Edward Witten, one of the leading experts in Quantum Field Theory (QFT), describes QFT as the most difficult theory in Modern Physics, pushing aside even Einstein’s General theory of relativity to the sidelines. Now, setting aside the argument of what exactly Witten meant by that, and just absorbing the apparent meaning of it, it can’t be denied that QFT, being a theory on the very frontiers of our current understanding of Physics, is indeed very difficult. The prerequisite knowledge necessary for a full-fledged understanding of the theory is undoubtedly formidable – leading to the deferment of its introduction to only very advanced courses in Physics, in most Universities. Notwithstanding all the “formidable-ness” of the theory, we, a bunch of postgraduate students at the University of Mysore, decided to attend a talk on QFT by our University professor, Prof. K. S. Mallesh (also the incumbent head of the University Physics Department), last Thursday, at the Regional Institute of Education, Mysore.
Where QFT fits into the picture
The professor began by mentioning the four broad domains that cover all the explanations of nature – the big (macro), the small (micro), the fast (relativistic) and the slow (non-relativistic). Anything that’s studied in Physics falls into one or more of these four domains. For example, the professor pointed out, classical mechanics deals with objects that are big and move slow; while Einstein’s special relativity, with objects which are big and may move fast or slow; and Quantum Mechanics, with the small and slow. And by Quantum Mechanics (here), it is meant the Quantum theory that was formulated between 1900 and 1927 – which the professor called the “Golden period of Quantum Mechanics”.
Making the above classification a bit more rigorous, he explained the meaning of the adjectives used for the four domains. Anything that can’t be seen by the naked eye is the Micro (more precisely, micro objects are ones that are prone to be affected even if you try to observe them gently), the rest all being the Macro, he quipped. And anything moving at speeds close to the speed-of-light-in-vacuum, ‘c’, is relativistic, while anything slower is non-relativistic. And as for the inter relationships between the three theories – classical mechanics can be viewed as a limiting case of both the special relativity and quantum mechanics.
Moving out of the triad-theories, Prof. Mallesh called our attention to the fourth combination of the domains we’d been quiet about all this while – the situation in which objects are small and also move fast – and this is such an arena that Quantum mechanics and Relativity do explain substantially but not completely, by themselves. But a theory that merges the two (relativity and quantum physics), along with its own radical viewpoints, seems to explain the domain quite precisely – and it is this new outlook that is termed the Quantum Field Theory (QFT).
Quantum Mechanics alone, not enough
Picking up the simplest problem in quantum mechanics – the hydrogen atom – the professor pointed out that the speed of the orbiting electron, assuming the Bohr model, is roughly one hundredth the vacuum speed of light, c. Hundred times slower than light, is still not slow enough, the professor exclaimed – the speed of the electron is still an enormous 3000 km/s. Therefore, the Schrodinger equation does not explain the whole picture, on its own. And it’s not not just the speed, the Schrodinger equation can’t handle completely (in the hydrogen atom); there are also the processes of mass-energy inter-conversions that the equation is mute about. For instance, when defining the normalization of the state vector, it is said that the condition must necessarily be satisfied, so that we’re sure that the particle under observation is definitely present at least somewhere in all of space (and so has always existed). By doing so, we’re overlooking the processes such as the particle creation and annihilation – a photon converting itself into a particle and its anti-particle, and vice-versa. Moreover, the very concept of the photon is ill-defined in quantum mechanics (again, here we speak of the theory formulated before 1927).
The professor then mentioned another major shortcoming of quantum mechanics – it couldn’t convincingly explain spontaneous emissions in atoms. In contrast to spontaneous emissions, the stimulated emissions can be simply explained by quantum mechanics, as the processes caused due to an external electromagnetic field. However, emissions without the presence of a field are inexplicable by the Schrodinger wave equation.
This major shortcoming, along with the very interesting fact the professor mentioned – that while examining an electron passing through, say, an electric field, the quantum theory takes good care of treating the electron quantum-mechanically, while unhesitatingly assumes a classical description of the electric field – calls for a new way of looking at force-fields.
The professor then elaborated that the limitations that were discussed so far aren’t just limited to the quantum theory prior to 1927; even Dirac’s relativistic quantum mechanics deals only with material particles, and doesn’t speak much on force fields. And what makes the idea more bothersome is that even classical mechanics, the theory that quantum mechanics had trounced heavily, explains fields correctly. For quantum mechanics to be a complete theory, it must explain the fields as well, satisfactorily, the professor stressed. Therefore, there is a logical necessity for a quantum theory of fields.
Establishing the realness of fields, Prof. Mallesh, then underlined the fact that experiments have decisively proved that electromagnetic radiations that are just a manifestation of electric and magnetic fields of accelerated charges, posses properties such as energy, momentum, angular momentum, etc. Therefore, fields are as much physical as particles are. But that doesn’t mean we can have a one-to-one correspondence between a field and a particle, since there’s a crucial difference between the two – localization. A particle is localized, meaning that if we hook a particle to a specific position, any number of measurements of its position will always yield the same result (under identical conditions). In contrast, the field is distributed in space – we never speak of fields as existing only at a particular point in space – fields aren’t localized (at least in the classical description). The professor pointed out that we can, to some extent compare fields to a many-body-system of classical mechanics. A many body system is one that is comprised of a number of material particles (or bodies, if you may call them so) whose motion depends on the details of the system, such as the number and nature of “constraints”. The more and more the particles of the system are restricted to move in specified ways (i.e., the more they are constrained), the lesser number of “degrees of freedom” they posses. Making use of this analogy the professor defined the field as a physical system that has a continuous infinity of degrees of freedom. Infinite number of degrees of freedom – for the space we consider is always assumed to be continuous – and so even a finite volume of the space contains infinite points.
Quantizing the field
The professor continued explaining the radical view of fields with some simple mathematics. In classical mechanics we define a function for a free particle, called the Lagrangian, that is chosen to represent the true motion of the particle. Note that the Lagrangian is a function of and , called the generalized coordinate and generalized velocity of the particle, respectively (generalized coordinates, , are simply numbers that tell us the position of the particle in space, completely; and generalized velocities, , are simply time derivatives of ). In analogy to this description, we might want to define a Lagrangian for a field in a certain volume of space. But before we do that, we ought to make sure we know what kind of field we’re speaking of! Firstly, we’re considering a simple, general field , that may be different at different positions and time. Note that is a scalar (in the sense that it has no direction associated); a physical example of which could be the temperature in a room (there’s just one number associated with every point in the room, with no meaning for a direction of the temperature!). For simplicity, we consider a free field – the part of the field that doesn’t contain its source. And since the field is continuous, we divide it into small units and define the Lagrangian density (in analogy with the classical description of the free particle) as . Where, plays the role analogous to the generalized coordinate, and that of the generalized velocity. And the crucial point here, that Prof. Mallesh emphasized, is that has, as its components, the 4 Minkowskian coordinates , and in , . This is where the relativistic formalism creeps into the description.
The total Lagrangian will simply be the Lagrangian density defined above, integrated over the entire volume under consideration: . And as in the classical mechanics, the quantity , and is termed the “momentum cannonically conjugate to “.
Coming back to the definition of the Lagrangian for the field, the professor plugged it into the equation for the “action”, again following suit from classical mechanics,
Where, the integration has been performed between times and , during which the field changes its configuration.
Writing in terms of the Lagrangian density that was defined earlier,
The Hamilton’s principle says that this quantity, action, , is the same for any path that is at extreme close quarters to the true path of the field as it changes its configuration in the time interval. Mathematically, we say,
Since we very well know that the Lagrangian density is a function of and , its variation may be written as,
Putting this in to Eq.(2),
Integrating the second term on the right, by parts, and using the fact that at and ; and the field itself vanishes at the edge of the region of space considered, we get the final expression,
Since is arbitrary, may be varied to get different neighborhoods, resulting in the integrand itself vanishing, i.e.,
This is just the Euler-Lagrange equation for the field, in complete analogy with the approach of classical mechanics.
To obtain specific equations for different fields, we need to put in the appropriate field strengths and the corresponding Lagrangian density , into the equation.
As a passing note, the professor also mentioned that just as in classical mechanics, the Hamiltonian density for fields may be defined as,
[Compare this with the familiar equation, ]
And the total Hamiltonian of the entire field (so far we have been speaking only of small units of volumes of the fields, in terms of densities),
So far we have seen a lot of analogies from classical mechanics with the relativistic 4-dimensional coordinates (the covariant formulation) incorporated in, but as promised earlier, QFT has a lot to do with quantum mechanics too! We realized this connection when Prof. Mallesh pointed out that just as in the conventional quantum mechanics and are operators that operate on state vectors belonging to a Hilbert space; and are the operators that operate on the quantum states of the field in the so called “Fock space”. (And here begins the analogy with quantum mechanics!) Just like we have the commutation relations in QM, , we have here,
And with and defined, we can essentially construct any dynamical variable using them.
Putting all of it together
After taking us all through this wonderful glimpse of the mathematical working of QFT, Prof. Mallesh listed the essence and conclusions of the entire talk, summing up the meaning of QFT in a few sentences. He pointed out that just like particles in quantum mechanics are said to be in a ground state, with a minimum energy; fields too with minimum energy are found in a ground state. Supply energy to a ground-state particle and it gets into an excited state. Same way, a field in the ground state too can be excited by supplying it with external energy. If the energy is supplied so as to bring the field to its first excited state, that is when the field is detected as particles. Supply further energy, in the second excited state, the field manifests as two particles; and so on. Therefore, the excited states of the field manifest themselves as identical particles of a definite kind such as photons, gravitons, etc (but with different energies, momenta, polarizations, etc, in general). To every observed field is a particle associated and to every particle, a field. Therefore, there are as many kinds of fields as there are distinct particles in the universe. Prof. Mallesh ended the fascinating talk by stating that QFT has gone so far as to remove the distinction between particles and fields – everything that we observe in the universe is simply different manifestations of different fields. And so, all of Physics may be simply described as only a study of various interacting fields.
- Webpage, “QFT“, for a brief outline of the theory.
- Website, “QFT lectures“, for study material to learn QFT further.
Cover image by Pascal Bovet