1. why QFT

1.1 classical field theory

principle of locality

Newtonian Mechanic: action of a distance

Faraday (1830): All points in space participate in the physical process; effects propagate from point to neighboring points

Field: a mathmatical device (vehicle) that the principle of locality is at work

• associated with each point in the space
• a dynamical variable

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ex1:

• electric field: ${\vec{E}}_{\vec{x}}(t)\equiv \vec{E}(\vec{x},t)$ ($\vec{x}$ is just a lebel)
• magnetc field: ${\vec{B}}_{\vec{x}}(t)$
• Maxwell eqn:
  • containts $\vec{E}$ and $\vec{B}$ depends at same position and time (is cocal)

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whether local:

• at same pt
• finite # of deriative

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ex2: Einstein’s gravaty

tensor: $g_{\mu \nu }(\vec{x},t)$ (metric)

• $\mu ,\nu =0,1,2,3$
• Euclidean metric: $\mathrm{d}{s}^2=\mathrm{d}{t}^2+\mathrm{d}{x}^2+\cdots$
  • $g=\mathrm{diag}(1,1,1,1)$
• Lorentz metric: $\mathrm{d}{s}^2=-\mathrm{d}{t}^2+\mathrm{d}{x}^2+\cdots$
  • $g=\mathrm{diag}(-1,1,1,1)$

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types of fields

1. scalar field: $\varphi (\vec{x},t)$
   • temperature: $T(\vec{x},t)$
2. vector field:
   • $\vec{E}(\vec{x},t),\vec{B}(\vec{x},t)$
   • vector potential: $A_{\mu }(X^{\nu })$ (A-vector)
3. tensor field:
   • metric: $g_{\mu \nu }(X^{\lambda })$
4. spinner:
   • ${\Phi }_{\alpha }(\vec{x},t)$

From CFT to QFT

1. classical mechanics:
   • action: $\begin{array}{r}S=\int \mathrm{d}tL(x,\dot{x},t)\end{array}$
   • canonical momentum: $p=\frac{\partial L}{\partial \dot{x}}$
   • Hamiltonian: $H=p\dot{x}-L$
   • Equation of motion (EOM): extremize $S[x(t)]$
     • variation: $\delta S=0$ $\Rightarrow$ trajectory
2. classical field theory:
   • $\begin{array}{r}L=\int {\mathrm{d}}^{3}x\mathcal{L}({\varphi }_a,{\partial }_{\mu }{\varphi }_a)\end{array}$ (local)
     • is significantly constrained
     • $\mathcal{L}$ : lagrange density
     • ${\varphi }_a$ is a general field
   • remarks:
     • not allow $\begin{array}{r}L\sim \int {\mathrm{d}}^{3}x{\mathrm{d}}^{3}y(\vec{x},t)\varphi (\vec{y},t)\end{array}$ ($\vec{x}$ and $\vec{y}$ is different pt)
     • only allow one derivative ${\partial }_{\mu }\varphi$ $\Rightarrow$ EOM contains two time deriative
   • canonical momentum desnsity: ${\Pi }_a(\vec{x},t)=\frac{\partial \mathcal{L}}{\partial {\dot{\varphi }}_a(\vec{x},t)}$ or $\frac{\partial \mathcal{L}}{\partial ({\partial }_t{\varphi }_a)}$
   • Hamitonian density: $\mathcal{H}(\vec{x},t)={\Pi }_a{\dot{\varphi }}_a-\mathcal{L}$
     • Hamitonian: $\begin{array}{r}H=\int {\mathrm{d}}^{3}x\mathcal{H}(\vec{x},t)\end{array}$
   • Equation of motion: $\delta S=0$
     • action: $\begin{array}{r}S=\int \mathrm{d}tL=\int \mathrm{d}t{\mathrm{d}}^{3}x\mathcal{L}({\varphi }_a,{\partial }_{\mu }{\varphi }_a)=\int {\mathrm{d}}^{4}x\mathcal{L}({\varphi }_a,{\partial }_{\mu }{\varphi }_a)\end{array}$
     • $\begin{array}{rl}0=\delta S=& \int {\mathrm{d}}^{4}x\left[\frac{\partial \mathcal{L}}{\partial {\varphi }_a}\delta {\varphi }_a+\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}\delta ({\partial }_{\mu }{\varphi }_a)\right]\\ =& \int {\mathrm{d}}^{4}x\left[\frac{\partial \mathcal{L}}{\partial {\varphi }_a}\delta {\varphi }_a+\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}\delta ({\partial }_{\mu }{\varphi }_a)\right]\\ =& \int {\mathrm{d}}^{4}x\left[\frac{\partial \mathcal{L}}{\partial {\varphi }_a}\delta {\varphi }_a-{\partial }_{\mu }\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}\delta {\varphi }_a\right]+\cancel{\mathrm{boundaryterm}}\\ \Rightarrow 0=& \frac{\partial \mathcal{L}}{\partial {\varphi }_a}-{\partial }_{\mu }\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}\end{array}$
   • ex1: maxwell equation
     • dynamic variable: $A_{\mu }(x)$
     • field of Strength $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$
     • $\begin{array}{r}S=\int {\mathrm{d}}^{4}x\left(\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }\right)\end{array}$
   • ex2: Einstein-Hilbert action
     • $\begin{array}{r}S=\frac{1}{16\pi G_N}\int {\mathrm{d}}^{4}x\sqrt{-g}R\end{array}$
     • $g$ : metric
     • $R$ : Ricci scalar
   • ex3: scalar field theory
     • real value: $\varphi (x)$
     • $\begin{array}{r}S=\int {\mathrm{d}}^{4}x\left[-\frac{1}{2}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -v(\varphi )\right]\end{array}$
     • translation symmetry $\Rightarrow$ all parameters in $v(x)$ must be constant
     • $\Pi (x)=\frac{\partial \mathcal{L}}{\partial \dot{\varphi }}=\dot{\varphi }(x)$ (${\partial }_{\mu }\varphi {\partial }^{\mu }\varphi =-{\dot{\varphi }}^2+(\nabla \varphi {)}^2$)
     • $\mathcal{H}=\Pi \dot{\varphi }-\mathcal{L}=\frac{{\pi }^2}{2}+\frac{1}{2}(\nabla \varphi {)}^2+v(\varphi )$
     • EOM: ${\partial }^2\varphi -\frac{\partial v}{\partial \varphi }=0$ (${\partial }^2={\partial }_{\mu }{\partial }^{\mu }=-\partial t^2+{\nabla }^2$)
       • simplest case: $v(\varphi )=f\varphi$ ($f$ is a constant from principle of locality)
         • $\Rightarrow$ EOM: ${\partial }^2\varphi =f$ ($f$ is an external force)
       • $v(\varphi )=\frac{1}{2}{m}^2{\varphi }^2$ ($m$ : mass of particle)
         • $\Rightarrow$ EOM: ${\partial }^2\varphi -m^2\varphi =0$ (Klein-Gorden eqn)

symmetry and conversation

• invariant

  • translationally invariant (4)
  • lorentz invariant
    • rotationally invariant (3)
    • boosts (3)

• symmetry:

  1. transformation: ${\varphi }_a\to {\varphi }_a^{\prime }$ which leaves the action invariant
     • 4 continuous parameters
     • ex: $x^{\mu }\to x^{{\mu }^{\prime }}+a^{\mu }$ where $a^{\mu }$ is a const vector $\Rightarrow$ $\varphi (x)={\varphi }^{\prime }(x^{\prime })$
       • a scalar field transfroms under a general spacetime transformation
  2. lorentz transformation: $x^{\mu }\to x^{{\mu }^{\prime }}={\Lambda }_{\nu }^{\mu }{x}^{\nu }$
     • 6 continuous parameters
     • ${\Lambda }_{\nu }^{\mu }$ : cosntant lorentz transformation

types of symmetry:

• continuous symmetry: Both 1 and 2 are continuous symmetries

• discrete symmetry: $Z_2$ : parity、reflection

• global symmetries: transformation parameters are spacetime independent

• local symmetries: ……dependent

$$\begin{array}{rl}\mathrm{symmetry}& \Rightarrow \mathrm{conservation}\\ \mathrm{time}\mathrm{translation}& \to \mathrm{enegy}\mathrm{conservation}\\ \mathrm{spatial}\mathrm{translation}& \to \mathrm{momentum}\mathrm{conservation}\\ \mathrm{rotational}\mathrm{symmetry}& \to \mathrm{angular}\mathrm{momentum}\mathrm{conservation}\end{array}$$

Noether Theorem：any continuous global symmetry leads to conservation

Proof: for any continuous symmetry, it has an infinitesimal form: ${\varphi }_a\to {\varphi }_a^{\prime }={\varphi }_a+ϵf_a({\varphi }_b,{\partial }_{\mu }{\varphi }_b)$

• $ϵ$ : infinitesimal traansformation parameters
• $\hat{\delta }{\varphi }_a=ϵf_a$
• by definition: $\hat{\delta }S=0$
  • $\hat{\delta }\mathcal{L}=ϵ{\partial }_{\mu }{K}^{\mu }$ for some $K^{\mu }$ (which can be zero)

$$\begin{array}{rl}\hat{\delta }\mathcal{L}=& \frac{\partial \mathcal{L}}{\partial {\varphi }_a}\hat{\delta }{\varphi }_a-{\partial }_{\mu }\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}\hat{\delta }{\varphi }_a\\ =& {\partial }_{\mu }\left(\right)\end{array}$$

recall

1. Schrodinger picture: $\Psi (x,t)$ wave function
   1. $x$ : eigenvalue of $\hat{x}$
   2. $\Psi$ change with time; but $\hat{x}$ , $\hat{p}$ not
2. Heisenberg picture: $|\Psi ⟩$ wave function
   1. $\hat{x}(t)$ , $\hat{p}(t)$ change with time; but $|\Psi ⟩$ not
   2. prefer this in QFT

Special relativity + QM → QFT Relativistic QM (wrong)

$\hslash =1$ , $c=1$

non-relativistic QM: $E=\frac{{\hat{p}}^2}{2m}$

• $E=i{\partial }_t$ , $\vec{p}=-i\nabla$
• Schrodinger eqn: $i\frac{\partial }{\partial t}\Psi (\vec{x},t)=-\frac{1}{2m}{\nabla }^2\Psi (\vec{x},t)$

relativistic dispertion relation: $E^2={\vec{p}}^2+m^2$

• $-{\partial }_t^2\Psi =-{\nabla }^2\Psi +m^2\Psi$ $\Rightarrow$ ${\partial }_{\mu }{\partial }^{\mu }\Psi -m^2\Psi =0$ (4)
  • note: this is a wave function for a single particle at location $\vec{x}$ at time $t$, nothing relates to field

Some dificulties of intepreting: (4) as a wave eqn for a relativistic particle

1. no quantity can be used as probability density condition:
   1. non-negative:
   2. conserved
2. $E=\pm \sqrt{{\vec{p}}^2+m^2}$ leads to instability

fundamental reason: QM: finite number $\Psi (x_1,x_2,\cdots ,x_n,t)$

• fundanmental asymmetry between $t$ and $\vec{x}$
• at most, it’s an approximate description in situation where no particle creation/annihilation

QFT: Simplest scalar field theoty

• $S=-\frac{1}{2}\int {\mathrm{d}}^4({\partial }_{\mu }\varphi {\partial }^{\mu }\varphi +m^2{\varphi }^2)$ $\Rightarrow$ ${\partial }^2\varphi -m^2\varphi =0$
  • $\varphi (\vec{x},t)$ : classical field

$x_n-x_n^0={\eta }_n$

• $x_n^0$ is the equilibrium position
• $L=T-V={\sum }_n\left[\frac{1}{2}\mu {\dot{\eta }}_n^2-\frac{\lambda }{2}({\eta }_{n+1}-{\eta }_n{)}^2-\frac{\sigma }{2}{\eta }_n^2\right]$
  • $\mu$ : mass of particle
• $a\to 0$ chain become a continnum of particles
  • ${\eta }_n(t)\to \eta (x,t)$
  • $a{\sum }_n\to \int \mathrm{d}x$
  • $\begin{array}{rl}L& =a\left[\frac{1}{2}\frac{\mu }{a}{\dot{\eta }}_n^2-\frac{a\lambda }{2}\frac{({\eta }_{n+1}-{\eta }_n{)}^2}{a^2}-\frac{\sigma }{2}{\eta }_n^{2a}\right]\\ & =\int \mathrm{d}x\left(\frac{1}{2}\tilde{u}{\dot{\eta }}^2-\frac{\tilde{\lambda }}{2}({\partial }_x\eta {)}^2-\frac{\tilde{\sigma }}{2}{\eta }^2\right)\\ & =\tilde{u}\int \mathrm{d}x\left[\frac{1}{2}{\dot{\eta }}^2-\frac{v^2}{2}({\partial }_x\eta {)}^2-\frac{1}{2}{m}^2{\eta }^2\right]\end{array}$
  • $-\frac{v^2}{2}({\partial }_x\eta {)}^2-\frac{1}{2}{m}^2{\eta }^2=-\frac{1}{2}{\partial }_{\mu }\eta {\partial }^{\mu }\eta -\frac{m^2}{2}{\eta }^2$
  • Field theory as a limit of discrete systems

Summary: all paths lead to QFT

1. quantum dynamics of classical field $\vec{E}$, $\vec{B}$, gravity
2. unify SR + QM
3. large distance discription of discrete system