Field-Operator Applications

Nonrelativistic field operators turn mode-based many-body mechanics into a spatial language. The annihilation field $\psi(x)$ represents particle removal at the coordinate $x$, while $\psi^\dagger(x)$ creates one. Here $x$ may combine position and an internal label such as spin.

For a one-particle basis $\{\varphi_i(x)\}$,

$$\begin{aligned} \psi(x) &= \sum_i \varphi_i(x)a_i, \\ \psi^\dagger(x) &= \sum_i \varphi_i^*(x)a_i^\dagger. \end{aligned}$$

This is not a new physical theory. It is the position-space representation of the same Fock-space operators used in the occupation-number description.

Helpful background. Field operators and mode expansions define the canonical algebra and basis normalization used here.

Spatial fields and retained modes

Use $x=(\mathbf r,\sigma)$, with $\int dx=\sum_\sigma\int d^dr$, for position and any internal component. Fields at a common time satisfy a commutator for bosons or anticommutator for fermions; write $[A,B]_\eta=AB-\eta BA$, with $\eta=+1$ or $-1$ respectively. Field operators explains their distributional meaning and mode expansions derives the basis map.

For a complete continuum basis,

$$[\psi(x),\psi^\dagger(x')]_\eta=\delta(x-x').$$

For a retained mode set $\mathcal M$, the field $\psi_{\mathcal M}(x)=\sum_{i\in\mathcal M}\varphi_i(x)a_i$ instead obeys

$$[\psi_{\mathcal M}(x),\psi_{\mathcal M}^\dagger(x')]_\eta =P_{\mathcal M}(x,x')=\sum_{i\in\mathcal M}\varphi_i(x)\varphi_i^*(x').$$

This is a projector kernel, not the full delta distribution. It is essential when deriving equations after a band or momentum cutoff.

Removal amplitudes and number density

An exact-point field is not a bounded operator. A well-defined wavepacket operator is $\psi[f]=\int dx\,f^*(x)\psi(x)$; its adjoint creates the one-particle wavefunction $f$ from vacuum. Smeared fermionic fields are bounded for square-integrable $f$, whereas bosonic ones remain unbounded and require a domain, such as finite-particle vectors.

For a sufficiently regular fixed-$N$ wavefunction, annihilation at $x$ gives the unnormalized $(N-1)$-particle amplitude $\sqrt N\Psi_N(x,x_2,\ldots,x_N)$. Consequently,

$$\langle n(x)\rangle=N\int dx_2\cdots dx_N\,|\Psi_N(x,x_2,\ldots,x_N)|^2.$$

The equality holds as a density (almost everywhere, or after spatial integration), not as a promise that point evaluation exists for every Hilbert-space vector. An arbitrary superposition of number sectors is lowered sector by sector, rather than becoming one fixed-number state. The coordinate-to-Fock map fixes its factorials and fermionic operator order.

One-Body Continuum Hamiltonians

Let the one-particle Hamiltonian be

$$h_0 = -\frac{\hbar^2}{2m}\nabla^2 + V_{\mathrm{ext}}(\mathbf r).$$

Its many-body lift is

$$H_0 = \int d^dr\, \psi^\dagger(\mathbf r) h_0 \psi(\mathbf r).$$

The differential operator acts on the field to its right. Under boundary conditions that remove surface terms, the kinetic energy can also be written

$$T = \frac{\hbar^2}{2m} \int d^dr\, \nabla\psi^\dagger \boldsymbol\cdot \nabla\psi.$$

For fields with internal components,

$$H_0 = \sum_{\sigma,\sigma'} \int d^dr\, \psi_\sigma^\dagger(\mathbf r) [h_0]_{\sigma\sigma'} \psi_{\sigma'}(\mathbf r).$$

The matrix $[h_0]_{\sigma\sigma'}$ may contain spin coupling, Zeeman terms, spin-orbit coupling, or other one-body internal structure.

Returning to a Discrete One-Body Matrix

Insert the mode expansion into the continuum one-body Hamiltonian. The result is

$$H_0 = \sum_{i,j} h_{ij} a_i^\dagger a_j,$$

where

$$h_{ij} = \int dx\, \varphi_i^*(x) h_0 \varphi_j(x).$$

If the modes diagonalize $h_0$, then

$$h_{ij} = \epsilon_i\delta_{ij},$$

and

$$H_0 = \sum_i \epsilon_i a_i^\dagger a_i.$$

The field and mode forms therefore encode the same one-body operator. One is spatially transparent; the other may be computationally or spectrally convenient.

Finite-range pair potentials

For a real symmetric pair potential $v(x,x')$, the number-conserving interaction is

$$H_{\mathrm{int}} ={} \frac12 \int dx\,dx'\, \psi^\dagger(x) \psi^\dagger(x') \times v(x,x') \psi(x') \psi(x).$$

The factor $1/2$ prevents double counting unordered pairs. The annihilation order is part of the convention and must be kept fixed for fermions.

Expansion in modes gives

$$H_{\mathrm{int}} = \frac12 \sum_{i,j,k,l} V_{ij;kl} a_i^\dagger a_j^\dagger a_l a_k,$$

with

$$V_{ij;kl} ={} \int dx\,dx'\, \varphi_i^*(x) \varphi_j^*(x') \times v(x,x') \varphi_k(x) \varphi_l(x').$$

This is the continuum origin of the two-body matrix elements. The potential couples two positions but is diagonal in the two-particle coordinate basis; a genuinely nonlocal interaction kernel would have four coordinate arguments and four integrations.

Bosonic Contact Interaction

For a spinless bosonic field, a zero-range effective interaction is written

$$H_{\mathrm{contact}}^{(B)} = \frac{g_B}{2} \int d^dr\, \psi^\dagger \psi^\dagger \psi \psi.$$

Every field in the integrand is evaluated at the same position. Equivalently,

$$H_{\mathrm{contact}}^{(B)} = \frac{g_B}{2} \int d^dr\, {:}n^2(\mathbf r){:},$$

where

$$n(\mathbf r) = \psi^\dagger(\mathbf r) \psi(\mathbf r).$$

Normal ordering removes the self-contraction implicit in $n^2$, but it does not by itself solve every ultraviolet problem of a continuum contact theory.

The Weakly Interacting Bose Gas Preview uses this operator model to identify the dilute parameter, mean-field equation of state, healing scale, and Bogoliubov excitation spectrum.

Two-Component Fermionic Contact Interaction

For two fermionic components, a standard intercomponent contact term is

$$H_{\mathrm{contact}}^{(F)} = g_F \int d^dr\, \psi_\uparrow^\dagger \psi_\downarrow^\dagger \times \psi_\downarrow \psi_\uparrow.$$

No factor $1/2$ appears because the displayed term counts one ordered pair of distinct components. An equivalent spin-summed expression may use $1/2$ and sum over both component orders.

A same-component zero-range $s$-wave term vanishes formally because

$$\psi_\sigma(\mathbf r) \psi_\sigma(\mathbf r) = 0,$$

but this should be interpreted with the same distributional and regularization care as every coincident-point identity. Identical fermions can still interact through finite-range forces and higher partial waves.

Finite-Volume Momentum Expansion

In a periodic box of volume $V=L^d$,

$$\psi(\mathbf r) = \frac{1}{\sqrt V} \sum_{\mathbf k} e^{i\mathbf k\cdot\mathbf r} a_{\mathbf k},$$

with

$$\mathbf k = \frac{2\pi}{L} \mathbf n, \qquad \mathbf n \in \mathbb Z^d.$$

The inverse is

$$a_{\mathbf k} = \frac{1}{\sqrt V} \int_V d^dr\, e^{-i\mathbf k\cdot\mathbf r} \psi(\mathbf r).$$

The kinetic Hamiltonian becomes diagonal:

$$T = \sum_{\mathbf k} \frac{\hbar^2k^2}{2m} a_{\mathbf k}^\dagger a_{\mathbf k}.$$

The box normalization and the continuum normalization must not be mixed. The Fourier-transform conventions page states the normalization choices, while Momentum-Space Representation develops the complete many-body box-to-continuum dictionary.

Contact Interaction in Momentum Space

For periodic spinless bosons,

$$H_{\mathrm{contact}}^{(B)} ={} \frac{g_B}{2V} \sum_{\mathbf k_1,\mathbf k_2,\mathbf q} a_{\mathbf k_1+\mathbf q}^\dagger a_{\mathbf k_2-\mathbf q}^\dagger \times a_{\mathbf k_2} a_{\mathbf k_1}.$$

The wave-vector transfer is $\mathbf q$; physical momentum transfer is $\hbar\mathbf q$. Each term conserves total momentum because

$$(\mathbf k_1+\mathbf q) + (\mathbf k_2-\mathbf q) = \mathbf k_1+\mathbf k_2.$$

For a translationally invariant nonlocal potential, $g_B$ is replaced by its Fourier transform $\widetilde v(\mathbf q)$.

Projection onto Lattice Orbitals

Let $w_i(\mathbf r)$ be localized orthonormal orbitals. A single-band projection writes

$$\psi_\sigma(\mathbf r) \approx \sum_i w_i(\mathbf r) c_{i\sigma}.$$

The approximation sign records the discarded bands or orbitals. The one-body matrix elements are

$$h_{ij} = \int d^dr\, w_i^*(\mathbf r) h_0 w_j(\mathbf r).$$

If the lattice Hamiltonian is written with $-t_{ij}c_i^\dagger c_j$, then $t_{ij}=-h_{ij}$ off diagonal. For a weak contact interaction and localized orbitals, the leading onsite coefficient is

$$U_i = g \int d^dr\, \lvert w_i(\mathbf r)\rvert^4.$$

$U_i$ is site independent only when the orbital shapes are equivalent. Neglecting intersite interaction integrals and higher-band corrections is an additional approximation; strong scattering can require matching beyond this leading overlap formula. The continuum field and lattice operators are not different particle species; they are descriptions at different resolutions.

Density and Correlation Previews

The local number-density operator is

$$n(x) = \psi^\dagger(x) \psi(x),$$

and the total number is

$$N = \int dx\, n(x).$$

The one-body density matrix is

$$G^{(1)}(x,x') = \left\langle \psi^\dagger(x) \psi(x') \right\rangle.$$

At equal points,

$$G^{(1)}(x,x) = \langle n(x)\rangle.$$

Four-field products probe pair correlations. Their ordering, connected parts, and time dependence belong to the dedicated correlation-function pages; a mean density alone does not determine them.

Heisenberg Equation for a Bosonic Contact Model

For the formal local contact Hamiltonian, the Heisenberg equation reads

$$i\hbar \frac{\partial\psi}{\partial t} ={} \left( -\frac{\hbar^2\nabla^2}{2m} + V_{\mathrm{ext}} \right) \psi + g_B \psi^\dagger \psi \psi.$$

Every field is evaluated at $(\mathbf r,t)$. Coincident products require the stated contact prescription, and the coupling must match that prescription. This is not an exact local equation for an arbitrarily truncated continuum field. For a mode projector, the nonlinear term is instead

$$g_B\int d^dr'\,P_{\mathcal M}(\mathbf r,\mathbf r')\, \psi_{\mathcal M}^\dagger(\mathbf r')\psi_{\mathcal M}(\mathbf r')\psi_{\mathcal M}(\mathbf r').$$

The one-body term is projected in the same way. Taking an expectation value still leaves a three-field correlation; factorizing it in a suitable dilute-condensate regime produces a Gross–Pitaevskii equation.

Global U(1) Symmetry and Number Conservation

The number operator satisfies

$$\begin{aligned} [N,\psi(x)] &= -\psi(x), \\ [N,\psi^\dagger(x)] &= \psi^\dagger(x). \end{aligned}$$

Therefore

$$e^{i\theta N} \psi(x) e^{-i\theta N} = e^{-i\theta} \psi(x).$$

A number-conserving Hamiltonian is invariant under this global phase rotation and obeys

$$[H,N] = 0.$$

Allowed pairing terms, with distinct components or a nonlocal antisymmetric kernel for fermions, break particle-number U(1) symmetry in an effective description, though fermion parity may remain conserved.

Engineering Dimensions

The equal-time delta algebra fixes the spatial dimension of the field:

$$[\psi] = L^{-d/2}.$$

For a contact term

$$\frac{g}{2} \int d^dr\, \psi^\dagger \psi^\dagger \psi \psi,$$

the coupling has dimension

$$[g] = \text{energy}\times L^d = \frac{\hbar^2}{m} L^{d-2}.$$

A dimensionless measure at wave-number scale $k$ is therefore proportional to

$$\frac{mg}{\hbar^2} k^{d-2}.$$

This dimensional statement does not replace a renormalization calculation, but it identifies $d=2$ as classically marginal for the nonrelativistic contact coupling.

Contact Couplings and Regularization

Coincident-point products probe arbitrarily short distances. A continuum contact Hamiltonian therefore requires a physical prescription, such as:

• a momentum cutoff with a cutoff-dependent bare coupling;
• a lattice spacing;
• dimensional regularization and matching;
• a pseudopotential or boundary condition matched to scattering data.

For a dilute three-dimensional Bose gas at low energy,

$$g_B = \frac{4\pi\hbar^2a_s}{m}$$

is the standard leading coupling in terms of the $s$-wave scattering length $a_s$. Its use assumes the dilute, low-energy pseudopotential regime. It is not a universal microscopic identity valid at arbitrary cutoff, density, range, or dimension.

Matching two-body scattering alone need not define every many-body zero-range model: the resonant three-dimensional identical-boson problem also requires three-body information, as Braaten and Hammer explain. In two dimensions the low-energy coupling has logarithmic scale dependence. In one dimension confinement and transverse modes can change the effective coupling. The model and matching prescription must be stated.

Boundary Conditions Matter

The field algebra itself does not select boundary conditions. They enter through the one-particle domain and completeness relation.

In a periodic box, the delta is periodic and momentum is discrete. With hard walls, the modes vanish at the boundary. Integrating the kinetic term by parts is valid only when the boundary term vanishes or is explicitly retained.

Taking $V\to\infty$ changes sums to integrals and Kronecker deltas to Dirac deltas together with normalization factors. It is not valid to change only one side of that dictionary.

Why This Is Called Nonrelativistic QFT

The operator formulation uses local fields, Fock space, and field products in a Hamiltonian, so it is often called nonrelativistic quantum field theory. Within a fixed set of nonrelativistic particle species, it is equivalent to many-particle quantum mechanics sector by sector.

It is not yet a relativistic QFT. The present framework has:

• nonrelativistic dispersion; traps and lattices can also break Galilean invariance;
• equal-time canonical algebra;
• a preferred time coordinate;
• no automatic antiparticle sector;
• particle-number conservation in many standard models;
• no relativistic spacelike microcausality requirement.

Relativistic QFT changes the representation theory, vacuum structure, locality requirements, and relation between fields and particles. Why Many-Body QM Leads to QFT explains this connection and distinguishes microscopic, auxiliary, collective, and relativistic fields. Nonrelativistic Field Theory from Many-Body QM continues from the operator Hamiltonian to the Schrödinger-field action, number current, propagator, and contact-EFT matching.

Exercises

Remove one particle from a two-particle state

Let

$$\lvert\Psi_2\rangle ={} \frac{1}{\sqrt2} \int dx\,dy\, \Psi_2(x,y) \times \psi^\dagger(x) \psi^\dagger(y) \lvert0\rangle,$$

where $\Psi_2$ has the symmetry appropriate to the statistics. Show that

$$\psi(z)\lvert\Psi_2\rangle = \sqrt2 \int dy\, \Psi_2(z,y) \psi^\dagger(y) \lvert0\rangle.$$

Solution

Commuting or anticommuting $\psi(z)$ through the two creation fields gives two delta-function terms. For bosons they add because $\Psi_2(x,y)=\Psi_2(y,x)$. For fermions the operator interchange contributes a minus sign and the antisymmetry of $\Psi_2$ contributes a second minus sign, so the terms again add.

Each term equals

$$\frac{1}{\sqrt2} \int dy\, \Psi_2(z,y) \psi^\dagger(y) \lvert0\rangle.$$

Adding both terms gives the stated factor $\sqrt2$.

Taking the norm yields

$$\langle n(z)\rangle = 2 \int dy\, \left\lvert \Psi_2(z,y) \right\rvert^2.$$

Transform the contact interaction to momentum space

Use the periodic-box mode expansion to derive the bosonic contact interaction in momentum space and identify the conserved quantity.

Solution

Insert four mode expansions into

$$H_{\mathrm{contact}}^{(B)} = \frac{g_B}{2} \int_V d^dr\, \psi^\dagger \psi^\dagger \psi \psi.$$

The spatial integral gives

$$\int_V d^dr\, e^{i(\mathbf k_1+\mathbf k_2-\mathbf k_3-\mathbf k_4)\cdot\mathbf r} = V \delta_{\mathbf k_1+\mathbf k_2,\mathbf k_3+\mathbf k_4}.$$

Using $\mathbf q=\mathbf k_3-\mathbf k_1$ gives

$$H_{\mathrm{contact}}^{(B)} ={} \frac{g_B}{2V} \sum_{\mathbf k_1,\mathbf k_2,\mathbf q} a_{\mathbf k_1+\mathbf q}^\dagger a_{\mathbf k_2-\mathbf q}^\dagger \times a_{\mathbf k_2} a_{\mathbf k_1}.$$

Total momentum is conserved in every scattering term.

References

• A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press (2010).
• E. Braaten and H.-W. Hammer, “Universality in few-body systems with large scattering length,” Physics Reports 428, 259–390 (2006), doi:10.1016/j.physrep.2006.03.001.
• H. Bruus and K. Flensberg, Many-Body Quantum Theory in Condensed Matter Physics, Oxford University Press (2004).
• P. Coleman, Introduction to Many-Body Physics, Cambridge University Press (2015).
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003).
• J. W. Negele and H. Orland, Quantum Many-Particle Systems, Westview Press (1998).
• C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases, 2nd ed., Cambridge University Press (2008).
• L. Pitaevskii and S. Stringari, Bose–Einstein Condensation and Superfluidity, Oxford University Press (2016).