Quantum field theory

Please note that the photos were written in portuguese. However, there is much math which can be understood independently.

Quantum field theory (QFT) is a branch of quantum mechanics that describes the behavior (including interactions) of elementary particles. Elementary particles are those that cannot be subdivided, that is, they are fundamental units, not being composed of smaller pieces. Although the mathematical description of QFT is extremely rigorous, its physical concept can be understood without much difficulty. Before going into quantum field theory properly, we must ask ourselves what is a field in the context of physics? Briefly, a field is an entity that permeates space and associates a value in each point of space. A field can evolve/change over time (thus being a function with spatial and temporal dependence) or it can remain stationary (constant) as time flows. In general, fields can be classified as scalar or vector. Scalar fields are those that, for each point in space, the field will present a scalar quantity (number) in response; examples are temperature, pressure and humidity fields (see photo 2). Vector fields are those that, for each point in space, a vector, with direction, orientation and module, will be assigned. The most common examples of vector fields are those of force (see photo 3). The human being does not have the sensitivity to “visualize” the fields properly, but we can infer their existence with effects that we perceive, as in the case of approaching two magnets, being that their magnetic fields will interact in order to generate an attraction or repulsion effect (depending on the arrangement of the poles of the magnets). Still, for the case of the magnetic field, we can, in a way, visualize it with the use of some ferromagnetic material (such as iron powder), which will be shaped according to the magnetic field in matter. To begin our discussion, our focus must be on a special type of field, the electromagnetic field, which is a combination of two vector fields: the electric field and the magnetic field, those being perpendicular to each other. The electromagnetic field, for being composed of two underlying vector fields, can be represented by another type of field (differentiating from scalar and vector), this being a tensor field. Classically, the electromagnetic field is generated by moving electrical charges and interacts with electrical charges and currents. With the key ideas of what fields are, we can begin our discussion about QFT. There is not only a single quantum field theory, each elementary particle has its own "version" of the theory, but the basic concepts are the same. What was the need that led to the development of the first quantum field theory? The quantum mechanics developed by Schrodinger, through his equation, has certain limitations, as in the case of systems with multiple particles and identical particles (as will be explained) and, therefore, a new form of quantum theory was required. It was the brilliant theoretical physicist Paul Dirac who adapted, from the quantum mechanics developed up to then, classical electromagnetism, introducing quantum concepts to the eletromagnetic field.

One of the main differences between the previous description of the particle mechanism for the new one was that, before, the particles were described by states (wave functions) and were “tracked” individually (each particle was considered), in a way we could obtain their respective observables, such as momenta and position. However, as fundamental particles (such as electrons) are indistinguishable from each other (there is no way to differentiate two electrons, as all their intrinsic properties are the same), when we try to count them up, there is a risk of counting the same particle - or set of particles - twice, since there is no difference between those that make up the considered system (hence the inefficiency of the Schrodinger equation for multiparticle systems). To calculate the probability of interactions between particles (which can happen in different ways), it is necessary to know the quantity of particles present in the system and, with repeated counting, the result is incorrect, thus providing the wrong probabilities. In order to correct the problem, Dirac stopped "tracking" the particles individually, he no longer tried to quantify the singular properties of each one (such as the aforementioned momenta and position); instead, the physicist quantized, that is, parameterized the electromagnetic field itself. In this way, the mathematics behind the calculations maintains contact/tracks the field itself, and no longer the particles. The new interpretation looks at particles as oscillations/disturbances in the underlying field; such oscillations, consequently, have properties such as the momenta, position and frequency. Thus, “double counting” is not committed and the particle's properties can still be obtained. Dirac developed this ideia for the electromagnetic field, with photons being the particles/oscillations corresponding to that field (we can now have a better intuiton about the wave-particle duality).

Such formulation resulted in the design of the first quantum field theory, quantum electrodynamics (QED), which will be explored in the future.

Essentially, the fields in QFT are imagined to be composed of an infinite number of harmonic oscillators (called “modes”), so that the oscillations, with an angular frequency ω, propagated by these modes are the particles in locomotion in the field. Their physical properties are described according to the behavior of the set of oscillators (which can be thought of as spring species), such as the frequency and amplitude that they oscillate. It is emphasized that the set of oscillators constitutes the field itself and, therefore, it oscillates with a certain frequency and amplitude. For the electromagnetic field, the oscillations correspond to photons (light), as mentioned. The behavior of electromagnetic modes as harmonic oscillators can be proved according to photos 4,5,6 7 and 8.

In more advanced terms, we take the Hamiltonian (operator corresponding to the total energy of the system) from the field, deriving it for the electromagnetic mode. The result is to obtain a Hamiltonian identical to that of the quantum harmonic oscillator, which results in the same expression for the energy of a state with n photons (see photo 9, where λ is the specification of the mode). One of the primary concepts of quantum mechanics is Heisenberg's uncertainty principle, which postulates that, fundamentally, one cannot know certain observables simultaneously, some have an uncertainty relationship, so that the more is known about one less is known about the other. In technical terms, the entities with an uncertainty relation correspond to operators with commutators that return a result other than 0 (as shown in photo 10). Let us turn our attention again to the definition of mode. Think of the harmonic oscillator/mode as a pendulum, which will oscillate (if not at rest) around its equilibrium position with a certain frequency. However, because our pendulum is “quantum” it must obey the uncertainty principle and one of its formulations says that it is not possible to know, simultaneously, the position and momenta of a particle. This implies that our pendulum can never be at rest, otherwise it will have a defined position and momenta (rest position and zero speed). Thus, the pendulum will always be oscillating, even if the amplitude and frequency are low. Remember that the pendulums represent the harmonic oscillators in the field.

Usually the vacuum is defined as the absence of matter. However, the quantization of the electromagnetic field has very interesting implications that totally change our concept of vacuum. The absence of particles in the field causes it to be in its lowest energy state (corresponding to the lowest frequency); however, given the uncertainty principle, even without the presence of oscillation in the field, the modes of the field cannot be at rest, resulting in numerous low-energy oscillations. Another (more direct) way of thinking is by associating another form of the uncertainty principle, which relates the uncertainties between energy and time, indicating that, for infinitesimal time intervals, amounts of energy (particles) can "arise" essentially out of nowhere and then disappear, returning the apparently created energy (so the principle of energy conservation is preserved in the long run). These oscillations from the void are called quantum vacuum fluctuations. The photon-free state of a mode can be seen in photo 11. Another factor that differs from Schrodinger's previous formulation from the description of field theory is the mechanism of particle creation and destruction, absent in Schrodinger's version. A practical example of this new description is the situation in which an electron and a positron (anti-particle) can, in an encounter, annihilate themselves, resulting in a release of energy in the form of photons. For such description, the use of creation and annihilation operators (see photo 11) is made, which have many interesting properties such as an operation which returns the number of particles present in the state (see photo 12). As in chemistry, in which elements/molecules interact with each other, subatomic particles also do, through particles called bosons. There are 5 “general” types of bosons: photons, gluons (which have 8 sub-types), Z boson, W boson (with 2 variations W ^+ and W ^-) and the Higgs boson. Bosons are force-transmitting particles, thus being responsible for intermediating the fundamental forces of nature (strong nuclear force, weak nuclear force and electromagnetic force), with the exception of gravity, whose unifying quantum theory has not yet been effectively obtained. The interaction mechanism in QFT occurs according to the interaction between the underlying fields through the bosons. In a simplified way, bosons are emitted by one particle, reach the other, thus delivering an “impact”. For the case of the electromagnetic field, whose corresponding boson is the photon, how will there be an “impact” if the photon is a massless particle? The answer can be obtained using de Broglie's relation, which enables the association of momenta (p) for photons, as shown in photo 13. Keep in mind that bosons also have associated fields, being oscillations within these.

In QFT, particles can have an infinite degree of freedom, which means that a state cannot be described completely with finite variables. Fields have infinite position points (x1,y1); (x2,y2)... and, as in QFT we work with fields, soon the infinitudes begin to appear in situations which finite values were expected; physicists soon realized that they needed to “adjust” the mathematics involved in some way. The way found to solve (even if not fundamentally) the problem was the development of a series of methods aimed at "removing" the obtained infinites, coming to be called renormalization. Given the complicated technical nature of the QFT, the extraordinary theoretical physicist Richard Feynman developed a series of diagrams which were intended to illustrate, very simply, what happened in complicated mathematical descriptions of interacting particles. In photo 14 we can see one of the simplest diagrams, representing the repulsion between two electrons, through a virtual photon (boson responsible for the transmission of the electromagnetic force). Note that in the diagram shown, time runs vertically, from bottom to top. Still, we can highlight the facts that the quantum field theory is compatible with special relativity and is so precise that it is able to predict even the small energy separations of electrons in atoms, coming from the Lamb deviation. Finally, the model capable of describing the behavior and interaction of all elementary particles gains the name of "standard model", being the governing mechanism of particle physics. The pillars of the standard model are often pushed to extremes in the LHC (large hadron collider), a particle accelerator present on the border between Switzerland and France. In the future, articles on QED (quantum electrodynamics) and QCD (quantum chromodynamics) will be published.

Photo 1: cover Photo 2: scalar field Photo 3: vector field Photo 4-8: proof of the behavior of harmonic oscillator for the electromagnetic mode Photo 9: energy of a harmonic oscillator Photo 10: commutator for operators that do not commute among themselves Photo 11: state for zero photons Photo 12: number operator Photo 13: Louis de Broglie's momenta relationship Photo 14: Feynman`s diagram

Reference material: Quantum mechanics for scientits and engineers (David A. B. Miller) Modern quantum mechanics (J.J. Sakurai and Jim Napolitano) 50 Quantum Physics Ideas You Really Need to Know (Joanne Baker) Quantum Field Theory (Frank Wilczek) Quantum theory a brief introduction (John Polkinghorne)