# Numerical calculation of a quantum field's observables

Okay so QFT is definitely beautiful and elegant theory, its mathematics is rich and ingenious, but there is so much one can do with symbolic manipulations of mathematical objects only, how can I actually do calculations that is useful in experimental reality? How can I do simulations using some programming language? Okay to make my question more clear, I will add an example: $$\hat{H}=\int{d^3p(\frac{\mathbf{p}^2}{2m}\hat{a}^\dagger_\mathbf{p}\hat{a}_\mathbf{p})}+\int{d^3pd^3q(\tilde{V}(\mathbf{p}-\mathbf{q})\hat{a}^\dagger_\mathbf{p}\hat{a}_\mathbf{q})},$$ Where: $$\tilde{V}(\mathbf{p}-\mathbf{q})=\int{d^3x\frac{1}{(2\pi)^3}V(\mathbf{x})e^{i(\mathbf{p}-\mathbf{q})\cdot\mathbf{x}}}$$ For some potential $$V(\mathbf{x})$$, this is the Hamiltonian for the non-relativistic limit of some massive complex scalar field, How do I make an actual numerical calculation out of this? Say for Coulomb potential or whatever, like if I have a vacuum state of $$\left|0\right>$$ -which I have no idea how to construct- how do I get the expectation value of the energy of that state?

This is just an example, a more general question is how to work numerically with operators and statevectors and these things in QFT to make simulations or predictions? Things was way clearer numerically in Schrödinger's Equation and single-particle QM. Apologies if this is obvious to some of you and a trivial question, but unfortunately I am self-studying, anyways, thanks!

• This is not really my field and I don't know much about it (hence a comment rather than an answer), but there's a whole field devoted to numerical solutions to field theory equations for QCD where the standard perturbative methods don't work. A search on "lattice QFT" may point you in that direction. Commented Apr 7, 2020 at 16:43
• Particle decays are described by Fermi's golden rule, and the transition elements in in are computed through QFT, as detailed in any text. Try "muon decay" in your favorite such. Lattice gauge theory computes such transition elements for breakfast. Commented Apr 7, 2020 at 19:46

If you're looking for a formulation of QFT that resembles Schrödinger's equation in single-particle QM and that can be solved on a (infinitely fast) computer, here it is:

## Scalar fields

For a single scalar field with Hamiltonian $$\newcommand{\pl}{\partial} H \sim \int d^3x \Big[\big(\dot\phi(x)\big)^2 + \big(\nabla\phi(x)\big)^2 + V\big(\phi(x)\big)\big], \tag{1}$$ simply replace continuous space with a finite lattice, treat $$\phi(x)$$ as a collection of independent real variables (one for each lattice site $$x$$), and use $$\dot\phi(x)\propto \frac{\pl}{\pl \phi(x)} \tag{2}$$ normalized so that the (lattice version of the) canonical commutation relation holds. Then the equation of motion in the Schrödinger picture is $$i\frac{\pl}{\pl t}\Psi[\phi,t] = H\Psi[\phi,t] \tag{3}$$ where the complex-valued state-function $$\Psi$$ depends on time $$t$$ and on all of the field variables $$\phi(x)$$, one for each lattice site.

## Gauge fields

The lattice Schròdinger-functional formulation for gauge fields associates elements of the Lie group (not the Lie algebra!) with each link of the lattice (pair of neighboring sites). The analog of the differential operator (2), corresponding to the time-derivative of a gauge field, is nicely described in section 3.3 of https://arxiv.org/abs/1810.05338. The state-function $$\Psi$$ is a function of all of these group-valued link variables, and gauge invariance is expressed by a Gauss-law constraint.

## Fermion fields

The lattice Schròdinger-functional formulation for fermion fields is conceptually the easiest of all, because the anticommutativity of fermion fields (Pauli exclusion principle) means that you only have a finite number of possible values associated with each lattice site (instead of, say, a continuous real variable like in the scalar-field case), so the Schròdiger equation (3) is just a gigantic matrix equation in this case. In practice, it's messy.

## Some complications

Of course, there are a few complications:

• Solving a partial differential equation (3) with a ga-jillion independent variables takes an awful lot of computer power, far more than we currently have, unless we settle for compromises like using a lattice with only a handful of sites in each dimension.

• As far as I know, we don't yet know quite how to put chiral non-abelian gauge theories on a lattice. In particular, we don't yet know quite how to put the Standard Model on a lattice. (X-G Wen has suggested that we actually do know how to do it, at least in principle, but I haven't seen it spelled out yet in terms that I understand.) But we know how to put QCD and QED on a lattice, and it's done routinely, subject to the limitation noted in the first bullet above.

• Figuring out which states $$\Psi$$ represent single-particle states is a difficult problem, not to mention the multi-particle states that you'd need to do scattering calculations or to study bound-state properties (like hydrogen in quantum electrodynamics). There are QFTs in which its easy, like non-relativistic QFTs and trivial relativistic QFTs, but it's surprisingly difficult in relativistic theories like quantum electrodynamics where an electron is always accompanied by an electric field and an arbitrary number of arbitrarily low-energy photons. This difficulty is related to the fact that observables in QFT are tied to regions of spacetime, not to particles. Particles are phenomena that the theory predicts, not inputs to the theory's definition.

For more detail, I recommend the book Quantum Fields on a Lattice. There are several books about lattice QFT, but this is one of the most comprehensive.

## Strictly non-relativistic QFT

Things are easier in strictly non-relativistic QFT, like the example shown in the question. In that case, the number of particles is fixed, and the model decomposes into separate non-relativistic QM for each fixed number of particles. Each one of these separate non-relativistic models has far fewer independent variables, proportional to the number of particles instead of proportional to the number of points in space.

• This resembles DFT, correct me if I am mistaken about that. You mentioned: "observables in QFT are tied to regions of spacetime", can you please elaborate more on this? how can observable of a quantum state exist somewhere where no particles exist? like how I can measure a spin without their being a particle? Also, is this the only way to do QFT numerically? then what good it is? what about the example I stated, is this question really nonsensical to ask in QFT? Is it all just mathematical objects' relations on infinite space that only give intuitive physics insights? Commented Apr 7, 2020 at 23:01
• @AGawish In relativistic QFT, trying to tie an observable to a particle is like trying to tie an observable to the crest of a wave. The wave can dissipate or split up, and then what would happen to the observable? It doesn't work. The relationship between observables and particles in QFT is more like the relationship between measurements and particles in the real world. We can use observables in QFT to represent detectors sitting in particular locations. The detector (observable) is sensitive to the presence of particles with particular properties in that location, like user1504 said. Commented Apr 8, 2020 at 0:27
• @AGawish The model can be made more interesting by using two factors of $a^\dagger$ and two factors of $a$ in the interaction term, like you would for the Coulomb interaction. Then the particles do interact with each other, but their number is still fixed, so the theory decomposes into separate non-relativistic QM models for each number of particles. Then, by considering just a single fixed number $N$ of particles, the number of independent variables is greatly reduced: just $3N$ for spinless particles in 3-d space, for example. Commented Apr 8, 2020 at 0:29
• @AGawish The main tool in particle physics applications is perturbation theory: we express the theory of interest as a small perturbation of an exactly-solvable (usually non-interacting) theory and then expand in powers of that small perturbation, up to some low order (like first or second, or maybe even third if we're really ambitious). Perturbation theory works well as long as we have a good guess about what some of the theory's particles are and how to relate them to the field operators, but even this involves some subtleties of its own, and it can cause problems when used too naively. Commented Apr 8, 2020 at 0:53
• @AGawish The existence of the Higgs field wasn't a prediction; it was an input to the Standard Model that made its predictions work well with a relatively simple set of inputs. The Higgs boson (particle) was predicted as a feature of certain scattering processes, using perturbation theory. Commented Apr 8, 2020 at 0:55

Yes, QFT books are unfortunately often vague about the connection to reality until the chapters where they discuss scattering, in my experience. The measurement-related postulates in QFT are the same as in QM but with some difficulties. $$| \langle p | \psi \rangle |^2$$ is the probability density of getting a momentum $$p$$ in an experiment measuring momentum; this is used in the derivation of the scattering matrix. From that, it follows that $$\langle \psi | P^\mu | \psi \rangle$$ is the expectation value of momentum in a given state, for example. Whether you then put the time dependence in $$|\psi \rangle, P^\mu$$, or both is up to the picture being used (schrödinger/heisenberg/interaction).

One big difference to QM is that position space doesn't seem to have a satisfactory description in QFT (no lorentz invariant position space state for >1 particle unless you have multi-time arguments). It is often argued that a position-space description is not necessary, with something vague like "no localization possiblie in QFT". I happen to not believe that is true, but I warn you that my view is not the majority view in that regard.

• So the only use of QFT is when we are discussing scattering? Why happens things like the hydrogen atom? how can I work such system using the QFT formulation? Or the example that I posted, how can I evaluate that? Commented Apr 7, 2020 at 16:51
• I have only seen QFT used in scattering calculations, despite trying to find other applications. There are QFT-like theories used in quantum optics in non-scattering scenarios, but they are not the same theory as the standard model. I believe it is impossible to fully describe the hydrogen atom in QFT without mixing the QFT calculation with elements of nonrelativistic QM, or taking analogies from QM to try to pull out results from scattering amplitudes even when the situation itself may not involve scattering. So it seems to be essentially a scattering theory at the moment. Commented Apr 8, 2020 at 13:06
• That may diminish QFT in your mind, but the more you think about practical experiments, the more you realize how many experiments are in fact described by scattering. However of course not every experiment is a scattering experiment. Therefore there are certainly some holes if you want to take QFT as a theory on its own without QM or analogies to other theories. I think it would do physics a lot of good to openly admit to such issues, which are sort of brushed under the rug at times. Commented Apr 8, 2020 at 13:08