# 物理代写|弦论代写string theory代考|MAST90069

## 物理代写|弦论代写string theory代考|Field Quantisation and Quantum Field Theory

As Exercise 3.1.1 has shown, the Klein-Gordon equation can be viewed as the implementation of the constraint $p^2+m^2=0$ which selects the physical states in the Fock space of a relativistic particle. With this interpretation the Klein-Gordon equation is analogous to the Schrödinger equation, in the sense that it is a condition imposed on quantum states. There is an alternative interpretation of the KleinGordon equation as a classical field equation, analogous to the Maxwell equations. We will now quantise the classical complex Klein-Gordon field and compare the result to the quantisation of a relativistic particle. The Klein-Gordon equation
$$\left(-\square+m^2\right) \Phi(x)=0$$
follows from the action
$$S[\Phi]=\int d^D x\left(-\partial_\mu \Phi \partial^\mu \Phi^-m^2 \Phi \Phi^\right) .$$
The canonical momentum of this field theory is
$$\Pi(x)=\frac{\partial L}{\partial \partial_0 \Phi}=\partial_0 \Phi^* .$$
The Fourier representation of the general solution can be parametrised in the following form:
$$\Phi(x)=\int d^D k\left(\theta\left(k^0\right) \delta\left(k^2+m^2\right) \tilde{\phi}{+}(k)+\theta\left(-k^0\right) \delta\left(k^2+m^2\right) \tilde{\phi}{-}^(k)\right) .$$ Here we used the step functions $\theta\left(\pm k^0\right)$ to separate the two components of the hyperboloid $k^2+m^2=0$. The $\delta$-function can be used to carry out the $k^0$-integration: $$\Phi(x)=\frac{1}{(2 \pi)^{(D-1) / 2}} \int \frac{d^{D-1} \vec{k}}{2 \omega_{\vec{k}}}\left(\phi_{+}(k) e^{i k x}+\phi_{-}^(k) e^{-i k x}\right){k^0=\omega{\vec{k}}} .$$
Here $\phi_{\pm}(k)$ are rescaled versions of $\tilde{\phi}{\pm}(k)$. We now quantise the complex KleinGordon field by declaring $\Phi(x)$ to be an operator ${ }^2$ satisfying the canonical commutation relation $$[\Phi(x), \Pi(y)]{x^0=y^0}=i \delta^{D-1}(\vec{x}-\vec{y}) .$$
Since $\Phi(x)$ depends on time $x^0$, we are in the Heisenberg picture, where operators depend on time while states are time-independent. We only need to specify the commutator at equal times, because the commutator at other times is fixed by time evolution. The spatial coordinate is treated as a continuous index labelling degrees of freedom located at different points. The operator $\Phi(x)$ can be represented as a Fourier integral, with the Fourier coefficients $\phi_{\pm}(k)$ promoted to operators. Complex conjugated quantities are now interpreted has Hermitian conjugate operators, and denoted $\Phi^{\dagger}(x), \phi_{\pm}^{\dagger}(k)$. The Fourier modes satisfy the relations: ${ }^3$

## 物理代写|弦论代写string theory代考|Quantised Relativistic Strings

In the final section of this chapter, we will outline the problem of quantising relativistic strings, and thus motivate why we need to develop certain tools in Part II before being able to formulate a quantum theory of strings in Part III. The classical solutions found in the previous chapter have shown that the degrees of freedom of a free relativistic string in Minkowski space combine those of a relativistic particle with an infinite set of harmonic oscillators. This gives us a clear idea of how the Hilbert space should look like, and we could postulate canonical relations for $x^\mu, p^v$ and the Fourier coefficients $\alpha_m^\mu, \tilde{\alpha}m^\mu$. To be more systematic we start by imposing canonical commutation on the string coordinates $X^\mu\left(\sigma^0, \sigma^1\right)$ and the canonical momenta $\Pi^v\left(\sigma^0, \sigma^1\right)$. We work with the Polyakov action in the conformal gauge, and can interprete $X^\mu$ either as embedding coordinates for a string in spacetime M, or as a set of scalar fields on the world-sheet $\Sigma$. In the conformal gauge $\Pi^\mu=T \dot{X}^\mu$. Both $X^\mu$ and $\Pi^v$ are time-dependent operators, and thus we are in the Heisenberg picture of quantum mechanics. Canonical commutators are imposed at equal world-sheet times, and $\sigma^1$ is treated as a continuous index, similar to the discrete index $\mu$. For concreteness we consider periodic boundary conditions. Then the canonical commutation relations are: $$\left[X^\mu\left(\sigma^0, \sigma^1\right), \Pi^v\left(\sigma^{\prime 0}, \sigma^{\prime 1}\right]{\sigma^0=\sigma^{\prime 0}}=i \eta^{\mu v} \delta_\pi\left(\sigma^1-\sigma^{\prime 1}\right),\right.$$
where
$$\delta_\pi\left(\sigma^1\right)=\frac{1}{\pi} \sum_{k=-\infty}^{\infty} e^{-2 i k \sigma^1}=\delta_\pi\left(\sigma^1+\pi\right)$$
is the periodic $\delta$-function with period $\pi$ (see Appendix C). One can view this as the canonical commutation relations of a two-dimensional field theory on a world-sheet $\Sigma$, which has the topology of a cylinder.

## 物理代写|弦论代写string theory代考|Field Quantisation and Quantum Field Theory

$$\left(-\square+m^2\right) \Phi(x)=0$$

$$\Pi(x)=\frac{\partial L}{\partial \partial_0 \Phi}=\partial_0 \Phi^* .$$

$$\Phi(x)=\int d^D k\left(\theta\left(k^0\right) \delta\left(k^2+m^2\right) \tilde{\phi}+(k)+\theta\left(-k^0\right) \delta\left(k^2+m^2\right) \tilde{\phi}-(k)\right) .$$

$$\left.\Phi(x)=\frac{1}{(2 \pi)^{(D-1) / 2}} \int \frac{d^{D-1} \vec{k}}{2 \omega_{\vec{k}}}\left(\phi_{+}(k) e^{i k x}+\phi_{-}^{(} k\right) e^{-i k x}\right) k^0=\omega \vec{k}$$

$$[\Phi(x), \Pi(y)] x^0=y^0=i \delta^{D-1}(\vec{x}-\vec{y})$$

## 物理代写|弦论代写string theory代考|Quantised Relativistic Strings

$$\left[X^\mu\left(\sigma^0, \sigma^1\right), \Pi^v\left(\sigma^{\prime 0}, \sigma^{11}\right] \sigma^0=\sigma^{\prime 0}=i \eta^{\mu v} \delta_\pi\left(\sigma^1-\sigma^{11}\right),\right.$$

$$\delta_\pi\left(\sigma^1\right)=\frac{1}{\pi} \sum_{k=-\infty}^{\infty} e^{-2 i k \sigma^1}=\delta_\pi\left(\sigma^1+\pi\right)$$

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