(\phi,\partial\phi)+\epsilon\left(\frac{\partial \mathcal{L}}{\partial\phi}\Delta \phi+\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\partial_\mu\Delta\phi\right)=\epsilon\left( \frac{\partial\mathcal{L}}{\partial\phi}-\partial_\mu\left(\frac{\partial \mathcal{L}}{\partial(\partial_\mu\phi)}\right)\right)\Delta\phi+\epsilon\partial_\mu\left[\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\Delta\phi\right] first term vanish with EOM, \partial_\mu j^\mu=0 with j^\mu=\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\Delta\phi-\Theta^\mu
Example) \mathcal{L}=\frac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi) have "shift symmetry" \phi\rightarrow \phi'=\phi+\epsilon S[\phi+\epsilon(x)]-S[\phi]=\int d^Dx\frac{1}{2}(\partial_\mu(\phi+\epsilon))^2-\frac{1}{2}(\partial_\mu\phi)^2=\int (\partial_\mu\phi)(\partial^\mu\epsilon)=-\int\epsilon\partial_\mu\partial^\mu\phi so we conclude j^\mu=\partial^\mu\phi and \partial^\mu j_\mu=0 satisfy from EOM.
With Q=\int j^0, \delta\phi\equiv \phi'-\phi=\epsilon =i\epsilon[Q,\phi]=i\epsilon[\int d^dy\partial_t \phi(y),\phi(x)]=i\epsilon \int d^dx[\pi(y),\phi(x)]=\epsilon when use \dot{\phi}=\frac{\partial\mathcal{L}}{\partial\phi}=\pi, [\pi,\phi]=-i\delta.
Example) Complex scalar field \phi\equiv =\phi_1+i\phi_2 \mathcal{L}=\partial_\mu\phi\partial^\mu\phi^*-m^2\phi^*\phi-V(|\phi|^2) \phi\rightarrow e^{i\alpha}\phi\\ \phi^*\rightarrow e^{-i\alpha}\phi e^{i\alpha}\in U(1), expand with \phi\rightarrow \phi'=\phi+i\alpha\phi \delta S=S[\phi+i\alpha\phi]-S[\phi]=\int d^Dx\alpha\partial_\mu(i\phi^*\partial^\mu\phi-i\phi\partial^\mu\phi^*) which derive current.
With quantization. j^\mu(x)=\int \frac{d^dk}{(2\pi)^d}(a^\dagger_ka_k-b^\dagger_kb_k)=N_p-N_\bar{p}
Spacetime translation
x^\mu\rightarrow x'^\mu=x^\mu-a^\mu \Rightarrow \phi(x)\rightarrow \phi(x')=\phi(x)+a^\nu\partial_\nu\phi+O(a^2) Current became for each direction \nu, \partial_\mu T^\mu_\nu=0 which called stress-energy tensor.
Symmetry: \mathcal{L}(\phi(x),\partial\phi(x))\rightarrow \mathcal{L}(\phi(x+a),\partial_\mu\phi(x+a))=\mathcal{L}+a^\mu\frac{d\mathcal{L}}{dx^\mu}=\mathcal{L}+a^\nu \frac{d}{dx^\mu}(\delta^\mu_\nu\mathcal{L}) \Rightarrow T^\mu_\nu=\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}(\partial_\nu\phi)-\mathcal{L}\delta^\mu_\nu T^0_0 is Hamiltonian, which is \frac{\partial\mathcal{L}}{\partial\dot{\phi}}\dot{\phi}-\mathcal{L}=\pi\dot{\phi}-\mathcal{L}=\mathcal{H}. P_i is momentum, \int T^0_id^dx=\int \left(\frac{\partial\mathcal{L}}{\partial \dot{\phi}}(\partial_i\phi)\right)d^dx=\int \pi\partial_i\phi d^dx which is momentum carried by fields
Scale transformation
x^\mu\rightarrow \lambda x^\mu \phi(x^\mu)\rightarrow \lambda^{d_\phi}\phi(\lambda x^\mu) which is called dilatation, d_\phi is scaling dim of \phi.
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RG flows & Scale invariance
Slogan: Every theory is an effective theory
\bullet we need to specify the relevant scale.
\bullet QFT defined UV-cutoff \Lambda
Renormalization group flow \Rightarrow a way to parametrize our ignorance of physics beyound \:ambda in terms of low energy dof.
\bullet Single scalar in (Euclidean) d-dim \mathcal{L}=\frac{1}{2}(\partial_\mu\phi)^2+\frac{1}{2}m^2\phi^2+\frac{\lambda}{4!}\phi^4 Z=\int [\mathcal{D}\phi]_\Lambda e^{-S} [\mathcal{D}\phi]_\lambda=\prod_{|\vec{k}|<\lambda} d\phi(\vec{k}) \phi(x)=\int \frac{d^dk}{(2\pi)^d}e^{ik\cdot x}\phi(k)
We would like to know how the description in terms of low E dof depending on \Lambda.
Pick 0<b<1 \hat{\phi}(k)=\begin{cases}\phi(k)&\mbox{for } b\Lambda\le k<\Lambda\\ 0& \mbox{for }|k|<b\Lambda\end{cases} \phi_{new}(k)=\begin{cases}\phi(k)& \mbox{for }k<b\Lambda\\ 0 & \mbox{otherwise}\end{cases} and \phi(k)=\hat{\phi}(k)+\phi_{new}(k). Z=\int [\mathcal{D}\phi]_\Lambda e^{-\int d^dx\, \mathcal{L}(\phi,\partial\phi)}=\int [\mathcal{D}\phi]_{b\Lambda}[\mathcal{D}\hat{\phi}]_\Lambda e^{-\int d^dx\, \mathcal{L}(\phi+\hat{\phi})}=\int [\mathcal{D}\phi]_{b\Lambda} e^{-\int d^dx\, \mathcal{L}(\phi)}\int [\mathcal{D}\hat{\phi}]_\Lambda e^{-\int d^dx(\frac{1}{2}(\partial\hat{\phi})^2+\frac{1}{2}m^2\hat{\phi}^2+\frac{\lambda}{6}\phi^3\hat{\phi}+\cdots)}
\bullet Assume m, \lambda small, & expand order by order
\bullet Integrate out \hat{\phi}, term by term.
\rightarrow Recast everything as an exponential for the low E dof.
\Rightarrow Z=\int [\mathcal{D}\phi]_{b\Lambda}e^{-\int d^dx\, \mathcal{L}_{eff}(\phi)} \mathcal{L}_{eff}=\frac{1}{2}(\partial_\mu \phi)^2+\frac{m^2}{2}\phi^2+\frac{\lambda}{4!}\phi^4+\frac{1}{2}\mu^2\phi^2+\frac{\xi}{4!}\phi^4+\Delta C(\partial_\mu\phi)^2+\Delta D(\phi^6)+\cdots
\bullet RG flow
Z=\int [\mathcal{D}\phi]_{b\Lambda}e^{-\int \mathcal{L}_{eff}} rescale distance x'=xb\quad k'=k/b\quad |k'|<\Lambda \int d^dx\, \mathcal{L}_{eff}=\int d^dx'\, b^{-d}(\frac{1}{2}(1+\Delta z)b^2(\partial_\mu\phi)^2+\frac{1}{2}(m^2+\Delta m^2)\phi^2+\cdots) & Rescale \phi\Rightarrow \phi'=(b^{2-d}(1+\Delta Z))^{1/2}\phi \int d^dx\, \mathcal{L}_{eff}=\int d^dx(\frac{1}{2}(\partial_\mu \phi)^2+\frac{1}{2}m'^2\phi^2+\frac{\lambda'}{4!}\phi^4+\cdots) when m'^2=(m^2+\Delta m^2)(1+\Delta Z)^{-1}b^{-2} \lambda'=(\lambda+\Delta\lambda)(1+\Delta Z)^{-2}b^{d-4} c'=(c+\Delta c)(1+\Delta Z)^{-3}b^d \Rightarrow Z=\int [\mathcal{D}\phi]_\Lambda e^{-S'_{eff}}
\bullet b: S\rightarrow S'_{eff} The process of integrating out high-E doe =transform of \mathcal{L}.
\bullet b=1-\delta \delta\ll 1 \Rightarrow "RG flow"
Scale-invariant theory
eg) Gaussian fixed point :\mathcal{L}_0=\frac{1}{2}(\partial_\mu\phi)^2 \rightarrow does not RG flow "fixed point": Scale invariant theory
Consider \mathcal{L} close to \mathcal{L}_0. m'^2=m^2b^{-2} \lambda'=\lambda b^{d-4} c'=cb^d when b=1-\delta, \delta\ll 1.
3 categories of couplings
\bullet (\ )\times b^{#<0} \Rightarrow grows under RG flow \Rightarrow away from the FP "relevant" ex) \phi^2
\bullet (\ )\times b^{#>0} \Rightarrow decays along RG. \Rightarrow "irrelevant" ex) \phi^4 for d\ge 4
\bullet (\ )\times b^0 \Rightarrow "marginal"
For m, \lambda, when d\ge 4, m bigger and bigger, \lambda smaller and smaller. We assume there is fixed point in high m.
For d<4, \phi^4 is relevant, m and \lambda grow. We assume another fixed point in high \lambda. This called Wilson-Fisher Fixed Point, which is non-trivial CFT.
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Conformal transformations
\mathcal{M} is spacetime, g_{\mu\nu} is metric on \mathcal{M} then ds^2=g_{\mu\nu}(x)dx^\mu dx^\nu
Def: Conformal transformation of coordinates is an invertible map x\rightarrow x'(x) which leaves the metric invariant up to a local rescaling g'_{\mu\nu}(x')=\Lambda(x)g_{\mu\nu}(x)
Remarks
i) \Lambda(x)=1 \Rightarrow isometry
eg) If \mathcal{M}=\mathbb{R}^{3,1}, g_{\mu\nu}=\eta_{\mu\nu} isometries = Poincare group
ii) \Lambda(x)=const. "scale transformation" or dilatations x^\mu\rightarrow \lambda x^\mu\quad \lambda\in\mathbb{R}>0
For an infinitesimal coord. tr. x^\mu\rightarrow x'^\mu=x^\mu+\epsilon^\mu(x) g_{\mu\nu}\rightarrow g'_{\mu\nu}=\left( \frac{\partial x^\alpha}{\partial x'^\mu}\right)\left(\frac{\partial x^\beta}{\partial x'^\nu}\right)g_{\alpha\beta}=g_{\mu\nu}-(\partial_\mu \epsilon_\nu+\partial_\nu\epsilon_\mu)+O(\epsilon^2) Let \partial_\mu\epsilon_\nu+\partial_\nu\epsilon_\mu=f(x)g_{\mu\nu} 2\partial^\mu\epsilon_\mu=f(x)\eta^{\mu\nu}\eta_{\mu\nu} (we set Euclidean g_{\mu\nu}=\eta_{\mu\nu} and \eta^{\mu\nu}\eta_{\mu\nu}=d) \Rightarrow f(x)=\frac{2(\partial\cdot \epsilon)}{d} \partial_\rho\partial_\mu\epsilon_\nu+\partial_\rho\partial_\nu\epsilon_\mu=(\partial_\rho f)\eta_{\mu\nu} we can get 2\partial_\mu\partial_\nu\epsilon_\rho=\eta_{\mu\rho}\partial_\nu f+\eta_{\nu\rho}\partial_\mu f-\eta_{\mu\nu}\partial_\rho f 2\partial^2\epsilon_\mu=(2-d)\partial_\mu f (2-d)\partial_\mu \partial_\mu f=2\partial^\rho\partial_\rho(\partial_\nu \epsilon_\mu)=\eta_{\mu\nu}\partial^2 f finally, (1-d)\partial^2 f=0
Assume d>2, let \partial_\mu\partial_\nu f=0 and f=A+B_\mu x^\mu \epsilon_\mu =a_\mu+b_{\mu\nu}X^\nu+C_{\mu\nu\rho}x^\nu x^\rho we put this in \partial_\mu\epsilon_\nu+\partial_\nu\epsilon_\mu=f(x)g_{\mu\nu} then we can find
- a_\mu no constraints \rightarrow translation (\dim=d)
- b_{\mu\nu}=\alpha\eta_{\mu\nu}+m_{\mu\nu} when \alpha is scale tr. and m_{\mu\nu}=-m_{\nu\mu} Lorentz tr.
- C_{\mu\nu\rho}=\eta_{\mu\rho}b_\nu+\eta_{\mu\nu}b_\rho-\eta_{\nu\rho}-\eta_{\nu\rho}b_\mu when b_\mu vector
conclusion, x^\mu\rightarrow x^\mu +a^\nu\quad\dim=d x^\mu\rightarrow (1+\alpha) x^\mu \quad\dim=1 x^\mu\rightarrow M^\mu_\nu x^\nu\quad\dim=\frac{1}{2}d(d-1) x^\mu\rightarrow x^\mu+2(x-\cdot b)x^\mu-b^\mu x^2\quad\dim=d last term called special conformal transformation. In finite version(not infinitesimal) x^\mu=\frac{x^\mu-b^\mu x^2}{1-2b\cdot x+b^2x^2}
# of generators = \dim of conformal group = \frac{1}{2}(d+1)(d+2)
If we define inversion, I:x^\mu\rightarrow x'^\mu=\frac{x^\mu}{x^2} then I^2=1 then SCT=IT_bI
Conformal Group
x\rightarrow x'=x'(x) \Phi(x)=\Phi'(x') x'^\mu=x^\mu+\omega)a\frac{\delta x^\mu}{\delta \omega^a}\quad |\omega_a|\ll 1 \delta_\omega\Phi\equiv \Phi'(x)-\Phi(x)=-i\omega_a G_a\Phi \Rightarrow iG_a\Phi=\frac{\delta x^\mu}{\delta \omega_a}\partial_\mu\Phi
eg) Translation \frac{\delta x^\mu}{\delta x^\nu}=\delta^\mu_\nu \Rightarrow Generator P_\mu=-i\partial_\mu
eg) Lorentz tr. x'^\mu=x^\mu+\omega^\mu_\nu x^\nu=x^\mu+\omega_{\rho\sigma}\eta^{\rho\mu}x^\rho (\omega_{\rho\sigma}=-\omega_{\sigma\rho}
then \frac{\delta x^\mu}{\delta \omega_{\rho\sigma}}=\frac{1}{2}(\eta^{\rho\mu}x^\sigma-\eta^{\rho\mu}x^\rho) then L_{\mu\nu}=i(X_\mu\partial_\nu-X_\nu\partial_\mu) other things are D=-ix^\mu\partial_\mu SCT=K_\mu=-i(2x^\mu x^\nu \partial_\nu-x^2\partial_\mu)
Conformal algebra
Generators D,P_\mu,K_\mu,L_{\mu\nu} satisfies commutation relation [D,L_{\mu\nu}]=[P_\mu,P_\mu]=[K_\mu,K_\nu]=0 [D,P_\mu]=iP_\mu\\ [D,K_\mu]=-iK_\mu\\ [K_\mu,P_\nu]=2i(\eta_{\mu\nu}D-L_{\mu\nu})\\ [L_{\mu\nu},P_\rho]=-i(\eta_{\mu\rho}P_\nu-\eta_{\nu\rho}P_\mu)\\ [L_{\mu\nu},K_\rho]=-i(\eta_{\mu\rho}K_\nu-\eta_{\nu\rho}K_\mu)\\ [L_{\mu\nu},L_{\rho\sigma}]=i(L_{\mu\rho}\eta_{\nu\sigma}+(\mbox{perm}))
Defining J_{\mu,\nu}=L_{\mu\nu}\\ J_{-1,\mu}=\frac{1}{2}(P_\mu-K_\mu)\\ J_{-1,0}=D\\ J_{0,\mu}=\frac{1}{2}(P_\mu+K_\mu)\\ a,b=-1,0,1,\cdots,d J_{a,b}=-J_{b,a} J_{a,b}\Rightarrow generates the Lorentz group in (d+1,1) SO(d+1,1)\quad (\mbox{Euclidean})\\ SO(d,2)\quad (\mbox{Minkowski})
Is scale invariance (Inv. in D) gives conformal invariance(Inv. in D,K_\mu)? With Poincare symmetry, yes.
[Homework 1]
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Conformal Algebra
Conformal Algebra \supset Scale symm. gp \simeq Poincare group \ltimes \{D\}
\bullet Any unitary, Poincare inv, QFT w/ a scale symmetry \rightarrow Conformal symmetry
(proven in d=2,3,4 but there is conterexample otherwise.)
Action on operators /fields
Any symmetry should be realized as operators acting on \mathcal{H} (Schrodinger) or on local operators (Heisenberg picture)
\phi_\alpha(x)=e^{-ip\cdot x}\phi_\alpha(0)e^{ip\cdot x} \partial_\mu \phi_\alpha(x)=e^{-ip\cdot x}(-iP_\mu \phi_\alpha(0)+\phi_\alpha(0)(iP_\mu)e^{-p\cdot x}=-i[P_\mu,\phi_\alpha(x)]
\bullet [P_\mu,\phi_\alpha(x)]=i\partial_\mu\phi_\alpha(x)
other generators? L_{\mu\nu},D,K_\mu \rightarrow leave origin inv.
\bullet Consider \Phi_\alpha(0). satisfying [D,\Phi_\alpha(0)]=i\Delta \Phi_\alpha(0) [L_{\mu\nu},\Phi_\alpha(0)]=i(S_{\mu\nu})_\alpha^\beta \Phi_\beta(0) (\Phi_\alpha is a irreducible representation of SO(d)) [K_\mu,\Phi_\alpha(0)]=0
\Delta: Scaling dim. of \Phi_\alpha
S_{\mu\nu}: irred. rep of Lorentz gp that acts on \Phi_\alpha
then \Phi_\alpha(x) primary operator/field
then \Phi_\alpha(x)=e^{-ip\cdot x}\Phi_\alpha(0)e^{ip\cdot x}
[D,\phi_\alpha(x)]=De^{-ip\cdot x}\phi_\alpha(0)e^{ip\cdot x}-e^{-ip\cdot x}\phi_\alpha(0)e^{ip\cdot x}D=e^{-ip\cdot x}(e^{ip\cdot x}De^{-ip\cdot x}\phi_\alpha(0)-\phi_\alpha(0)e^{ip\cdot x}De^{-ip\cdot x})e^{ip\cdot x}=e^{-ip\cdot x}[\hat{D},\phi_\alpha(0)]e^{ip\cdot x} \hat{D}=e^{ip\cdot x}De^{-ip\cdot x}
\hat{D}=(1+ixp-\frac{(xp)^2}{2}+\cdots)D(1-ixp-\frac{(xp)^2}{2}+\cdots)=D+ix^\mu [x_\mu,D]-\frac{1}{2}x^\mu x^\nu [x_\mu,[x_\nu,D]]=D+x^\mu p_\mu
[D,\Phi_\alpha(x)]=i(\Delta+x^\mu \partial_\mu)\Phi_\alpha(x)
[L_{\mu\nu},\Phi_\alpha(x)]=-i(x_\mu\partial_\nu-x_\nu\partial_\mu)\Phi_\alpha(x)+i(S_{\mu\nu})_\alpha^\beta \Phi_\beta(x).
[K_\mu,\phi_\alpha(x)]=2ix^\mu \Delta \Phi_\alpha(x)+i(2x^\mu x^\nu \partial_\nu -x^2\partial_\mu)\Phi_\alpha(x)+2ix^\rho(S_{\rho\mu})_\alpha^\beta \Phi_\beta(x)
eg) \varphi(x): scalar primary w/ scaling dim \Delta
\varphi(x)\rightarrow \varphi(x')=\left| \frac{\partial x'}{\partial x}\right|^{-\Delta/d}\varphi(x)
\left| \frac{\partial x'}{\partial x}\right| =\Lambda(x)^{-d/2}(\mbox{Jacobian})
\varphi(x')=\Lambda(x)^{\Delta/2}\phi(x)
In QM we want to know eigenstates (values of H=P^0)
In CFT, we focus on "eigenstates(primary)" of D, & eigenvalues \Delta
[D,P_\mu]=iP_\mu
[D,K_\mu]=-iK_\mu
Suppose \mathcal{O}_\Delta is a primary D\cdot \mathcal{O}\equiv [D,\mathcal{O}_\Delta]=i\Delta \mathcal{O}_\Delta D\cdot P_\mu\mathcal{O}=([D,P_\mu]+P_\mu D)\cdot \mathcal{O}=i(\Delta+1)P_\mu \mathcal{O}
P_\mu \rightarrow raising op.
K_\mu \rightarrow lower op.
"Ground state" \rightarrow annihilated by K_\mu K_\mu \mathcal{O}_\Delta=0 primary states/op.
\Rightarrow P_\mu \mathcal{O}_\Delta \rightarrow D=\Delta+1\\ P_\mu P_\nu \mathcal{O}_\Delta \rightarrow \Delta+2
For a given primary \mathcal{O}_\Delta \rightarrow can form "descendants" P_{\mu_1}\cdots P_{\mu_r}\mathcal{O}_\Delta \sim \partial_{\mu_1}\cdots\partial_{\mu_r}\mathcal{O}_\Delta(x)
Dilatation op. can be thought of as a Hamiltonian via certain map
In \mathbb{R}^d, ds^2=dr^2+r^2d\Omega_{d-1}=r^2\left[\frac{dr^2}{r^2}+d\Omega_{d-1}\right] let t=\log r, then \frac{\dr^2}{r^2}=dt^2. ds^2=dt^2+d\Omega_{d-1}:\mathbb{R}\times S^{d-1}
\Rightarrow By conformal symmetry, CFT on \mathbb{R}^d \Leftrightarrow CFT on \mathbb{R}_t\times S^{d-1}
(that picture)
State-operator correspondence
\mathbb{R}^d\quad \mathbb{R}\times S^{d-1}\\ D\leftrightarrow H\\ \mathcal{O}_\Delta(x)\leftrightarrow |\Delta,j\rangle \in \mathcal{H},\ K^\dagger=P
[3/20]
Consequences of Conformal Symmetry
Noether's theorem: To \forall continuous symmetry, \exists a current j_\mu, \partial^\mu j_\mu=0 x^\mu\rightarrow x'^\mu=x^\mu+\omega_a\frac{\delta x^\mu}{\delta \omega_a} \phi\rightarrow \phi'(x')=\phi(x)+\omega_a\frac{\delta F}{\delta \omega_a} Symmetry: \delta S=0 when \omega_a\rightarrow \omega_a(x) \delta S=\int d^dx\, j^\mu_a(\partial_\mu \omega^a)=-\int d^dx (\partial_\mu j^\mu_a)\omega^a=0 by EOM, so \partial_\mu j^\mu_a=0.
j^\mu_a=\left( \frac{\mathcal{L}}{\partial(\partial_\mu\phi)}\partial_\mu \phi-\delta^\mu_\nu \mathcal{L}\right)\times \frac{\delta x^\nu}{\delta \omega^a}-\frac{\partial \mathcal{L}}{\partial (\partial_\mu\phi)}\frac{\delta F}{\delta \omega^a}
Q_a=\int d^{d-1}_{x^0=const}x\, j^0_a
j^\mu_a\rightarrow j^\mu_{new,a}=j^\mu_a+\partial_\nu \beta^{\mu\nu}_a (\beta^{\mu\nu}=-\beta^{\nu\mu}) we call this "improvement"
\partial_\mu j^\mu_{new,a}=0
Stress-energy tensor
=Noether current for the translation symmetry x^\mu\rightarow x^\mu +\epsilon^\mu
\frac{\delta x^\mu}{\delta x^\nu}=\delta^\mu_\nu,\quad \frac{\delta F}{\delta \epsilon}=0
T^{\mu\nu}_c=\frac{\partial \mathcal{L}}{\partial(\partial_\mu\phi)}\partial^\nu \phi-\eta^{\mu\nu}\mathcal{L}
\partial_\mu T^{\mu\nu}_c=0 p^\nu=\int d^{d-1}x\, T^{0\nu} H=p^0=\int d^{d-1}T^{00}=\int d^{d-1}x\left( \frac{\partial\mathcal{L}}{\partial\phi}\dot{\phi}-\mathcal{L}\right) which is Hamiltonian.
For a Poincare invariant theory, we can "improve" T^{\mu\nu}_c\rightarrow T^{\mu\mu}
such that T^{\mu\nu}=T^{\nu\mu}
x^\mu\rightarrow x^\mu=x^\mu+\epsilon^\mu(x)
\delta S=-\int d^dx T^{\mu\nu}(\partial_\mu \epsilon_\nu)=-\frac{1}{2}\int d^dx T^{\mu\nu}(\partial_\mu \epsilon_\nu+\partial_\nu\epsilon_\mu)
T^{\mu\nu} generates the deformations under general coordinates transformations.
(이거 아노말리 렉노에서 나온 \mathacal{J} 랑 같은말인가?)
\bullet Translation \partial_\mu T^{\mu\nu}=0
Poincare T^{\mu\nu}=T^{\nu\mu}
For a conf. tr. \partial_\mu \epsilon_\nu+\partial_\nu \epsilon_\mu=f(x)\eta_{\mu\nu} \delta S=-\frac{1}{2}\int d^dx \,T^\mu_\mu f(x) Scale transformations is f(x)=\alpha=const. \delta S\sim \alpha\int d^dx T^\mu_\mu=0 T^\mu_\mu is a total derivative, T^\mu_\mu=\partial_\nu T^\nu for some J.
If the theory is invariant under SCT, then T^\mu_\mu=0
\Leftrightarrow Conformally invariant.
ex) free boson. S[\varphi]=\frac{1}{2}\int d^dx\, (\partial_\mu\varphi)(\partial^\mu \varphi)
T^{\mu\nu}_c=(\partial^\mu \varphi)(\partial^\nu \varphi)-\frac{1}{2}\eta^{\mu\nu}(\partial_\rho\varphi)(\partial^\rho\varphi) T^{\mu\nu}_c=T^{\mu\nu}_c
T^\mu_\mu=(\partial^\mu\varphi)(\partial_\mu\varphi)-\frac{d}{2}(\partial_\rho\varphi)(\partial^\rho\varphi)=\frac{2-d}{2}(\partial^\mu\varphi)(\partial_\mu\varphi)=0 for d=2 conformal. =#\partiall(\varphi\partial\varphi) for d\ne 2
Can add an improvement term s.t. \tilde{T}^\mu_\mu=0
T^{\mu\nu}=\frac{2}{\sqrt{g}}\left. \frac{\delta S}{\delta g^{\mu\nu}}\right|_{g^{\mu\nu}=\eta^{\mu\nu}} (g=\det g_{\mu\nu}) S[\varphi,g_{\mu\nu}]=\frac{1}{2}\int d^dx\sqrt{g}g^{\mu\nu}\partial_\mu \varphi\partial_\nu\varphi g_{\mu\nu}\rightarrow g_{\mu\nu}+\delta g_{\mu\nu}
Implication for correctors
Correlation functions \langle \phi(x_1)\phi(x_2)\cdots \phi(x_n)\rangle=\frac{1}{Z}\int [\mathcal{D}\phi]e^{-S[\phi]}\phi(x_1)\cdots \phi(x_n)
\bullet Suppose the action is inv. under a symmetry S[\phi]\rightarrow S[F(\phi)]
\bullet [\mathcal{D}\phi]\rightarrow [\mathcal{D}\phi']=[\mathcal{D}\phi] for a scale-symmetry \mathcal{D}\phi'\ne \mathcal{D}\phi \exists uv cut-off \Lambda.
at RG-fixed point, scale symmetry becomes exact!
\langle \phi(x_1')\phi(x_2')\cdots\phi(x_n')\rangle =\langle \phi'(x_1')\phi'(x_2')\cdots\phi'(x_n')\rangle=\langle F(\phi(x_1))F(\phi(x_2))\cdots F(\phi(x_n))\rangle F(\phi(x_1))=\phi'(x_1)
eg) Translation x'=x+a, F(\phi)=\phi.
\langle \phi(x_1+a)\cdots\phi(x_n+a)\rangle=\langle \phi(x_1)\cdots\phi(x_n)\rangle
eg) Rotation of scalars x'^\mu=\Lambda^\mu_\nu x^\nu \phi\rightarrow \phi \langle \phi(\Lambda x_1)\cdots\phi(\Lambda x_n)\rangle=\langle \phi(x_1)\cdots\phi(x_n)\rangle
\bullet In a CFT \rightarrow all operators are primary(\phi) of descendant (\partial_\mu\partial_\nu\cdots\phi.
For primary scalars \langle \phi(x_1')\cdots\phi(x_n')\rangle=\left|\frac{\partial x_1'}{\partial x_1}\right|^{-\Delta^1/d}\cdots \left| \frac{\partial x_n'}{\partial x_n}\right|^{-\Delta^n/d}\times \langle \phi(x_1)\cdots\phi(x_n)\rangle
x\rightarrow \lambda x\\ \phi(\lambda x)=\lambda^{-1}\phi(x) (for d=4, free masless scalar.)
2-point function
(\phi: primary of scale dim. \Delta_i, which is K_\mu\phi(0)=0,\ D\phi(0)=\Delta\phi(x))
Translation: \langle \phi(x_1)\phi(x_2)\rangle =f(|x_1-x_2|)
Scale tranformation x \rightarrow \lambda x \langle \phi(x_1)\phi(x_2)\rangle=\lambda^{\Delta_1+\Delta_2}\langle \phi(\lambda x_1)\phi(\lambda x_2)\rangle f(x)=\lambda^{\Delta_1+\Delta_2}f(\lambda x)
\langle \phi(x_1)\phi(x_2)\rangle=\frac{C_{12}}{|x_1-x_2|^{\Delta_1+\Delta_2}}
SCT \vec{x}'_\mu=\frac{\vec{x}_\mu-\vec{b}_\mu x^2}{1-2(b\cdot x)+b^2x^2} |x'_1=x'_2|=\frac{|x_1-x_2|}{\gamma^{1/2}_1\gamma^{1/2}_2}, \gamma_i\equiv 1-2b\cdot x_i+b^2x_i^2: \langle \phi(x_1)\phi(x_2)\rangle =\left| \frac{\partial x_1'}{\partial x_1}\right|^{\Delta_1/d}\left| \frac{\partial x_2'}{\partial x_2}\right|^{\Delta/2}\langle \phi(x_1')\phi(x_2')\rangle=\frac{1}{\gamma_1^{\Delta_1}}\frac{1}{\gamma_2^{\Delta_2}}\langle \phi(x_1')\phi(x_2')\rangle \frac{c_{12}}{|x_1-x_2|^{\Delta_1+\Delta_2}}=\frac{1}{\gamma_1^{\Delta_1}\gamma_2^{\Delta_2}}\frac{c_{12}}{|x_1'-x_2'|^{\Delta_1+\Delta_2}}=\frac{\gamma_1\gamma_2)^{(\Delta_1+\Delta_2)/2}}{\gamma_1^{\Delta_1}\gamma_2^{\Delta_2}}\frac{c_{12}}{|x_1-x_2|^{\Delta_1+\Delta_2}}
\langle \phi(x_1)\phi(x_2)\rangle =\begin{cases}\frac{1}{|x_1-x_2|{2\Delta_1}}& (\Delta_1=\Delta_2)\\ 0& (\Delta_1\ne \Delta_2)\end{cases} c_{12} can be normalized to 1.
3-point function
\langle \phi(x_1)\phi(x_2)\phi(x_3)\rangle=\frac{c_{123}}{|x_{12}|^a|x_{23}|^b|x_{31}|^c} when x_{ij}=x_i-x_j.
scale-invariance \Rightarrow a+b+c=\Delta_1+\Delta_2+\Delta_3
SCT \Rightarrow a=\Delta_1+\Delta_2-\Delta_3,\ b=\Delta_2+\Delta_3-\Delta_1,\ c=\Delta_3+\Delta_1-\Delta_2 and c_{123}\rightarrow not fixed by symmetry! contains dynamics of CFT.
and all the n-point function determined by 3-point function.
[3/25]
CFT data is made from spectrum of primary operators (\Delta,l) and OPE coefficients.
Spin-1 field 2 point function
\langle J_\mu(x)J_\nu(y)\rangle=\frac{\alpha_{\mu\nu}(x-y)}{|x-y|^{2\Delta}}
where \alpha_{\mu\nu}: symmetric \mu\leftrightarrow \nu.
\alpha_{\mu\nu}(x)=\eta_{\mu\nu}+\alpha\frac{X_\mu X_\nu}{X^2}
J'(x')=\frac{\partial x^\nu}{\partial x'^\mu}J_\nu(x)
\langle_\mu(x')J_\nu(y')\rangle=\langle J'_\mu(x')J'_\nu(y')\rangle \rightarrow \alpha=2\, \alpha_{\mu\nu}(x)=I_{\mu\nu}(x)=\eta_{\mu\nu}-2\frac{x_\mu x_\nu}{x^2} where I_{\mu\nu} is invariant tensor. (I_{\mu\nu}I^{\nu\rho}=\delta^\rho_\mu)
Conserved currents
If J_\mu is a conserved current \partial^\mu J_\mu=0 \Rightarrow \langle \partial^\mu_x J_\mu(x)J_\nu(y)\rangle=0 \partial^\mu_x\left( \frac{I_{\mu\nu}(x-y)}{|x-y|^{2\Delta}}\right)=0\quad\Rightarrow \Delta=d-1 generally, \Delta\ge d-1, so \partial^\mu J_\mu=0\Leftrightarrow \Delta=d-1
For a scalar primary \Delta_\phi\ge \frac{d-2}{2} then \Delta=\frac{d-2}{2}\Leftrightarrow \partial^2\phi=0
then \phi is a free scalar.
EM tensor 2 point function
\langle T_{\mu\nu}(x)T_{\rho\sigma}(y)\rangle=\frac{c}{|x-y|^{2\Delta}}\left(\frac{1}{2}I_{\mu\sigma}(x-y)I_{\nu\rho}(x-y)+I_{\mu\rho}(x-y)I_{\nu\sigma}(x-y)-\frac{1}{d}\eta_{\mu\nu}\eta_{\rho\sigma}\right) c is unambiguous "central charge" or "weyl anomaly". This measure DoF.
\partial_\mu T^{\mu\nu}=0\quad \Leftrightarrow \Delta=d
4-point function
\langle \phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\rangle=\frac{I(\mu,\nu)}{\prod_{i<j}|x_{ij}|^{\delta_{ij}}} where \sum_{j\ne i}\delta_{ij}=\Delta_i
corss-ratios u=\frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2},\ v=\frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2}
Radial Quantization
In a QFT, spacetime is foliation of time sheets \Sigma_{t_i}, Hilbert space defined each slice \mathcal{H}_{\Sigma_i}. Let incoming state O_1,O_2,O_3, denote as |\psi_{in}\rangle and out state O'_1,O'_2,O'_3 denote as \langle \psi_{out}| then Correlators is \langle O'\cdots O\rangle\sim \langle \psi_{out}|u|\psi_{in}\rangle,\quad u=e^{-iP_0\Delta t} when Hamiltonian P_0 commute with any quantum number.
In CFT, sheets become spheres \Sigma. (Euclidian space) In CFT, we foliate the space by S^{d-1} with the origin at the center.
In Hilbert space \mathcal{H}_\Sigma, in state |\psi_{in}\rangle means operators O_1,O_2,O_3 are in \Sigma, out state \langle \psi_{out}| means operators O'_1,O'_2,O'_3 are out \Sigma. Then correlator is \langle \psi_{out}|u|\psi_{in}\rangle,\quad u=e^{iD\Delta \tau},\ \tau=\log r all states on \mathcal{H}_\Sigma are classified in terms of (\Delta,l). D|\Delta,l\rangle=i\Delta |\Delta,l\rangle\\ L_{\mu\nu}|\Delta,l\rangle_\alpha=i(S_{\mu\nu})^\alpha_\beta|\Delta,l\rangle_\beta when S_{\mu\nu} is generators of angular momentum operator L_{\mu\nu}. also which is equal to primary operator.
[4/1]
State operator correspondence
\psi\in \mathcal{H}(S^{d-1})
\leftrightarrow inserting op. inside S^{d-1}
ex) |0\rangle vacuum state (\Delta=0\ l=0) \leftrightarrow no insertion (annihilated by \forall generators P_\mu,\ K_\mu,L_{\mu\nu},D)
ex) Insert a primary op. at the origin
\phi_\Delta(0)\ \leftrightarrow \ |\Delta\rangle=\phi_\Delta(0)|0\rangle
D|\Delta\rangle=D\phi_\Delta(0)|0\rangle\\ =[D,\phi_\Delta(0)]|0\rangle\\ =i\Delta\phi_\Delta(0)|0\rangle\\ =i\Delta |\Delta\rangle |\Delta\rangle is an eigentate of D.
WTS: |\Delta\rangle is a primary state. (K_\mu|\Delta\rangle=0)
K|\Delta\rangle=K\phi_\Delta(0)|0\rangle=[K,\phi_\Delta(0)]|0\rangle=0
|\Delta,z\rangle=\phi_\Delta(z)|0\rangle Not a primary.
Because |psi\rangle=\phi_\Delta(x)|0\rangle=e^{-ixp}\phi_\Delta(0)e^{ixp}|0\rangle\\ =e^{-ixp}\phi_\Delta(0)|0\rangle=(1-ixp+\cdots)\phi_\Delta(0)|0\rangle\\ =|\Delta\rangle-ix\cdot p|\Delta\rangle+\cdots
D(P_\mu|\Delta\rangle)=([D,P_\mu]+P_\mu D)|\Delta\rangle=i(\Delta+1)P_\mu|\Delta\rangle
Descendant state of |\Delta\rangle
P_\mu|\Delta\rangle is a Not primary state. (K_\nu P_\mu|\Delta\rangle\ne 0)
Because K_\nu P_\mu|\Delta\rangle=([K_\nu,P_\mu]+P_\mu K_\nu)|\Delta\rangle=[K_\nu,P_\mu]|\Delta\rangle\ne 0
P_\mu on |\Delta\rangle \rightarrow scaling dim =\Delta+1
P_\mu P_\nu on |\Delta\rangle \rightarrow \Delta+2
Looks like P is raising op, K is lowering op. and |\Delta\rangle is ground state.
Operator Product Expansion
O_1(x)O_2(y)=\sum (x-y)^n\partial_n O(x)\sim \sum_j C^{(x,y)}_{Rj}O_{Rj}(x)
In CFT, |\psi\rangle=\phi_1(x)\phi_2(0)|0\rangle |psi\rangle=\sum_n C_n(x)|E_n\rangle where |E_n\rangle is eigenstate of D.
|E_n\rangle is a linear conbination of primaries & descendants \phi_1(x)\phi_2(0)|0\rangle=\sum C_\Delta(x,\partial)\phi_\Delta(0)|0\rangle for primary operators \phi_\Delta. This is more constraint than generic QFT.
\phi_1(x)\phi_2(0)|0\rangle=\frac{const}{|x|^k}(\phi_\Delta(0)+\cdots)|0\rangle
D\phi_1(x)\phi_2(0)|0\rangle=i(\Delta_1+x^\mu\partial_\mu)\phi_1(x)\phi_2(0)|0\rangle
[D\phi_1(x)]\phi_2(0)|0\rangle+\phi_1(x)D\phi_2(0)=i(\Delta_1+x^\mu\partial_\mu)\phi_1(x)\phi_2(0)|0\rangle+\phi_1(x)(i\Delta_2)\phi_2(0)|0\rangle=i(\Delta_1-\Delta_2-k)\frac{const}{|x|^k}(\cdots)
D(RHS)=i\Delta \frac{const}{|x|^k}(\phi_\Delta(0)+\cdots)|0\rangle\\ k=\Delta_1-\Delta_2+\Delta In conclusion, we can fix primary part (other parts are descendents) by OPE \phi_1(x)\phi_2(0)=\frac{consst}{|x|^{\Delta_1-\Delta_2+\Delta}}(\phi_\Delta(0)+\alpha x^\mu \partial_\mu \phi_\Delta+\cdots)
We can fix first descendent term from [K_\mu,\phi_\Delta(x)]=2ix_\mu\Delta \phi_\Delta(x)+i(2x_\mu x^\nu\partial_\nu-x^2\partial_\mu)\phi_\Delta(x) [K_\mu,\phi_1(x)]\phi_2(0)=ix_\mu(\Delta_1-\Delta_2+\Delta)\frac{const}{|x|^{\Delta_1+\Delta_2-\Delta}}\times (\phi_\Delta(0)+\cdots)|0\rangle
K_\mu(RHS)=\cdots \alpha=\frac{\Delta_1-\Delta_2+\Delta}{2\Delta}
Another way to computing is C_\Delta(x,\partial) by 3-point function \langle\phi_1(x)\phi_2(x)\phi_\Delta(z)\rangle=\sum_{\Delta'\in prim}C_{R\Delta'}C_{\Delta'}(x,\partial)\langle \phi_{\Delta'}(y)\phi_\Delta(z)\rangle \Delta=\Delta' then =C_{12\Delta}C_\Delta(x,\partial)\frac{1}{|x-y|^{2\Delta}} =\frac{C_{12\Delta}}{x^{\Delta_1+\Delta_2-\Delta}z^{\Delta_2+\Delta-\Delta_1}-|x-z|^{\Delta_1+\Delta-\Delta_2}}
[4/8]
OPE in CFT. |\psi\rangle\in \mathcal{H}_{S^{d-1}} |\psi\rangle=\phi_1(x)\phi_2(0)|0\rangle |\psi\rangle=\sum_n c_n(x)|E_n\rangle \phi_1(x)\phi_2(0) =\sum_{\phi\in prim\ in \phi_1\times \phi_2}C_\Delta(x,\partial)\phi_\Delta(0) "Operator algebra" C_\Delta is fixed by conformal symmetry
Consider \langle \phi_1(x)\phi_2(0)\phi_\Delta(z)\rangle=\sum_{\Delta'\in \phi_1\times \phi_2}C_{12\Delta'}C_{\Delta'}(x,\partial_y)\langle \phi_{\Delta'}(y)\phi_\Delta(z)\rangle|_{y=0} where \langle \phi_{\Delta'}(y)\phi_\Delta(z)\rangle|_{y=0}=\delta_{\Delta \Delta'}\times \frac{1}{|y-z|^{2\Delta}}
LHS is \frac{C_{12\Delta}}{|x|^{\Delta_1+\Delta_2-\Delta}|z|^{\Delta_2+\Delta-\Delta_1}|x-z|^{\Delta_1+\Delta-\Delta_2}} and RHS is \left.\frac{C_\Delta(x,\partial_y)}{|y-z|^{2\Delta}\right|_{y=0}
Look at 4-point function
scalar primary with scaling dimension \Delta \langle \phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\rangle=\frac{G(u,v)}{|x_{12}^{2\Delta_E}|x_{34}|^{2\Delta E}} where u=\frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}\quad v=\frac{x_{14}^2x_{23}^2}{x_{12}^2x_{34}^2} are invariant of conformal group.
We have to fix G(u,v).\phi(x_1)\phi(x_2)=\sum_{\Delta \in \phi\times \phi}c_\Delta C_\Delta (x_{12},\partial_y)\phi_\Delta(y)|_{y=\frac{x_1+x_2}{2}} \phi(x_2)\phi(x_4)=\sum_{\Delta \in \phi\times \phi}c_\Delta C_\Delta (x_{34},\partial_z)\phi_\Delta(z)|_{z=\frac{x_3+x_4}{2}} c_\Delta: "OPE coeff": \phi_{\Delta_E}\phi_{\Delta_#}\phi_\Delta
\langle \phi\phi\phi\phi\rangle= \sum_{\Delta\in\phi\times\phi}C_\Delta^2[C_\Delta(x_{12},\partial_y)C_\Delta(x_{34},\partial_z)\langle \phi_\Delta(y)\phi_\Delta(z)\rangle |_{y=\frac{x_1+x_2}{2},z=\frac{x_3+x_4}{2}}] we just need C_\Delta. other things are completely fixed by symmetry. =\sum_\Delta C_\Delta^2 \frac{G_{\Delta,l}(u,v)}{|x_{12}|^{2\Delta_E}|x_{34}|^{2\Delta_E}} and G_{\Delta,l}(u,v) are completely fixed by symmetry and it called 4-point conformal block. this depends op "external ops" and intermediate ops.
C^2_\Delta and \Delta\in\phi\times \phi are data from 3-point functions.
2-dim CFT
infinitesimal coordinate transformation x\rightarrow {x'}^\mu=x^\mu+\epsilon^\mu(x) g_{\mu\nu}\rightarrow g_{\mu\nu}'=\frac{\partial x^\alpha}{\partial {x'}^\mu}\frac{\partial x^\beta}{\partial {x'}^\nu}g_{\alpha\beta}=(\delta^\alpha_\mu-\partial_\mu \epsilon^\alpha)(\delta^\beta_\nu-\partial_\nu\epsilon^\beta)\eta_{\alpha\beta}=\eta_{\mu\nu}-(\partial_\mu\epsilon_\nu+\partial_\nu\epsilon_\mu)+O(\epsilon^2)
Conformal tr: g_{\mu\nu}\rightarrow g_{\mu\nu}'=\Lambda(x)g_{\mu\nu} when \Lambda(x)\simeq 1-f(x) g^{\mu\nu}(\partial_\mu\epsilon_\nu+\partial+\nu\epsilon_\mu)=(f(x)g_{\mu\nu})g^{\mu\nu} 2\partial_\mu\epsilon^\mu=f(x)d
\partial_\mu\epsilon_\nu+\partial_\nu\epsilon_\mu=\frac{2}{d}(\partial_\rho\epsilon^\rho)\eta_{\mu\nu}
when d=2, \partial_0\epsilon_0=\partial_1\epsilon_1,\quad \partial_0\epsilon_1=-\partial_1\epsilon_0 "Cauchy-Riemann eqn."
z\equiv x^0+ix^1,\quad \bar{z}=x^0-ix^1
\partial\equiv \partial_z=\frac{1}{2}(\partial_0-i\partial_1)
\bar{\partial}\equiv \partial_{\bar{z}}=\frac{1}{2}(\partial_0+i\partial_1)
\epsilon(z)=\epsilon^0+i\epsilon^1, \bar{\epsilon}(z)=\epsilon^0-i\epsilon^1 then \epsilon(z) is arbitrary holomorphic, \bar{\epsilon}(z) is anti-holomorphic.
2d confirmal tr. consider w/ analytic coord tr. z\rightarrow z'=f(z)\\ \bar{z}\rightarrow \bar{z}'=\bar{f}(\bar{z})
$$\Rightarrow$ $\infty$-dim'l algebraic of conformal tranformation.
Consider z'=z+\epsilon(z) when \epsilon(z)=\sum_{n\in\mathbb{Z}} C_nz^{n+1}
\phi(z,\bar{z))\rightarrow spin-less \Delta=0
\phi'(z',\bar{z}')=\phi(z,\bar{z}) \delta\phi=\phi'(z,\bar{z})-\phi(z,\bar{z})\\ =\phi(z-\epsilon,\bar{z}-\epsilon)-\phi(z,\bar{z})\\ \sum_n(c_n l_n+\bar{c}_n\bar{l}_n)\phi(z,\bar{z}) with l_n\equiv -z^{n+1}\partial_z\\ \bar{l}_n\equiv -\bar{z}^{n+1}\partial_\bar{z}
Witt algebra: [l_m,l_n]=(m-n)l_{m+n} [\bar{l}_m,\bar{l}_n]=(m-n)\bar{l}_{m+n} [l_m,\bar{l}_n]=0 Conf. algebra in 2d = Widd \oplus Witt (Classically)
\{l_n,\bar{l}_n\} generates "local" conformal algebra.
Not globally well-defined on the Riemann sphere. S^2=\mathbb{C}\cup \{\infty\}
[Homework 2]
1. Consider the canonical energy-momentum tensor for the free boson in d > 2. Find an improvement term that makes it classically traceless without spoiling classical conservation.
Sol)
\begin{align}T^{\mu\nu}_c=-\eta^{\mu\nu}\mathcal{L}+\frac{\partial\mathcal{L}}{\partial(\partial_\mu\Phi)}\partial^\nu\Phi\end{align} Free boson is \mathcal{L}=\partial_\mu \varphi\partial^\mu\varphi Then canonical energy momentum tensor is T^{\mu\nu}_c=-\eta^{\mu\nu}\partial_\rho\varphi\partial^\rho\varphi+\partial_\mu\varphi\partial^\nu\varphi Spin is 0. We can make Belinfante tensor and virial:
2. Consider two-dimensional Liouville theory whose Lagrangian density is given as \mathcal{L}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi+\frac{1}{2}m^2e^\phi. Write down the canonical energy-momentum tensor and verify that it is conserved. Add a term so that it is also traceless without spoiling the conservation.
Sol)
Belinfante tensor and virial:
3. Prove the following property under special conformal transformations |{x'}_i-{x'}_j|=\frac{|x_i-x_j|}{\gamma^{1/2}_i\gamma^{1/2}_j} where \gamma_i=1-2b\cdot x_i+b^2x_i^2.
Sol)
4. Consider the inverse tensor I_{\mu\nu}(x)\equiv \eta_{\mu\nu}-2\frac{x_\mu x_\nu}{x^2}. Show that I_{\mu\alpha}(x)I^{\alpha\beta}(x-y)I_{\beta\nu}(y)=I_{\mu\nu}(x'-y'), where (x')^\mu=x^\mu/x^2, (y')^\mu=y^\mu/y^2.
Reference
(Song, Jaewon)[PH754] Advanced Particle Physics<Conformal and Topological Field Theory>