Let us now address quantum states in a quantum field theory. We startby recalling some notions and concepts from quantum mechanics ofnon-relativistic particles and will then transfer the different conceptsto field theory.
Canonical quantization
A set of harmonicoscillators
We start with a set of mechanical harmonic oscillators. We writetheir 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\[H = \sum_{j=1}^N \left\{ \frac{1}{2}\pi_j^2(t) + \frac{1}{2} \omega_j^2 \phi_j^2(t) \right\}.\] Theclassical 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 \(\{ \phi_m(t), \phi_n(t) \} = \{\pi_m(t), \pi_n(t) \} = 0\).
Canonicalquantization 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 Poissonbrackets 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 thatwe are working here in the Heisenberg representation of quantummechanics 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 \[\begin{split} \hat \phi_j(t) = & \sqrt{\frac{\hbar}{2\omega_j}} \left[e^{-i\omega_j t} a_j(t) + e^{i\omega_j t} a_j^\dagger(t) \right], \\ \hat \pi_j(t) = & \sqrt{\frac{\hbar \omega_j}{2}} \left[ - ie^{-i\omega_j t} a_j(t) + i e^{i\omega_j t} a_j^\dagger(t) \right].\end{split}\label{eq:phipidecomposition}\] The equal timecommutation 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 equationsare solved by the ansatz in eq.\(\eqref{eq:phipidecomposition}\) when \(a_j(t) = a_j\) and \(a_j^\dagger(t) = a_j^\dagger\) areindependent of time \(t\). We restrictto this case in the following.
Hamiltonian,ground state and excitated state
We can write the Hamiltonian as \[H =\sum_{j=1}^N \frac{1}{2} \hbar \omega_j \left[ a^\dagger_j a_j + a_ja^\dagger_j \right] = \sum_{j=1}^N \hbar \omega_j \left[a^\dagger_j a_j + \frac{1}{2} \right].\] In the last line weused 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 stateor vacuum state. The energy is given there by zero-pointfluctuations \(\hbar w_j/2\) for eachmode.
One can also create excited states where the oscillator with index\(j\) has occupation number \(n_j \in \mathbb{N}_0\). Up to anormalization 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-relativisticcomplex 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 conjugateto \(\phi(t, \mathbf{x})\) is \(\pi(t, \mathbf{x}) = i \hbar \phi^*(t,\mathbf{x})\). The non-vanishing elementary classical Poissonbracket 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 orsecond 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 Poissonbracket, \[{\big [}\hat\phi(t, \mathbf{x}),\hat \phi^\dagger(t, \mathbf{y}) {\big ]} = \delta^{(3)}(\mathbf{x} -\mathbf{y}).\] This can be solved by writing the free Heisenbergfield 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 \(w_\mathbf{p}=\mathbf{p}^2/(2m)\) andtime-independent \(a_\mathbf{p}\) forthe solution to the free evolution equations, and the commutationrelations \[{\big [}a_\mathbf{p},a^\dagger_\mathbf{q} {\big ]} = (2\pi)^3 \delta^{(3)}(\mathbf{p} -\mathbf{q}), \quad\quad\quad {\big [}a_\mathbf{p}, a_\mathbf{q} {\big ]}= {\big [}a^\dagger_\mathbf{p}, a^\dagger_\mathbf{q} {\big ]} =0.\]
Particles as quantumexcitations
The operators \(a_\mathbf{p}\) and\(a_\mathbf{p}^\dagger\) have the sameproperties as the annihilation and creation operators for the energylevels of the harmonic oscillator. But what they annihilate and createare actually particles! This provides a new possibility to understandmany-particle states in quantum mechanics as corresponding excitationsof a vacuum state.
Vacuum states
In the present formalism (for non-interacting fields) one can takethe vaccum state \(|0 \rangle\) in thefield 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 momentumeigenstate by using the creation operator, \[| \mathbf p \rangle \sim a_\mathbf{p}^\dagger | 0\rangle.\] We discuss the normalization and related issues lateron. In a similar way one can construct two-particle states, \[| \mathbf{p}, \mathbf{q} \rangle \sima_\mathbf{p}^\dagger a_\mathbf{q}^\dagger | 0 \rangle,\] and as aconsequence of the commutation relations it is autmotically symmetric,\[| \mathbf{p}, \mathbf{q} \rangle = |\mathbf{q}, \mathbf{p} \rangle.\]
Schrödingerfunctional representation of quantum states
The algebraic method we used for canonical quantization of harmonicoscillators and the free non-relativistic field works in a similar wayfor many free or non-interacting quantum field theories. It shows thatin a quantum field theory particles can be seen as excitations.
To address more general situations it is often useful to have aconcrete representation of the quantum states and operators. We developthis 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 andSchrödinger picture coincide. We use the field theoretic analoge of theposition space representation of quantum mechanics to describe quantumstates. For a field theory, the Schrödinger wave function will become aSchrödinger functional \(\Psi[\phi]\).
The density matrix
Recall that in the position space representation of quantum mechanicsfor \(N\) degrees of freedom, such asparticle 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 generalmixed state needs to be described by a density matrix or adensity operator. For a mixture of states \(|\Psi_j \rangle\) with probability \(p_j\) such that \(\sum_j p_j = 1\), the density operator isformally given by \[\rho_t = \sum_j p_j |\Psi_j\rangle\langle \Psi_j |.\] The concept of a mixed state isneeded if one does not know the state with certainty but has only aprobailistic description available. Mixed states are also needed if onewould like to describe degrees of freedom that are not fully isolatedbut entangled with other degrees of freedom. This is actuallythe general situation for the local description of a quantumfield theory in some subvolume of space.
Expectation values
From the density operator, one can calculate expectation values attime \(t\) 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, forthe 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 bewritten 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 tocalculate the expectation value of the momentum component \(P_k\). It corresponds to a derivativeoperator in the position space representation we use. In our notation itcan be written as a distribution, \[P_k(\mathbf x, \mathbf y) = -i\frac{\partial}{\partial x^k} \delta^{(3)}(\mathbf x - \mathbfy).\] With this one finds with a few steps involving partialintegration \[\begin{split}\langle P_k(t) \rangle = \int_{\mathbf x, \mathbf y} \rho_t(\mathbf x,\mathbf y) \left[ -i \frac{\partial}{\partial y^k} \delta^{(3)}(\mathbfy - \mathbf x) \right] = \int_\mathbf{x} \sum_j p_j \psi_j^*(\mathbf x)\left[ -i \frac{\partial}{\partial x^k} \psi_j(\mathbf x) \right],\end{split}\nonumber\] which is the expression familiar fromquantum mechanics.
