«

QFT I topics

时间:2023-2-20 07:38     作者:JourinTown     分类: 2022秋


The energy-momentum tensor Tνμ:=jνμ=νϕL(muϕ)ηνμL and μTμνT^\mu_\nu:=j^\mu_\nu=\partial_\nu\phi\frac{\partial \mathcal{L}}{\partial(\partial_mu \phi)}-\eta^\mu_\nu \mathcal{L} ~\text{and}~ \partial_\mu T^{\mu\nu}(Not symmetric)

Relation: P0=d3xT00=d3xH=HP^0=\int d^3x T^{00}=\int d^3x \mathcal{H}=H

with canonical momenta: H=π0ϕL\mathcal{H}=\pi\partial_0\phi -\mathcal{L}

What are the canonical momenta?

π=L(0ϕ)\pi=\frac{\partial\mathcal{L}}{\partial(\partial_0 \phi)}(in real scalar field 0ϕ(x)\partial_0 \phi(\mathbf{x}))
The commutation relation is:
[ϕ(x),π(y)]=iδ(xy)[ϕ(x),ϕ(y)]=[π(x),π(y)]=0[\phi(\mathbf{x}),\pi(\mathbf{y})]=\mathrm{i}\delta(\mathbf{x-y})\qquad[\phi(\mathbf{x}),\phi(\mathbf{y})]=[\pi(\mathbf{x}),\pi(\mathbf{y})]=0

scalar complex scalar spinor gauge QED
L\mathcal{L} μϕμϕm2ϕ2\partial_\mu \phi \partial^\mu \phi-m^2 \phi^2 μϕμϕm2ϕϕ\partial_\mu \phi \partial^\mu \phi^*-m^2 \phi \phi^* ψˉ(im)ψ\bar{\psi}(\mathrm{i} \cancel{\partial}-m) \psi 14FμvFμv-\frac{1}{4} F_{\mu v} F^{\mu v} 12trfFμνFμv+ψˉ(iDm)ψ-\frac{1}{2} \operatorname{tr}_f F_{\mu \nu} F^{\mu v}+\bar{\psi}(\mathrm{i} \cancel{D}-m) \psi
EoM (2+m2)ϕ(x)=0\left(\partial^2+m^2\right) \phi(x)=0 - iσˉμμψL=0iσμμψR=0\begin{aligned} \mathrm{i} \bar{\sigma}^\mu \partial_\mu \psi_L=0 \\ \mathrm{i} \sigma^\mu \partial_\mu \psi_R=0\end{aligned} ρρAv=0\partial_\rho \partial^\rho A^v=0 -

How are scalar/spinor/gauge fields quantised?

  1. Scalar field

    • Normal field OP
      ϕ(x)=d3p(2π)312ωp(a(p)eipx+a(p)eipx)π(x)=id3p(2π)3ωp2(a(p)eipxa(p)eipx)\begin{aligned} \phi(x) & =\int \frac{\mathrm{d}^3 p}{(2 \pi)^3} \frac{1}{\sqrt{2 \omega_{\mathbf{p}}}}\left(a(\boldsymbol{p}) e^{-\mathrm{i} p x}+a^{\dagger}(\boldsymbol{p}) e^{\mathrm{i} p x}\right) \\ \pi(x) & =-\mathrm{i} \int \frac{\mathrm{d}^3 p}{(2 \pi)^3} \sqrt{\frac{\omega_{\mathbf{p}}}{2}}\left(a(\boldsymbol{p}) e^{-\mathrm{i} p x}-a^{\dagger}(\boldsymbol{p}) e^{\mathrm{i} p x}\right)\end{aligned}
    • Fourier Transformed OP
      ϕ~(p)=d3xeipxϕ(x)=12ωp(a(p)+a(p))π~(p)=d3xeipx0ϕ(x)=iωp2(a(p)a(p))\begin{aligned} & \tilde{\phi}(\boldsymbol{p})=\int \mathrm{d}^3 x e^{-\mathrm{i} \boldsymbol{p} x} \phi(\boldsymbol{x})=\frac{1}{\sqrt{2 \omega_{\boldsymbol{p}}}}\left(a(\boldsymbol{p})+a^{\dagger}(-\boldsymbol{p})\right) \\ & \tilde{\pi}(\boldsymbol{p})=\int \mathrm{d}^3 x e^{-\mathrm{i} \boldsymbol{p} x} \partial^0 \phi(\boldsymbol{x})=-i \sqrt{\frac{\omega_{\mathbf{p}}}{2}}\left(a(\boldsymbol{p})-a^{\dagger}(-\boldsymbol{p})\right)\end{aligned}
    • Creation and annihilation operators
      a(p)=ωp2ϕ~(p)+i12ωpπ~(p)a(p)=ωp2ϕ~(p)i12ωpπ~(p).\begin{aligned} a(\boldsymbol{p}) & =\sqrt{\frac{\omega_{\mathbf{p}}}{2}} \tilde{\phi}(\boldsymbol{p})+\mathrm{i} \frac{1}{\sqrt{2 \omega_{\mathbf{p}}}} \tilde{\pi}(\boldsymbol{p}) \\ a^{\dagger}(-\boldsymbol{p}) & =\sqrt{\frac{\omega_{\mathbf{p}}}{2}} \tilde{\phi}(\boldsymbol{p})-\mathrm{i} \frac{1}{\sqrt{2 \omega_{\mathbf{p}}}} \tilde{\pi}(\boldsymbol{p}) .\end{aligned}
    • Hamiltonian
      H=d3p(2π)312[π~(p)π~(p)+ωp2ϕ~(p)ϕ~(p)]=d3p(2π)3ωpa(p)a(p)H=\int \frac{\mathrm{d}^3 p}{(2 \pi)^3} \frac{1}{2}\left[\tilde{\pi}(\boldsymbol{p}) \tilde{\pi}(-\boldsymbol{p})+\omega_{\mathbf{p}}^2 \tilde{\phi}(\boldsymbol{p}) \tilde{\phi}(-\boldsymbol{p})\right]=\int \frac{\mathrm{d}^3 p}{(2 \pi)^3} \omega_{\mathbf{p}} a^{\dagger}(\boldsymbol{p}) a(\boldsymbol{p})
    • Relation
      • [ϕ(t,x),π(t,y)]=iδ(xy)[ϕ(t,x),ϕ(t,y)]=0=[π(t,x),π(t,y)]\begin{aligned} & {[\phi(t, \boldsymbol{x}), \pi(t, \mathbf{y})]=\mathrm{i} \delta(\boldsymbol{x}-\mathbf{y})} \\ & {[\phi(t, \boldsymbol{x}), \phi(t, \mathbf{y})]=0=[\pi(t, \boldsymbol{x}), \pi(t, \mathbf{y})]}\end{aligned}
      • [ϕ~(p),π~(q)]=i(2π)3δ(p+q)[ϕ~(p),ϕ~(q)]=0=[π~(p),π~(q)]\begin{aligned} & [\tilde{\phi}(\boldsymbol{p}), \tilde{\pi}(\mathbf{q})] =\mathrm{i}(2 \pi)^3 \delta(\boldsymbol{p}+\mathbf{q}) \\ & [\tilde{\phi}(\boldsymbol{p}), \tilde{\phi}(\mathbf{q})]=0=[\tilde{\pi}(\boldsymbol{p}), \tilde{\pi}(\mathbf{q})]\end{aligned}
      • [a(p),a(q)]=(2π)3δ(pq)[a(p),a(q)]=0=[a(p),a(q)]\begin{aligned} &[a(\boldsymbol{p}), a^{\dagger}(\mathbf{q})]=(2 \pi)^3 \delta(\boldsymbol{p}-\mathbf{q}) \\ & [a(\boldsymbol{p}), a(\mathbf{q})]=0=\left[a^{\dagger}(\boldsymbol{p}), a^{\dagger}(\mathbf{q})\right]\end{aligned}
  2. Complex Scalar field

    • Fourier Transformed OP
      ϕ(x)=d3p(2π)312ωp(a(p)eipx+b(p)eipx)π(x)=0ϕ(x)=id3p(2π)3ωp2(b(p)eipxa(p)eipx)\begin{aligned} & \phi(\mathbf{x})=\int \frac{\mathrm{d}^3 p}{(2 \pi)^3} \frac{1}{\sqrt{2 \omega_{\mathbf{p}}}}\left(a(\boldsymbol{p}) e^{\mathrm{i} \mathbf{p x}}+b^{\dagger}(\boldsymbol{p}) e^{-\mathrm{i} \mathbf{p x}}\right) \\ & \pi(\mathbf{x})=\partial^0 \phi^{\dagger}(x)=-\mathrm{i} \int \frac{\mathrm{d}^3 p}{(2 \pi)^3} \sqrt{\frac{\omega_{\mathbf{p}}}{2}}\left(b(\boldsymbol{p}) e^{\mathrm{i} \mathbf{p x}}-a^{\dagger}(\boldsymbol{p}) e^{-\mathrm{i} \mathbf{p x}}\right)\end{aligned}
      with a=12(a1+ia2),b=12(a1ia2)a=\frac{1}{\sqrt{2}}\left(a_1+\mathrm{i} a_2\right), \quad b=\frac{1}{\sqrt{2}}\left(a_1-\mathrm{i} a_2\right)
    • Hamiltonian
      H=12d3p(2π)3[π~(p)π~(p)+ωp2ϕ~(p)ϕ~(p)]=12d3p(2π)3ωp[a(p)a(p)+b(p)b(p)]H=\frac{1}{2} \int \frac{\mathrm{d}^3 p}{(2 \pi)^3}\left[\tilde{\pi}(\boldsymbol{p}) \tilde{\pi}^{\dagger}(\boldsymbol{p})+\omega_{\mathbf{p}}^2 \tilde{\phi}(\boldsymbol{p}) \tilde{\phi}^{\dagger}(\boldsymbol{p})\right]=\frac{1}{2} \int \frac{\mathrm{d}^3 p}{(2 \pi)^3} \omega_{\mathbf{p}}\left[a^{\dagger}(\boldsymbol{p}) a(\boldsymbol{p})+b^{\dagger}(\boldsymbol{p}) b(\boldsymbol{p})\right]
    • Relation
      • [ϕ(x),π(y)]=iδ(xy)[a(p),a(q)]=(2π)3δ(pq)[b(p),b(q)]=(2π)3δ(pq)\begin{aligned} &{[\phi(\boldsymbol{x}), \pi(\mathbf{y})] } =\mathrm{i} \delta(\boldsymbol{x}-\mathbf{y}) \\& {\left[a(\boldsymbol{p}), a^{\dagger}(\boldsymbol{q})\right] } =(2 \pi)^3 \delta(\boldsymbol{p}-\boldsymbol{q}) \\ &{\left[b(\boldsymbol{p}), b^{\dagger}(\boldsymbol{q})\right] } =(2 \pi)^3 \delta(\boldsymbol{p}-\boldsymbol{q})\end{aligned}
  3. Spinor field

    • Fourier Transformed OP
      ψ(x)=d3p(2π)312p0s[eipxas(p)us(p)+e+ipxbs(p)vs(p)]\psi(x)=\int \frac{\mathrm{d}^3 p}{(2 \pi)^3} \frac{1}{\sqrt{2 p^0}} \sum_s\left[e^{-\mathrm{i} p x} a_s(\boldsymbol{p}) u_s(\boldsymbol{p})+e^{+\mathrm{i} p x} b_s^{\dagger}(\boldsymbol{p}) v_s(\boldsymbol{p})\right]
    • Hamiltonian
      H=d3xψ(x)γ0(iγ+m)ψ(x)=d3p(2π)32p02p0p0s[as(p)as(p)bs(p)bs(p)]H =\int \mathrm{d}^3 x \psi^{\dagger}(\mathbf{x}) \gamma^0(\mathrm{i} \boldsymbol{\gamma} \boldsymbol{\partial}+m) \psi(\boldsymbol{x}) =\int \frac{\mathrm{d}^3 p}{(2 \pi)^3} \frac{2 p^0}{2 p^0} p^0 \sum_s\left[a_s^{\dagger}(\boldsymbol{p}) a_s(\boldsymbol{p})-b_s(\boldsymbol{p}) b_s^{\dagger}(\boldsymbol{p})\right]
    • Relation(mainly anti-commutation)
      • {as(p),ar(q)}=(2π)3δsrδ(pq){bs(p),br(q)}=(2π)3δsrδ(pq)\begin{aligned} & \left\{a_s(\boldsymbol{p}), a_r^{\dagger}(\boldsymbol{q})\right\}=(2 \pi)^3 \delta_{s r} \delta(\boldsymbol{p}-\boldsymbol{q}) \\ & \left\{b_s(\boldsymbol{p}), b_r^{\dagger}(\boldsymbol{q})\right\}=(2 \pi)^3 \delta_{s r} \delta(\boldsymbol{p}-\boldsymbol{q})\end{aligned}
      • [ψ(x),iψ(y)]=iδ(xy)\left[\psi(\boldsymbol{x}), \mathrm{i} \psi^{\dagger}(\boldsymbol{y})\right]=\mathrm{i} \delta(\boldsymbol{x}-\boldsymbol{y})
      • {ψξ(x),ψξ(y)}=δξξδ(xy),{ψξ(x),ψξ(y)}=0={ψξ(x),ψξ(y)}\left\{\psi_{\xi}(\boldsymbol{x}), \psi_{\xi^{\prime}}^{\dagger}(\boldsymbol{y})\right\}=\delta_{\xi \xi^{\prime}} \delta(\boldsymbol{x}-\boldsymbol{y}), \quad\left\{\psi_{\xi}(\boldsymbol{x}), \psi_{\xi^{\prime}}(\boldsymbol{y})\right\}=0=\left\{\psi_{\xi}^{\dagger}(\boldsymbol{x}), \psi_{\xi^{\prime}}^{\dagger}(\boldsymbol{y})\right\}
      • uˉr(p)us(p)=2mδrs,vˉr(p)vs(p)=2mδrs,uˉr(p)vs(p)=0=vˉr(p)us(p)\bar{u}_r(p) u_s(p)=2 m \delta_{r s}, \quad \bar{v}_r(p) v_s(p)=-2 m \delta_{r s}, \quad \bar{u}_r(p) v_s(p)=0=\bar{v}_r(p) u_s(p)
      • sus(p)ξuˉs(p)ξˉ=(+m)ξξˉ,svs(p)ξvˉs(p)ξˉ=(m)ξξˉ\sum_s u_s(p)_{\xi} \bar{u}_s(p)_{\bar{\xi}}=(\not p+m)_{\xi \bar{\xi}}, \quad \quad \sum_s v_s(p)_{\xi} \bar{v}_s(p)_{\bar{\xi}}=(\not p-m)_{\xi \bar{\xi}}
  4. Gauge field

    • Fourier Transformed OP
      Aμ(x)=d3k(2π)312k0[eikxaμ(k)+eikxaμ(k)]A_\mu(x)=\int \frac{\mathrm{d}^3 k}{(2 \pi)^3} \frac{1}{\sqrt{2 k^0}}\left[e^{-\mathrm{i} k x} a_\mu(\mathbf{k})+e^{\mathrm{i} k x} a_\mu^{\dagger}(\mathbf{k})\right]
    • Field OP that satisfies the canonical commutation relation
      Aμ(x)=d3k(2π)312k0λ=03[αλ(k)εμλ(k)eikx+αλ(k)εμλ(k)eikx]A_\mu(x)=\int \frac{\mathrm{d}^3 k}{(2 \pi)^3} \frac{1}{\sqrt{2 k^0}} \sum_{\lambda=0}^3\left[\alpha_\lambda(\mathbf{k}) \varepsilon_\mu^\lambda(k) e^{-\mathrm{i} k x}+\alpha_\lambda^{\dagger}(\mathbf{k}) \varepsilon_\mu^{\lambda^*}(k) e^{\mathrm{i} k x}\right]
    • Relation
      • [aμ(k),av(k)]=ημν(2π)3δ(kk)[aμ(k),av(k)]=0=[aμ(k),av(k)]\begin{aligned} & \left[a_\mu(\mathbf{k}), a_v^{\dagger}\left(\mathbf{k}^{\prime}\right)\right]=-\eta_{\mu \nu}(2 \pi)^3 \delta\left(\mathbf{k}-\mathbf{k}^{\prime}\right) \\ & \left[a_\mu(\mathbf{k}), a_v\left(\mathbf{k}^{\prime}\right)\right]=0=\left[a_\mu^{\dagger}(\mathbf{k}), a_v^{\dagger}\left(\mathbf{k}^{\prime}\right)\right]\end{aligned}
      • [αi(k),αj(k)]=δij(2π)3δ(kk)[α+(k),α(k)]=(2π)3δ(kk)[α±(k),α±()(k)]=0=[α±(k),αi()(k)]\begin{aligned}&{\left[\alpha_i(\mathbf{k}), \alpha_j^{\dagger}\left(\mathbf{k}^{\prime}\right)\right]=\delta_{i j}(2 \pi)^3 \delta\left(\mathbf{k}-\mathbf{k}^{\prime}\right)}\\&{\left[\alpha_{+}(\mathbf{k}), \alpha_{-}^{\dagger}\left(\mathbf{k}^{\prime}\right)\right]=-(2 \pi)^3 \delta\left(\mathbf{k}-\mathbf{k}^{\prime}\right)} \\ &{\left[\alpha_{\pm}(\mathbf{k}), \alpha_{\pm}^{(\dagger)}\left(\mathbf{k}^{\prime}\right)\right]=0=\left[\alpha_{\pm}(\mathbf{k}), \alpha_i^{(\dagger)}\left(\mathbf{k}^{\prime}\right)\right]}\end{aligned}

What are the quantisation relations for scalar/spinor fields and ladder operators?

See above

What is a symmetry of a Lagrangian?

A symmetry of a Lagrangian is a transformation of the Lagrangian that leaves the EoM invariant.

What is the statement of the Noether theorem?

Continuous symmetries of the action lead to a conserved current density and a conserved charge
-- Lecture note from Jan Pawlowski

Conservation in the certain part of the space-time symmetries with invariance in translation, rotation and boost.

How to calculate the Noether current and charge?

current: jrμ:=L(μϕi)ΔrϕiJrμ, with μjrμ=0j_r^\mu:=\frac{\partial \mathcal{L}}{\partial\left(\partial_\mu \phi_i\right)} \Delta_r \phi_i-J_r^\mu, \quad \text { with } \quad \partial_\mu j_r^\mu=0

charge: Qr(t):=d3xj0(t,x), with tQr(t)=0Q_r(t):=\int \mathrm{d}^3 x j^0(t, \mathbf{x}), \quad \text { with } \quad \partial_t Q_r(t)=0

e.g. for Energy-momentum tensor

current: Tvμ:=jμv=vϕL(μϕ)ημvL with μTμv=0T_v^\mu:=j^\mu{ }_v=\partial_v \phi \frac{\partial \mathcal{L}}{\partial\left(\partial_\mu \phi\right)}-\eta^\mu{ }_v \mathcal{L} \quad \text { with } \quad \partial_\mu T^{\mu v}=0

charge: Pμ=d3xT0μP^\mu=\int \mathrm{d}^3 x T^{0 \mu}

What is the Klein-Gordon equations?

(2+m2)ϕ(x)=0\left(\partial^2+m^2\right) \phi(x)=0

\toEoM of the real scalar field

What is the Dirac equation?

(im)ψ=0(\mathrm{i} \cancel{\partial}-m) \psi=0

What is the Clifford algebra?

{γμ,γν}=2ημν\left\{\gamma^\mu, \gamma^\nu\right\}=2 \eta^{\mu \nu}

What is time ordering and normal ordering?

Time ordering: T(ϕ(x1)ϕ(x2))={ϕ(x1)ϕ(x2)x10>x20ϕ(x2)ϕ(x1)x20<x10\begin{aligned} & T\left(\phi\left(x_1\right) \phi\left(x_2\right)\right) = \begin{cases}\phi\left(x_1\right) \phi\left(x_2\right) & x_1^0>x_2^0 \\ \phi\left(x_2\right) \phi\left(x_1\right) & x_2^0<x_1^0\end{cases} \end{aligned}, based on ordering the early and late OPs

Normal ordering: :a(p1)a(p2):=a(p2)a(p1): a\left(\mathbf{p}_{\mathbf{1}}\right) a^{\dagger}\left(\mathbf{p}_{\mathbf{2}}\right):=a^{\dagger}\left(\mathbf{p}_{\mathbf{2}}\right) a\left(\mathbf{p}_{\mathbf{1}}\right), all annihilation operators in a product of creation and annihilation operators are pulled to the right, expectation value 0 at vacuum state.

What is the time evolution operator?

U(t,t0)=e(iH(t0t))U(t,t_0)=e^(\mathrm{i}H(t_0-t)) and f(t)=eiHtf(t)|f(t)\rangle=e^{-\mathrm{i}Ht}|f(t)\rangle

we can see that

itU(t,t0)=Hint(t)U(t,t0)i \partial_t U\left(t, t_0\right)=H_{\mathrm{int}}(t) U\left(t, t_0\right)

and with integration and Taylor expansion

U(t,t0)=1it0tdt1Hint(t1)+(i)2t0tt0t1dt1dt2Hint(t1)Hint(t2)+=T{exp[it0tdtHint(t)]}U\left(t, t_0\right)=\mathbf{1}-i \int_{t_0}^t d t_1 H_{\mathrm{int}}\left(t_1\right)+(-i)^2 \int_{t_0}^t \int_{t_0}^{t_1} d t_1 d t_2 H_{\mathrm{int}}\left(t_1\right) H_{\mathrm{int}}\left(t_2\right)+\ldots=T\left\{\exp \left[-i \int_{t_0}^t d t^{\prime} H_{\mathrm{int}}\left(t^{\prime}\right)\right]\right\}

What is the statement of the Wick theorem?

Tϕ(x1)ϕ(xn)=:ϕ(x1)ϕ(xn)+T \phi\left(x_1\right) \cdots \phi\left(x_n\right)=: \phi\left(x_1\right) \cdots \phi\left(x_n\right)+ all contractions :

or

T(ϕ1ϕ2)=N(ϕ1ϕ2)+[ϕ,ϕ+]θ(x10x20)+[ϕ,ϕ+]θ(x20x10)T\left(\phi_1\phi_2\right)=N\left(\phi_1 \phi_2\right)+[\phi_-,\phi_+]\theta(x_1^0-x_2^0)+[\phi_-,\phi_+]\theta(x_2^0-x_1^0)

the transformation of time ordered and normal ordered results.

What is a cross section?

Collision events rate on effective area A\mathcal{A}

σ=Nevents (NBNA)/A\sigma=\frac{N_{\text {events }}}{\left(N_B \cdot N_A\right) / \mathcal{A}}

N-particle cross section

dσ=i=1nd3pi(2π)312ωpid2bp1pnSib2\mathrm{d} \sigma=\prod_{i=1}^n \frac{\mathrm{d}^3 p_i}{(2 \pi)^3} \frac{1}{2 \omega_{\mathbf{p}_{\mathbf{i}}}} \int \mathrm{d}^2 b\left|\left\langle\mathbf{p}_{\mathbf{1}} \cdots \mathbf{p}_{\mathbf{n}}|S| i_{\mathbf{b}}\right\rangle\right|^2

How to calculate a scattering amplitude?

Feynman rules and LSZ formalism treatment on time ordered products of fields.

What is a Feynman diagram?

Diagrammatic rules for interaction vertices and propagators.

What is the LSZ formalism

the LSZ formalism includes the LSZ reduction formula that provides an explicit way of expressing physical S matrix elements(i.e.scattering amplitudes)in terms of T-ordered correlation functions of an interacting field within a quantum field theory to all orders in perturbation theory. Thus, the goal is to express the SS matrix elements in terms of asymptotic free fields instead of the unknown interacting field. Therefore, given the Lagrangian of some quantum field theory, it leads to predictions of measurable quantities.
-- From Tommy Ohlsson - Relativistic Quantum Physics, Cambridge University Press (2011)

\to the reduction formular

p1pnSk1kmon-shell =i=1n d4xieipixij=1m d4yieikjyji=1n(xi2+m2)j=1m(yi2+m2)×[iZ1/2]n+mTϕ(x1)ϕ(xn)ϕ(y1)ϕ(yn)\left.\left\langle\mathbf{p}_1 \cdots \mathbf{p}_{\mathbf{n}}|S| \mathbf{k}_{\mathbf{1}} \cdots \mathbf{k}_{\mathbf{m}}\right\rangle\right|_{\text {on-shell }}=\int \prod_{i=1}^n \mathrm{~d}^4 x_i e^{\mathrm{i} p_i x_i} \prod_{j=1}^m \mathrm{~d}^4 y_i e^{-\mathrm{i} k_j y_j} \prod_{i=1}^n\left(\partial_{x_i}^2+m^2\right) \prod_{j=1}^m\left(\partial_{y_i}^2+m^2\right) \times\left[\frac{i}{Z^{1 / 2}}\right]^{n+m}\left\langle T \phi\left(x_1\right) \cdots \phi\left(x_n\right) \phi\left(y_1\right) \cdots \phi\left(y_n\right)\right\rangle

Only the amputated connected parts.

What is a gauge symmetry?

ψ(x)eieα(x)ψ(x)\psi(x) \rightarrow e^{\mathrm{i} e \alpha(x)} \psi(x)

\to The invariance of local rotation

How do fields transform under a gauge symmetry?

ϕ(x)eieα(x)ϕ(x)AμAμ+μα\phi(x) \rightarrow e^{\mathrm{i} e \alpha(x)} \phi(x) \quad A_\mu \rightarrow A_\mu+\partial_\mu \alpha

How do ϕ0\phi_0, ZZ, ϕ\phi scale under μ\mu and Λ\Lambda?

Re-parameterize ϕ0=Zϕ1/2ϕ\phi_0=Z_\phi^{1 / 2} \phi

We fix ZZ to a certain momentum scale p2=μ2p^2=\mu^2

we have μd{Phy. Ob}dμ=0\mu \frac{d\{\text{Phy. Ob}\}}{d \mu} =0

Λ\Lambda is the cutoff scale with the propagator wick rotated.

General scheme:

Procedure:

  1. Perturbation at ϕ\phi
  2. Re-parameterize g,m,ϕg,m,\phi
  3. Feynman rules, add counter term and remove the divergence
  4. Two method
    1. Λ\Lambda cutoff and get renormalization condition
      • [Tϕϕ(p)]p2=m21=0ip2[Tϕϕ(p)]p2=m21=1\begin{array}{r}{[\langle T \phi \phi\rangle(p)]_{p^2=m^2}^{-1}=0} \\ \mathrm{i} \frac{\partial}{\partial p^2}[\langle T \phi \phi\rangle(p)]_{p^2=m^2}^{-1}=1\end{array}
      • i[Tϕϕ(pi)]1Tϕ(p1)ϕ(p4)s=t=u=m2=iλ\left.\prod_i\left[\langle T \phi \phi\rangle\left(p_i\right)\right]^{-1} \cdot\left\langle T \phi\left(p_1\right) \cdots \phi\left(p_4\right)\right\rangle\right|_{s=t=u=m^2}=-\mathrm{i} \lambda
    2. Dimensional regularization at d=42ϵd=4−2\epsilon and remove the parameter μ\mu to get renormalization group equation.

标签: QFT

版权所有:《豪言亂語
文章标题:《QFT I topics
除非注明,文章均为 《豪言亂語》 原创
转载请注明本文短网址:https://article.benhaotang.cn/2022f/33.html  [生成短网址]