The Higgs Story-Part 6: Relativistic quantum fields

In the previous post in this series, I introduced the idea of fields and also said that while the Schrodinger equation and wave function overcame some of the problems with understanding how particles could also have wave properties, there were still difficulties in both interpretation and practice. The person who made the next major advance and created the framework for our present understanding of all matter was Paul Dirac (1902-1984) whose views on religion I wrote about over the weekend.

Dirac attempted to find a better equation than the Schrodinger equation that would at least make quantum mechanics consistent with relativity. What he came up with ended up succeeding much more than that. What is now known as the Dirac equation that he published in 1928 not only solved the relativity problem, it also solved the problem of the creation and destruction of particles and predicted the existence of anti-particles. The Dirac approach laid the groundwork for the development of a theory that unifies particles and waves in such an elegant way that it forms the basis for our current understanding of elementary particle physics.

Just as the Schrodinger equation was solved to get a wave function that was interpreted as the probability amplitude of a particle being found in a particular region of space, the solution of the Dirac equation for (say) an electron gives us something that also looks like a wave function but differs in a significant way. What it introduced was something that we call the relativistic quantum field.

The difference of those fields with the solutions of the Schrodinger equation is that the Dirac fields for elementary particles are considered fundamental. In other words, these fields are not made up of anything else nor are they descriptors of particles, that tell us how they behave. Instead there is an emerging consensus that these relativistic quantum fields are what everything in the world is made of. In other words, there are no particles, there are only fields and these fields exist throughout space all the time.

As far as the Standard Model goes, the 19 fields associated with the 19 elementary particles that we described in Part 3 (six quarks, six leptons, six force particles (photon, graviton, gluon, W+, W, Z) and the Higgs boson) are what everything in the universe is made up of. (The wave function obtained by solving the Schrodinger equation is also a quantum field since it too deals with the quantum world and has values over all space but the fact that it is not relativistic prevents it from having the additional properties that would enable it to play the role of a truly fundamental field.)

The theory of relativistic quantum fields can deal with ‘particles’ (which are really fields) with mass and without mass, is consistent with relativity, enables systems to exhibit both wave and particle properties, and allows for particles to interact with each other so that there can be changes in the number of particles and for them transform from one type of particle to another.

The relativistic quantum fields are extended objects, like waves. These fields interact with each other and as a result re-arrange themselves depending on those interactions. On occasion, these interactions will result in the fields clumping up in one small region of space, like a spike, and it is these clumps that we identify as particles. But unlike classical particles, which have a definite boundary, these fields do not. So even an electron field extends itself over all space, though it may be mostly clustered in one small region that we identify as its location. We can and do still use the word ‘particle’ provided we understand that it refers to a field that is localized to be largely (but not entirely) concentrated in a small region of space.

In the vacuum, these fields are in their lowest energy (ground) states. Vibrations of these fields correspond to excitations in energy that correspond to the number of particles present. So a one-electron state corresponds to an excitation of the electron field above its ground state. A two-electron state will correspond to an even higher excitation of the electron field. The arrangement of this excited field corresponds to the physical reality being described. So a two-electron field will have two peaks whose locations correspond to the locations of the electrons.

As an example of how you can visualize reactions using the idea of fields, suppose we consider the process in which a muon decays into an electron, an electron antineutrino, and a muon neutrino. So the initial state consists of a muon only. That means that the muon field is in an excited state such that it is peaked around the region where we say the muon ‘is’ (though it is technically everywhere) while the other three fields are in their lowest states. When the decay occurs, the muon field goes into its lowest state (i.e., the muon seems to disappear) and transfers its energy to the electron field, the electron antineutrino field, and the muon neutrino field. These three fields are now in excited states corresponding to each having one particle.

To sum up, the basic idea is that the 19 elementary particles each have a relativistic quantum field associated with them. In the vacuum, these fields are all in their ground, or lowest energy, states. Excitations of each of these fields correspond to the presence of particles, with more energy meaning more particles. Reactions and decays correspond to energy being transferred from some fields to others, corresponding to the disappearance of some particles and the appearance of others.

One important result to bear in mind is that the energy of these relativistic quantum fields in the vacuum is not zero. That is not allowed by Heisenberg’s uncertainty principle. Even in the vacuum, each field has what is known as zero-point energy so the quantum vacuum has a non-zero energy and many fields, even though it contains no particles at all. This is a big change from the classical idea of a vacuum as containing nothing at all. The quantum vacuum is quite a dynamic place, in which the fields could still interact producing ‘virtual’ particles that live briefly before disappearing again and, on occasion, even spawning new universes.

The nature of this quantum vacuum and whether it could be appropriately called ‘nothing’ is what lay at the heart of the disagreement between Lawrence Krauss and David Albert that I wrote about earlier, where the latter challenged the former’s claim that we now could answer the age-old question of how we could have arrived at ‘something’ (i.e., the current universe) while starting from ‘nothing’. Krauss’s ‘nothing’ was the quantum vacuum, while Albert argued that that begged the question, and that Krauss needed to start from the classical vacuum and first address the question of how the quantum vacuum emerged from the classical vacuum of pure nothingness.

I will not be getting into the weeds of that debate, but will continue the Higgs story using the quantum vacuum and its fields as the starting point.

Next: How fields behave.


  1. Rob Grigjanis says

    Krauss’s ‘nothing’ was the quantum vacuum, while Albert argued that that begged the question, and that Krauss needed to start from the classical vacuum…

    Hadn’t seen this kerfuffle before, and just read the Albert review. Sorry, OT, but I don’t think this is what Albert was saying. His point, I think, was that the quantum vacuum is a concept that arises from our current best picture of the universe (relativistic quantum fields), and that Krauss is going a step too far by claiming, in some sense, that the picture is the universe. And Dawkins’ afterword is just embarrassingly awful.

  2. One Day Soon I Shall Invent A Funny Login says

    It would be fun if you could write a sidebar addressing “tunneling” electronic devices, which exploit the quantum view of particles in macro devices for practical purposes.

  3. Mano Singham says

    Unfortunately that would be wandering quite far from the main story line of this series.

  4. invivoMark says

    I’m convinced that you made the post with the long quote from Dirac on religion just so this story could have a likeable protagonist. 🙂

  5. Mano Singham says

    Not quite. But it always helps in reading about abstract ideas to put some flesh on the people behind them. The general public knows quite a lot about some, like Einstein, but Dirac is not that well known outside of physics.

  6. MNb says

    Hmmm, I can totally see how a smart theologian can explain that Gen. 1:3 “let there be light” should be understood as “let there be relativistic quantumfields with non-zero energy”. Not that I buy it, but I wouldn’t know what to respond.
    So Krauss is wrong, I’m afraid.
    Maybe in the not too far future (ie after these series) you could devote an article about this subject?

  7. Mano Singham says

    It is close, if you reverse the analogy.

    In the sand speaker, there are actually particles of sand and the patterns they form as they move around look like waves.

    In contrast, in this case there are only waves (that are waves in space not of any medium) and when they happen to bunch up they look like particles.

  8. Glenn says

    I have read, reread, and enjoyed David Albert’s “Quantum Mechanics and Experience”, and, to his credit, did not find any supernatural references in it at all.

    I have read Lawrence Krauss’s book, also.

    I enjoyed both books. My universe is richer because of these books.

    Before science, mankind occupied a world dominated by supernatural beliefs. The natural has pushed the supernatural into ever more remote corners of the natural universe.

    Religion (re ligare), the rebinding of, and the formation of in-groups and out-groups, with their associated violence, have been the scourge of mankind.

    “Why Religion is Natural and Science is Not”, by Robert N. McCauley is also a book worth reading. It is natural for mankind to believe in the supernatural, but belief in the supernatural doesn’t prove its existence. It is simply human nature that makes it very easy for most people to wander from the counterintuitive ideas of science when responding intuitively to the world around them.

  9. Jeffrey Johnson says

    This is very exciting and interesting stuff. There is a minor confusion for me though.

    For each field type, say the electron field, is there only one electron field in the Universe that contains all electrons? Or is there a field for each electron or system of electrons, like all the electrons localized in a conductor, say? Or is the difference between these two just calculational convenience?

    The different fields, say the photon and electron fields, how do they overlap in space? Does every point of space contain 19 fields? Or are they somehow stacked in 4 or more dimensions, the way we might think of multiple planes being stacked in 3 dimensions if they were all near enough one another to be interacting at each point in a 2d space?

    Is there any more detail on how we think of an “excitation”? It seems unlikely it is simply a vibration. Do we just think of it as the presence of energy?

  10. Mano Singham says

    To your first question, Rob Grigjanis has already provided the answer in response to a similar question.

    As to your second question, yes they can and will overlap in space so that you have 19 fields at the same point but how they do so will depend on the nature of the interactions between the fields that cause the fields to rearrange appropriately. If the interactions of two particles forbid them from being in the same point at the same time, then only one of them will be in an excited state while the other will be in the ground state (i.e., no excitations.) You do not have to go into extra-dimensional space, at least not at this level of understanding.

    As to the third question, you can think of an increase in energy as an increase in the vibrations, as is the case for classical waves.

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