Quantum field theory 1, lecture 10

Quantum states

Let us now address quantum states in a quantum field theory. We start by recalling some notions and concepts from quantum mechanics of non-relativistic particles and will then transfer the different concepts to field theory.

Canonical quantization

A set of harmonic oscillators

We start with a set of mechanical harmonic oscillators. We write their amplitudes as \(\phi_j(t)\) where \(j=1,\ldots, N\) is a set of indices. The classical action is (for units where \(m=1\)) \[S = \int dt L(\phi_j(t), \dot \phi_j(t)) = \int dt \sum_{j=1}^N \left\{ \frac{1}{2} \dot \phi_j(t)^2 - \frac{1}{2}\omega_j^2 \phi_j(t)^2 \right\}.\] With the canonical conjugate momenta, \(\pi_j(t) = \partial L / \partial \dot \phi_j(t) = \dot \phi_j(t)\), we can write the Hamiltonian as \[H = \sum_{j=1}^N \left\{ \frac{1}{2} \pi_j^2(t) + \frac{1}{2} \omega_j^2 \phi_j^2(t) \right\}.\] The classical Poisson brackets are here defined for two functions \(A(\phi_j(t), \pi_j(t))\) and \(B(\phi_j(t), \pi_j(t))\) in phase space, \[\left\{ A, B \right\} = \sum_{j=1}^N \left\{ \frac{\partial A}{\partial \phi_j} \frac{\partial B}{\partial \pi_j} - \frac{\partial A}{\partial \pi_j} \frac{\partial B}{\partial \phi_j} \right\}.\] In particular one has \(\{ \phi_m(t), \pi_n(t)\} = \delta_{mn}\) and \(\{ \phi_m(t), \phi_n(t) \} = \{ \pi_m(t), \pi_n(t) \} = 0\).

Canonical quantization for harmonic oscillators

Canonical quantization works by promoting the oscillator amplitudes \(\phi_j(t)\) and \(\pi_j(t)\) to hermitian operators \(\hat \phi_j(t)\) and \(\hat \pi_j(t)\), and the classical Poisson brackets to commutation relations at equal times, \[{\big [}\hat \phi_m(t), \hat \pi_n(t) {\big ]} = i \hbar \delta_{mn}, \quad\quad\quad [\hat \phi_m(t), \hat \phi_n(t)] = [\hat \pi_m(t), \hat \pi_n(t)] = 0. \label{eq:canonicalCommutationRelationsHOEqualTimes}\] Note that we are working here in the Heisenberg representation of quantum mechanics where the operators are time-dependent.

It is beneficial to introduce the linear combination of operators \[\begin{split} a_j(t) = & \frac{e^{i\omega_j t}}{\sqrt{2\hbar \omega_j}} \left[ \omega_j \hat \phi_j(t) + i \hat \pi_j(t) \right],\\ a_j^\dagger(t) = & \frac{e^{-i\omega_j t}}{\sqrt{2\hbar \omega_j}} \left[ \omega_j \hat \phi_j(t) - i \hat \pi_j(t)\right], \end{split}\] so that one has \[\begin{split} \hat \phi_j(t) = & \sqrt{\frac{\hbar}{2\omega_j}} \left[ e^{-i\omega_j t} a_j(t) + e^{i\omega_j t} a_j^\dagger(t) \right], \\ \hat \pi_j(t) = & \sqrt{\frac{\hbar \omega_j}{2}} \left[ - i e^{-i\omega_j t} a_j(t) + i e^{i\omega_j t} a_j^\dagger(t) \right]. \end{split}\label{eq:phipidecomposition}\] The equal time commutation relations \(\eqref{eq:canonicalCommutationRelationsHOEqualTimes}\) are then equivalent to \[\big[a_m(t), a^\dagger_n(t) \big] = \delta_{mn}, \quad\quad\quad \big[a_m(t), a_n(t)\big] = \big[a^\dagger_m(t), a^\dagger_n(t) \big] = 0.\]

Time evolution

For free oscillators one can write the time evolution equations as \(d \hat \phi_j(t)/dt = \hat \pi_j(t)\) and \(d\hat \pi_j(t) / d t = d^2 \hat \phi_j(t) / dt^2 = -\omega_j^2 \hat \phi_j(t)\). These equations are solved by the ansatz in eq. \(\eqref{eq:phipidecomposition}\) when \(a_j(t) = a_j\) and \(a_j^\dagger(t) = a_j^\dagger\) are independent of time \(t\). We restrict to this case in the following.

Hamiltonian, ground state and excitated state

We can write the Hamiltonian as \[H = \sum_{j=1}^N \frac{1}{2} \hbar \omega_j \left[ a^\dagger_j a_j + a_j a^\dagger_j \right] = \sum_{j=1}^N \hbar \omega_j \left[ a^\dagger_j a_j + \frac{1}{2} \right].\] In the last line we used the commutation relation.

One starts now from a state \(| 0 \rangle\) with the property \(a_j | 0 \rangle = 0\) for all indices \(j\). This state is called the ground state or vacuum state. The energy is given there by zero-point fluctuations \(\hbar w_j/2\) for each mode.

One can also create excited states where the oscillator with index \(j\) has occupation number \(n_j \in \mathbb{N}_0\). Up to a normalization factor it is given by \[| n_1, \ldots, n_N \rangle = (a_1^\dagger)^n_1 \cdots (a_N^\dagger)^{n_N} | 0 \rangle.\]

The non-relativistic complex scalar field

We now discuss canonical quantization for a field theory, specifically the free, non-relativistic scalar field with action \[S[\phi] = \int dt \int d^3 x \left\{ i \hbar \phi^*(t, \mathbf{x}) \partial_t \phi(t, \mathbf{x}) - \frac{\hbar^2}{2m} \boldsymbol \nabla \phi^*(t, \mathbf{x}) \boldsymbol \nabla \phi(t, \mathbf{x}) - V(t, \mathbf{x}) \phi^*(t, \mathbf{x}) \phi(t, \mathbf{x}) \right\}.\] Here the momentum field conjugate to \(\phi(t, \mathbf{x})\) is \(\pi(t, \mathbf{x}) = i \hbar \phi^*(t, \mathbf{x})\). The non-vanishing elementary classical Poisson bracket with the field theoretic definition in eq. \(\eqref{eq:PoissonBracketDefinitionClassicalFieldTheory}\) evaluates here to \[{\big \{} \phi(t, \mathbf{x}), \pi(t, \mathbf{y}) {\big \}} = i\hbar \, {\big \{} \phi(t, \mathbf{x}),\phi^*(t, \mathbf{y}) {\big \}} = \delta^{(3)}(\mathbf{x} - \mathbf{y}).\]

Field quantization or second quantization

Field quantization now promotes \(\phi(t, \mathbf{x})\) to an operator \(\hat \phi(t, \mathbf{x})\) and \(\phi^*(t, \mathbf{x})\) to \(\hat\phi^\dagger(t, \mathbf{x})\) with the commutation relation given by \(i\hbar\) times their classical Poisson bracket, \[{\big [}\hat\phi(t, \mathbf{x}), \hat \phi^\dagger(t, \mathbf{y}) {\big ]} = \delta^{(3)}(\mathbf{x} - \mathbf{y}).\] This can be solved by writing the free Heisenberg field operator as \[\hat \phi(t, \mathbf{x}) = \int \frac{d^3 p}{(2\pi)^3} e^{-i\omega_\mathbf{p}t+ i \mathbf{p} \mathbf{x}} a_\mathbf{p}, \quad\quad\quad \hat \phi^\dagger(t, \mathbf{x}) = \int \frac{d^3 p}{(2\pi)^3} e^{i\omega_\mathbf{p}t- i \mathbf{p} \mathbf{x}} a^\dagger_\mathbf{p},\] with \(w_\mathbf{p}=\mathbf{p}^2/(2m)\) and time-independent \(a_\mathbf{p}\) for the solution to the free evolution equations, and the commutation relations \[{\big [}a_\mathbf{p}, a^\dagger_\mathbf{q} {\big ]} = (2\pi)^3 \delta^{(3)}(\mathbf{p} - \mathbf{q}), \quad\quad\quad {\big [}a_\mathbf{p}, a_\mathbf{q} {\big ]} = {\big [}a^\dagger_\mathbf{p}, a^\dagger_\mathbf{q} {\big ]} = 0.\]

Particles as quantum excitations

The operators \(a_\mathbf{p}\) and \(a_\mathbf{p}^\dagger\) have the same properties as the annihilation and creation operators for the energy levels of the harmonic oscillator. But what they annihilate and create are actually particles! This provides a new possibility to understand many-particle states in quantum mechanics as corresponding excitations of a vacuum state.

Vacuum states

In the present formalism (for non-interacting fields) one can take the vaccum state \(|0 \rangle\) in the field theory to be such that \[a_\mathbf{p} | 0 \rangle = 0,\] for all momenta \(\mathbf{p}\).

Single particle states

One can also construct now states for single particles in a momentum eigenstate by using the creation operator, \[| \mathbf p \rangle \sim a_\mathbf{p}^\dagger | 0 \rangle.\] We discuss the normalization and related issues later on. In a similar way one can construct two-particle states, \[| \mathbf{p}, \mathbf{q} \rangle \sim a_\mathbf{p}^\dagger a_\mathbf{q}^\dagger | 0 \rangle,\] and as a consequence of the commutation relations it is autmotically symmetric, \[| \mathbf{p}, \mathbf{q} \rangle = | \mathbf{q}, \mathbf{p} \rangle.\]

Schrödinger functional representation of quantum states

The algebraic method we used for canonical quantization of harmonic oscillators and the free non-relativistic field works in a similar way for many free or non-interacting quantum field theories. It shows that in a quantum field theory particles can be seen as excitations.

To address more general situations it is often useful to have a concrete representation of the quantum states and operators. We develop this now by appealing to concepts used in quantum mechanics.

We are interested in describing quantum states at a fixed time \(t=0\). In that case the Heisenberg and Schrödinger picture coincide. We use the field theoretic analoge of the position space representation of quantum mechanics to describe quantum states. For a field theory, the Schrödinger wave function will become a Schrödinger functional \(\Psi[\phi]\).

The density matrix

Recall that in the position space representation of quantum mechanics for \(N\) degrees of freedom, such as particle positions, one can represent an arbitrary pure state \(|\Psi\rangle\) at some time \(t\) in terms of a Schrödinger wave function \[\Psi_t(x_1,\cdots, x_N).\] A general mixed state needs to be described by a density matrix or a density operator. For a mixture of states \(|\Psi_j \rangle\) with probability \(p_j\) such that \(\sum_j p_j = 1\), the density operator is formally given by \[\rho_t = \sum_j p_j | \Psi_j\rangle\langle \Psi_j |.\] The concept of a mixed state is needed if one does not know the state with certainty but has only a probailistic description available. Mixed states are also needed if one would like to describe degrees of freedom that are not fully isolated but entangled with other degrees of freedom. This is actually the general situation for the local description of a quantum field theory in some subvolume of space.

Expectation values

From the density operator, one can calculate expectation values at time \(t\) as \[\langle A(t)\rangle = \text{Tr}\left\{\rho_t A \right\} = \sum_j p_j \text{Tr}\left\{|\Psi_j\rangle\langle\Psi_j|A \right\} = \sum_j p_j \langle \Psi_j |A|\Psi_j \rangle.\] Concretely, for the position space representation one would have \[\rho_t(x_1,\ldots, x_N; y_1,\ldots, y_N) = \sum_j p_j \Psi_j(x_1,\ldots, x_N)\Psi_j^*(y_1,\ldots,y_N).\] An arbitrary operator can be written as \[A(x_1,\ldots, x_N; y_1,\ldots, y_N),\] and the expectation value would be \[\begin{split} \langle A(t)\rangle = \text{Tr}\left\{ \rho_t A \right\} = \int_{x_1,\ldots, x_N} \int_{y_1,\ldots,y_N} \rho_t(x_1,\ldots, x_N; y_1,\ldots,y_N) A(y_1,\ldots,y_N; x_1,\ldots, x_N). \end{split}\]

Momentum operator

As an example let us consider just a single particle. We want to calculate the expectation value of the momentum component \(P_k\). It corresponds to a derivative operator in the position space representation we use. In our notation it can be written as a distribution, \[P_k(\mathbf x, \mathbf y) = -i \frac{\partial}{\partial x^k} \delta^{(3)}(\mathbf x - \mathbf y).\] With this one finds with a few steps involving partial integration \[\begin{split} \langle P_k(t) \rangle = \int_{\mathbf x, \mathbf y} \rho_t(\mathbf x, \mathbf y) \left[ -i \frac{\partial}{\partial y^k} \delta^{(3)}(\mathbf y - \mathbf x) \right] = \int_\mathbf{x} \sum_j p_j \psi_j^*(\mathbf x) \left[ -i \frac{\partial}{\partial x^k} \psi_j(\mathbf x) \right], \end{split}\nonumber\] which is the expression familiar from quantum mechanics.