2. free field theory

Quantization of Harmonic Oscillator in the Heisenberg Picture

$\ddot{x}+{\omega }^{2}x=0$ classical solution: $x(t)=\frac{1}{\sqrt{2}}(a{e}^{-i\omega t}+a^{\ast }{e}^{-i\omega t})$

${\partial }_t^2\hat{x}+{\omega }^2\hat{x}=0$ quantum solution: $\hat{x}(t)=\frac{1}{\sqrt{2}}(\hat{a}{e}^{-i\omega t}+{\hat{a}}^{†}{e}^{i\omega t})$ canonical momentum: $\hat{p}={\partial }_t\hat{x}$ canonical quantization condition: $[\hat{x}(t),\hat{p}(t)]=i$ $\Rightarrow$ $[\hat{a},{\hat{a}}^{†}]=1$ Hilbert space: $\hat{a}|0⟩=0$ , $|\hat{n}⟩=\frac{1}{\sqrt{n!}}({\hat{a}}^{†}{)}^n|0⟩$

summary: Steps of Quantization

1. classical eom → quantum operator eqn
2. find most general solution to classical eom
3. promote integration consts ($a,a^{\ast }$) to constant quantum operators $\Rightarrow$ full time evolution at quantum level
4. impose canonical quantization conditions
5. constant operators, which can be used to generalize Hilbert space

For free scalar field theory:

Fourier transformation: $\varphi (x)=e^{-iEt+i\vec{k}\cdot \vec{x}}$ (plane wave) $E^2={\vec{k}}^2+m^2\equiv {\omega }_k^2$ $\Rightarrow$ $u_{\vec{k}}=e^{-i{\omega }_{\vec{k}}t+i\vec{k}\cdot \vec{x}}$ (positive energy soln) and $u_{\vec{k}}^{\ast }=e^{i{\omega }_{\vec{k}}t-i\vec{k}\cdot \vec{x}}$ (negative energy soln) $\varphi (x)=\int \frac{{\mathrm{d}}^{3}k}{(2\pi {)}^3}\frac{1}{\sqrt{2{\omega }_{\vec{k}}}}(a_{\vec{k}}{u}_{\vec{k}}+a_{\vec{k}}^{\ast }{u}_{\vec{k}}^{\ast })$ $\Rightarrow$ $\hat{\varphi }$ with ${\hat{a}}_{\vec{k}},{\hat{a}}_{\vec{k}}^{\ast }$ , $\hat{\Pi }={\partial }_t\hat{\varphi }$ canonical commutation relation: $[\hat{\varphi }(x),\hat{\Pi }(x)]=?$ Hints: for multiple particle:

• $[{\hat{x}}_a(t),{\hat{p}}_b(t)]=i{\delta }_{ab}$ (kronecker delta)
• $[{\hat{x}}_a,{\hat{x}}_b]=0$ , $[{\hat{p}}_a,{\hat{p}}_b]=0$
• $\Rightarrow$
  • $[\hat{\varphi }(t,\vec{x}),\hat{\varphi }(t,{\vec{x}}^{\prime })]=0$
  • $[\hat{\Pi }(t,\vec{x}),\hat{\Pi }(t,{\vec{x}}^{\prime })]=0$
  • $[\hat{\varphi }(t,\vec{x}),\hat{\pi }(t,{\vec{x}}^{\prime })]=i{\delta }^{(3)}(\vec{x}-{\vec{x}}^{\prime })$ (dirac delta)
  • $\Rightarrow$
    • $[{\hat{a}}_{\vec{k}},{\hat{a}}_{{\vec{k}}^{\prime }}]=[{\hat{a}}_{\vec{k}}^{†},{\hat{a}}_{{\vec{k}}^{\prime }}^{†}]=0$
    • $[{\hat{a}}_{\vec{k}},{\hat{a}}_{{\vec{k}}^{\prime }}^{†}]=(2\pi {)}^3{\delta }_{(k)}^{(3)}(\vec{k}-{\vec{k}}^{\prime })$

$\hat{H}=\int {\mathrm{d}}^{3}x\mathcal{H}=\cdots =\frac{1}{2}\int \frac{{\mathrm{d}}^{3}k}{(2\pi {)}^3}{\omega }_{\vec{k}}({\hat{a}}_{\vec{k}}^{†}{\hat{a}}_{\vec{k}}+{\hat{a}}_{\vec{k}}{\hat{a}}_{\vec{k}}^{†})=\frac{1}{2}\int \frac{{\mathrm{d}}^{3}k}{(2\pi {)}^3}{\omega }_{\vec{k}}\left({\hat{a}}_{\vec{k}}^{†}{\hat{a}}_{\vec{k}}+\frac{1}{2}[{\hat{a}}_{\vec{k}},{\hat{a}}_{\vec{k}}^{†}]\right)$ zero-point energy: $E_0=\int \frac{{\mathrm{d}}^{3}k}{(2\pi {)}^3}{\omega }_{\vec{k}}\frac{1}{2}[{\hat{a}}_{\vec{k}},{\hat{a}}_{\vec{k}}^{†}]=\int \frac{{\mathrm{d}}^{3}k}{(2\pi {)}^3}{\omega }_{\vec{k}}\frac{1}{2}(2\pi {)}^3{\delta }^{(3)}(0)$ , which is divergent

Remarks:

1. no time-dependence since $H$ is conserved quantity
2. QFT of $\varphi$ reduces to a system of continum of h.o. labeled by $\vec{k}$ with frequency $w_{\vec{k}}=\sqrt{k^2+m^2}$

The other conserved quantities:

1. spatial translation: $p^i=-\int {\mathrm{d}}^{3}x\hat{\Pi }(t,\vec{x}){\partial }_i\hat{\varphi }(t,\vec{x})=\int \frac{{\mathrm{d}}^3\vec{k}}{(2\pi {)}^3}\vec{k}{\hat{a}}_{\vec{k}}^{†}{\hat{a}}_{\vec{k}}$
2. Lorentz symmetry: $M^{\mu \nu }$

$E_0=\int \frac{{\mathrm{d}}^{3}k}{(2\pi {)}^3}{\omega }_{\vec{k}}\frac{1}{2}(2\pi {)}^3{\delta }^{(3)}(0)E_0=\frac{1}{2}(2\pi {)}^3{\delta }^{(3)}(0)\int \frac{{\mathrm{d}}^{3}k}{(2\pi {)}^3}{\omega }_{\vec{k}}$

$(2\pi {)}^3{\delta }^{(3)}(\vec{k})=\int {\mathrm{d}}^3\vec{x}{e}^{i\vec{k}\cdot \vec{x}}$

$(2\pi {)}^3{\delta }^{(3)}(0)=\int {\mathrm{d}}^3\vec{x}=\text{Volume}$

$E_0=\mathrm{V}{ϵ}_0$

${ϵ}_0=\int \frac{{\mathrm{d}}^{3}k}{(2\pi {)}^3}\sqrt{k^2+m^2}$

how to deal with it?

${ϵ}_0={\int }^{\Lambda }\frac{{\mathrm{d}}^{3}k}{(2\pi {)}^3}\sqrt{k^2+m^2}\left(\left|\vec{k}\right|\ll \Lambda \right)$

• $\Lambda$ : cut-off
• large overall number → just ignore $E_0$ (current purples)