Prof. Dr. Stefan Floerchinger

Researcher and professor for quantum field theory.

Non-relativistic bosonic fields

From relativistic to non-relativistic scalar fields

In this section we go from a relativistic quantum field theory back to non-relativistic physics but in a quantum field theoretic formalism. This non-relativistic QFT is in the few-body limit equivalent to quantum mechanics for a few particles but also has interesting applications to condensed matter physics (many body quantum theory) and it is interesting conceptually. We start from the action of a complex, relativistic scalar field in Minkowski space \[S=\int dt d^3x \left\{-\partial_\mu\phi^*\partial^\mu\phi- m^2\phi^*\phi - \frac{\lambda}{2}(\phi^*\phi)^2\right\}.\] The quadratic part can be written in Fourier space with (\(px = -p^0x^0 + \mathbf{p}\mathbf{x}\)), \[\phi(x) = \int\frac{d^4 p}{(2\pi)^4} e^{ipx} \phi(p), \quad\quad\quad \phi^*(x) = \int\frac{d^4 p}{(2\pi)^4} e^{-ipx} \phi^*(p),\] as \[\begin{split} S_2 = & -\int \frac{d^4 p}{(2\pi)^4}\left\{\phi^*(p)\left[-(p^0)^2+\mathbf{p}^2+m^2\right]\phi(p)\right\} \\ = & -\int \frac{d^4 p}{(2\pi)^4}\left\{\phi^*(p)\left[-\left(p^0-\sqrt{\mathbf{p}^2+m^2}\right)\left(p^0+\sqrt{\mathbf{p}^2+m^2} \right)\right]\phi(p)\right\}. \end{split}\]

Two zero crossings

One observes that the so-called inverse propagator has two zero-crossings, one at \(p^0 = \sqrt{\mathbf{p}^2+ m^2}\) and one at \(p^0 = -\sqrt{\mathbf{p}^2+ m^2}\). At these points the quadratic part of the action become stationary in the sense \[\frac{\delta}{\delta \phi^*(p)}S_2 = 0.\] The zero-crossings also correspond to poles of the propagator. These so-called on-shell relations give the relation between frequency and momentum for propagating, particle-type excitations of the theory. In fact, \(p^0 = \sqrt{\mathbf{p}^2+ m^2}\) gives the one for particles, \(p^0 = -\sqrt{\mathbf{p}^2+ m^2}\) the one of anti-particles. In the non-relativistic theory, anti-particle excitations are absent. Intuitively, one assumes that the fields are close to fulfilling the dispersion relation for particles, \(p^0 = \sqrt{\mathbf{p}^2+ m^2}\) which is for large \(m^2\) rather far from the frequency of anti-particles. One can therefore replace in a first step \[p^0 + \sqrt{\mathbf{p}^2+ m^2} \to 2\sqrt{\mathbf{p}^2+ m^2} \approx 2m.\] Moreover, one can expand the dispersion relation for particles for \(m^2 \gg \mathbf{p}^2,\) \[p^0 = \sqrt{\mathbf{p}^2+ m^2} = m + \frac{\mathbf{p}^2}{2m} + \ldots\] This leads us to a quadratic action of the form \[S_2 = -\int\frac{d^4 p}{(2\pi)^4}\left\{\phi^*(p)\left(-p^0+m+\frac{\mathbf{p}^2}{2m}\right)2m \, \phi(p)\right\},\] or for the full action in position space \[S=\int dt d^3x \left\{\phi^*\left(i\partial_t - m + \frac{\boldsymbol{\nabla}^2}{2m}\right) 2m\;\phi - \frac{\lambda}{2}(\phi^*\phi)^2\right\}.\]

Rescaled fields and dispersion relation

It is now convenient to introduce rescaled fields by setting \[\phi(t,\mathbf{x}) = \frac{1}{\sqrt{2m}}e^{-i(m-V_0) t}\varphi(t,\mathbf{x}).\] The action becomes then \[S=\int dt d^3x \left\{\varphi^*\left(i\partial_t - V_0 + \frac{\boldsymbol{\nabla}^2}{2m}\right) \varphi - \frac{\lambda}{8m^2}(\varphi^*\varphi)^2\right\}. \label{eq:nonrelativisticactionScalar}\] The dispersion relation is now with \[\varphi(t,\mathbf{x}) = \int \frac{d\omega}{2\pi}\frac{d^3 p}{(2\pi)^3} e^{-i\omega t + i\mathbf{p}x} \varphi(\omega,\mathbf{p}),\] given by \[\omega = V_0 + \frac{\mathbf{p}^2}{2m}.\] This corresponds to the energy of a non-relativistic particle where \(V_0\) is an arbitrary normalization constant corresponding to the offset of an external potential. The action in equation \(\eqref{eq:nonrelativisticactionScalar}\) describes a non-relativistic field theory for a complex scalar field. As we will see, one can obtain quantum mechanics from there but it is also the starting point for a description of superfluidity.

Symmetries of non-relativistic theory

Global U(1) symmetry

The non-relativistic action in equation \(\eqref{eq:nonrelativisticactionScalar}\) has a number of symmetries that are interesting to discuss. First we have translations in space and time as well as rotations in space as in the relativistic case. There is also a global U\((1)\) internal symmetry, \[\varphi(x) \to e^{i\alpha}\varphi(x), \quad\quad\quad \varphi^*(x) \to e^{-i\alpha}\varphi^*(x).\] By Noether’s theorem this symmetry is related to particle number conservation (exercise).

Time-dependent U\((1)\) symmetry

There is also an interesting extension of the global U(1) symmetry. One can in fact make it time-dependent according to \[\varphi(x) \to e^{i(\alpha+\beta t)}\varphi(x), \quad\quad\quad \varphi^*(x) \to e^{-i(\alpha+\beta t)}\varphi^*(x).\] All terms in the action are invariant except for \[\varphi^* i\partial_t\varphi \to \varphi^* e^{-i(\alpha+\beta t)}\; i\partial_t \; e^{i(\alpha+\beta t)} \varphi(x) =\varphi^*(i\partial_t - \beta)\varphi.\] However, if we also change \(V_0 \to V_0-\beta\) we have for the combination \[\varphi^*(i\partial_t-V_0)\varphi \to \varphi^*(i\partial_t -\beta -V_0 + \beta)\varphi = \varphi^* (i\partial_t -V_0) \varphi.\] This shows that \[\varphi(x) \to e^{i(\alpha+\beta t)} \varphi,\quad\quad\quad \varphi^* \to e^{-i(\alpha+\beta t)}\varphi^*,\quad\quad\quad V_0 \to V_0-\beta,\] is in fact another symmetry of the action in equation \(\eqref{eq:nonrelativisticactionScalar}\). One can say here that \((i\partial_t-V_0)\) acts like a covariant derivative. This says that \((i\partial_t -V_0)\varphi\) transforms in the same (covariant) way as \(\varphi\) itself. The physical meaning of this transformation is a change in the absolute energy scale, which is possible in non-relativistic physics.

Galilei transformation

Note that the action in equation \(\eqref{eq:nonrelativisticactionScalar}\) is not invariant under Lorentz transformations any more. This is directly clear because derivatives with respect to time and space do not enter in an equal way. However, non-relativistic physics is invariant under another kind of space-time transformations, namely Galilei boosts, \[t\to t,\] \[\mathbf{x} \to \mathbf{x}+\mathbf{v}t.\] One can go to another reference frame that moves relative to the original one with a constant velocity. How is this transformation realized in the non-relativistic field theory described by equation \(\eqref{eq:nonrelativisticactionScalar}\)? This is a little bit complicated and we directly give the transformation law, \[\varphi(t,\mathbf x) \to \varphi'(t,\mathbf{x}) = e^{i\left(m\mathbf{v} \cdot\mathbf{x}-\frac{1}{2}m\mathbf{v}^2 t\right)}\varphi(t,\mathbf{x}-\mathbf{v}t).\] Indeed one can confirm that \[\left(i\partial_t + \tfrac{\boldsymbol{\nabla}^2}{2m}\right)\varphi(t,\mathbf{x}) \to e^{i\left(m\mathbf{v} \cdot \mathbf{x}-\frac{1}{2}m\mathbf{v}^2 t\right)}\left[\left(i\partial_t + \tfrac{\boldsymbol{\nabla}^2}{2m}\right)\varphi \right](t,\mathbf{x}-\mathbf{v}t),\] so that the action \(\eqref{eq:nonrelativisticactionScalar}\) is invariant under Galilei transformations.

Spontaneous symmetry breaking, Bose-Einstein condensation and superfluidity

Effective potential

One can write the action in \(\eqref{eq:nonrelativisticactionScalar}\) also as \[S=\int dt d^3x \left\{\varphi^*\left(i\partial_t + \tfrac{\boldsymbol{\nabla}^2}{2m}\right)\varphi- V(\varphi^*\varphi)\right\}, \label{eq:microscopicactionBoseGasPotential}\] with microscopic potential as a function of \(\rho = \varphi^*\varphi\), \[V(\rho)= V_0\rho + \frac{\lambda}{2}\rho^2 = -\mu\rho + \frac{\lambda}{2}\rho^2.\] At non-vanishing density one has \(V_0 = -\mu\), where \(\mu\) is the chemical potential. For, \(\mu > 0\) the minimum of the effective potential is at \(\rho_0>0\). In a classical approximation where the effect of fluctuation is neglected, one has the equation of motion following from \(\delta S = 0.\)

Bose-Einstein condensate

If the solution \(\varphi(x)= \phi_0\) is homogeneous (constant in space and time), it must correspond to a minimum of the effective potential. Without loss of generality we can assume \(\phi_0 \in \mathbb{R}\) and \[V'(\rho_0) = -\mu + \lambda\rho_0 = 0,\] leads to \[\phi_0 = \sqrt{\rho_0}= \sqrt{\frac{\mu}{\lambda}}.\] Assuming that it survives the effect of quantum fluctuations, such a field expectation value breaks the global U\((1)\) symmetry spontaneously, similar to magnetization. This phenomenon is known as Bose-Einstein condensation. One can see this as a macroscopic manifestation of quantum physics. The mode with vanishing momentum \(\mathbf{p}= 0\) has a macroscopically large occupation number, which is possible for bosonic particles. On the other side, it arises here in a classical approximation to the quantum field theory described by the action in eq. \(\eqref{eq:nonrelativisticactionScalar}\). In this sense, a Bose-Einstein condensate can also be seen as a \(\textit{classical}\) field, similar to the electro-magnetic field, for example.

Bogoliulov excitations

It is also interesting to study small perturbations around the homogeneous field value \(\phi_0\). Let us write \[\varphi(x) = \phi_0 + \frac{1}{\sqrt{2}}\left[\phi_1(x) + i\,\phi_2 (x) \right],\] with real fields \(\phi_1(x)\) and \(\phi_2(x)\). The action in eq. \(\eqref{eq:microscopicactionBoseGasPotential}\) becomes (up to total derivatives) \[S= \int dt\;d^3x \left\{\phi_2 \partial_t \phi_1 + \frac{1}{2}\sum_{j=1}^2 \phi_j \frac{\boldsymbol{\nabla}^2}{2m}\phi_j -V\left(\phi^2_0 + \sqrt{2}\phi_0\phi_1+ \tfrac{1}{2}\phi^2_1 + \tfrac{1}{2}\phi^2_2\right) \right\}.\] It is instructive to expand to quadratic order in the deviations from a homogeneous field \(\phi_1\) and \(\phi_2\). The quadratic part of the action reads \[S_2 = \int dt\;d^3x \left\{-\frac{1}{2}(\phi_1,\phi_2) \begin{pmatrix} -\frac{\boldsymbol{\nabla}^2}{2m}+2\lambda\phi^2_0 & \partial_t \\ -\partial_t & -\frac{\boldsymbol{\nabla}^2}{2m} \end{pmatrix} \begin{pmatrix} \phi_1 \\ \phi_2 \end{pmatrix} \right\}.\] In momentum space, the matrix between the fields becomes \[G^{-1}(\omega,\mathbf{p}) = \begin{pmatrix} \frac{\mathbf{p}^2}{2m}+ 2\lambda\phi^2_0 & -i\omega \\ i\omega & \frac{\mathbf{p}^2}{2m} \end{pmatrix}.\] In cases where the inverse propagator is a matrix, this holds also for the propagator. When the determinant of the inverse propagator has a zero-crossing, the propagator has a pole. This defines the dispersion relation for quasi-particle excitations, \[\det G^{-1}(\omega,\mathbf{p})= 0.\] Here this leads to \[-\omega^2 + \left(\frac{\mathbf{p}^2}{2m} + 2\lambda\phi^2_0\right)\frac{\mathbf{p}^2}{2m} = 0,\] or \[\omega= \sqrt{\left(\frac{\mathbf{p}^2}{2m}+2\lambda\phi^2_0\right)\frac{\mathbf{p}^2}{2m}}. \label{eq:BogoliubovDispersionRelation}\] This is known as Bogoliubov dispersion relation.

Linear and quadratic regimes

For small momenta, such that \[\frac{\mathbf{p}^2}{2m} \ll 2\lambda\phi^2_0,\] one finds \[\omega \approx \sqrt{\frac{\lambda\phi^2_0}{m}}|\mathbf{p}|. \label{eq:BogoliubovDispersionRelationLowMomentum}\] In contrast, for \[\frac{\mathbf{p}^2}{2m} \gg 2\lambda\phi^2_0,\] one recovers the usual dispersion relation for non-relativistic particles \[\omega \approx \frac{\mathbf{p}^2}{2m}. \label{eq:BogoliubovDispersionRelationLargeMomentum}\] The low-momentum region describes phonons (quasi-particles of sound excitations), while the large-momentum region describes normal particles.

Superfluidity

The fact that the dispersion relation is linear for small momenta is also responsible for another interesting phenomenon, namely superfluidity, a fluid motion without friction. To understand this consider an interacting Bose-Einstein condensate flowing past some body of through a capillary. If the energy and momentum of the fluid are \(E=E_0\) and \(\mathbf{p}=0\) in the fluid rest frame, they are \[E^\prime =E+ \mathbf{p} \mathbf{v} + \frac{1}{2}M \mathbf{v}^2 = E_0 + \frac{1}{2}M \mathbf{v}^2, \quad\quad\quad \mathbf{p}^\prime = \mathbf{p} + M \mathbf{v} = M \mathbf{v},\] in the rest frame of the body or capillary. We used here first the general transformation of energy \(E\) and momentum \(\mathbf{p}\) under Galilei boost transformations and then the particular values for the homogeneous fluid state.

Imagine now that we can create an excitation or quasi-particle in the fluid with energy \(\epsilon(\mathbf{p})\) and momentum \(\mathbf{p}\). In the fluid rest frame we have now \(E=E_0+\epsilon(\mathbf{p})\) and \(\mathbf{p}=\mathbf{p}\). The energy and momentum in the rest frame of the capillary are then \[E^\prime = E_0 + \epsilon(\mathbf{p}) + \mathbf{p} \cdot \mathbf{v} + \frac{1}{2}M \mathbf{v}^2, \quad\quad\quad \mathbf{p}^\prime = \mathbf{p} + M \mathbf{v}.\] Comparison to the corresponding relation for the homogeneous state shows that the energy and momentum associated to the excitation are in the rest frame of the capiliary \(\epsilon(\mathbf{p})+ \mathbf{p} \cdot \mathbf{v}\) and \(\mathbf{p}\), respectively.

Landau’s criterion for superfluidity

Now the point is that at small temperature, excitations will only be created in the fluid in appreciable numbers when it is energetically favorable, i.e. for \[\epsilon(\mathbf{p})+ \mathbf{p} \cdot \mathbf{v} < 0,\] such that the energy of the fluid is lowered. If this relation is not fulfilled for any momentum \(\mathbf{p}\), no excitations that could transport momentum out of a local fluid cell will be created. This means that there is no viscosity and the flow is superfluid. It follows that for friction to become possible, the fluid needs to have a fluid velocity larger than \[v_c = \underset{\mathbf{p}}{\text{min}} \frac{\epsilon(\mathbf{p})}{|\mathbf{p}|},\] known as critical velocity. For the Bogoliubov dispersion relation \(\eqref{eq:BogoliubovDispersionRelation}\) the critical velocity equals the velocity of sound.

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