Of Particular Significance

7a. How Bosons and Fermions Differ

© Matt Strassler [September 12, 2012]

I earlier wrote a simple descriptive and qualitative article about fermions and bosons that you might like to read. Here, I aim to follow up on the article about why particles are quanta.  The qualitative facts that you need to know is:

  • All elementary particles are either fermions or bosons
  • Fermions (including electrons, quarks and neutrinos) satisfy the Pauli exclusion principle: no two fermions of the same type can ever be doing exactly the same thing.
  • Bosons (including photons, W and Z particles, gluons, gravitons and Higgs particles) are different: two (or more) bosons of the same type may indeed do the same thing.

This last point is why lasers can be made from photons, which are bosons and can all do the same thing in lockstep, making a powerful narrow one-color beam, but not from electrons, which are fermions.

How does this distinction manifest itself in the language of the Particles and Fields articles of which this one is a pat? It turns out that all the formulas I’ve been giving you are true for bosons, and need a very small but very consequential modification for fermions. For bosons we had

  • E = (n + 1/2) h ν, where n = 0,1,2,3,4,…

which means each quantum has energy h ν. That implies that boson quanta can be made to do exactly the same thing; when n is greater than 1, the boson field has a wave made from more than one quantum which are oscillating and moving in lock-step. But for fermions

  • E = (n 1/2) h ν, where n = 0 or 1.

It is still true that each quantum has energy h ν, so all of the discussion about particles and their energies, momenta and masses is still correct. But the number of quanta of an electron wave can only be zero or one; ten electrons, unlike ten photons, cannot be organized into a wave of larger amplitude. So there are no fermion waves that are made from large numbers of fermions oscillating and moving in lock-step.

[And then there’s that interesting minus sign in the zero-point energy for fermions, compared to bosons. Zero-point energy, and the two different signs, are not important in this set of articles, but will be important in other sets later.]

17 Responses

  1. Hi, Matt. I was wondering if you had any information on how invariant-mass matter excludes (at least partially) the quantum vacuum field modes longer than the radius of that invariant-mass matter (the effect which causes a gravity ‘well’, which causes time to slow down in a gravity ‘well’, and indeed the effect which underlies the stability of invariant-mass matter) via vacuum polarization? I’m trying to wrap my brain around how fermions can exclude bosons, even if only partially. Is there an analogy to some physical process which would help in removing my brain-block?

    Is it a similar concept to the Pauli Exclusion Principle, or is there another principle which explains this phenomenon besides SR?

  2. Thanks so much for these articles! Feel like I’m missing something blindingly obvious, or maybe this is what the last [bracketed] paragraph refers to… but if the number of quanta of oscillation in a wave is 0 [n=0], shouldn’t that mean there is no wave? The formulas above, though, seem to suggest that a Class 0 wave with no quanta will still have energy equal to (1/2) h ν. Also that a Class 1 wave will actually have negative energy.

    Thanks again,

  3. Lasers or Bose-Einstein condensates imply a sophisticated technical procedure….
    Hence the question :do we have some examples of “spontaneous bosonic unanimity” in nature ?

  4. I see. Thank you for your answer, Dr. Strassler. So, between an electron and a positron there is no effect (no repulsive “nor attractive force”?) whatsoever due to the Pauli exclusion principle? And, the Pauli exclusion principle applies between an up quark and an up quark but NOT between up and down quark etc.(does this mean the distance btw u&u and u&d quark inside a proton is different due to the Pauli exclusion principle?)? And, no physical phenomena can violate/overcome the Pauli exclusion principle? such as the pressure created by supernova/big bang or electrons fell into a black hole? Electrons inside a black hole still cannot occupy the same state/position even under the enormous(infinite?) pressure due to gravity? Electrons or the exactly the same fermions cannot be crushed nor put together(superimposed) even at the singularity(if this exists)?? First of all how does the Pauli exclusion principle arises? why it only applies to “fermions of exactly the same type”? but not fermions of different types?

    1. At some level, these are inevitably technical questions, for which math is necessary… but I’ll do my best to answer non-technically.

      To answer your last questions first: the Pauli principle (like the Bose-Einstein principle for bosons) is not due to a force, but rather a consequence of the nature of the things themselves. If you try to make a state with two electrons doing the same thing, you will find that state equals zero; it’s not a state at all. So no physical phenomenon can violate the Pauli principle; no amount of pressure and gravity can change it.

      Why does it only impact fermions of the same type? Analogy: If two waves of similar shape collide, and two waves emerge from the collision, did the two waves pass through each other or did they bounce off each other? There’s no way to tell. When two identical particles collide, you cannot tell after the collision which one is which.

      Before X –> <– X

      After <– X X –>
      – – – (which X is which???)

      If the two particles were not identical to start with, you could tell the difference:

      Before: X –> <– Y

      After: <– X Y –> is different from After: <– Y X –>

      This potential ambiguity of which one is which is the key to the story, and that’s why the interesting properties of bosons and of fermions are only important for two bosons of the same type and for two fermions of the same type.

      All fundamental particles of nature are (or, more precisely, “are today described as”) ripples in quantum fields. And two ripples in the same field cannot be distinguished. In a world with three or more space dimensions, there are two types of fields; ripples in boson fields (like photons) must be symmetrically related, whereas ripples in fermion fields must be antisymmetrically related. No force is required to make this so; it is in the nature of the ripples. Since antisymmetrically related objects cannot be in the same place doing the same thing, the Pauli exclusion principle applies to them.

      Yes, this applies to up quarks, and separately, to down quarks. In fact the Pauli exclusion principle helps explain why, even though there are protons with two up quarks and a down quark (plus many other things, see http://profmattstrassler.com/articles-and-posts/largehadroncolliderfaq/whats-a-proton-anyway/) and there are neutrons with two down quarks and an up quark (plus many other things), there are no corresponding objects with three up quarks (plus many other things) or with three down quarks (plus many other things.) Objects with three up quarks can be formed, but they require the up quarks be doing three different things, which requires more energy, which means in turn that they have larger masses and are unstable, decaying in about a trillionth of a trillionth of a second.

      1. Your last paragraph here made so many things astonishingly clear. I’m reading the articles and comments as I go along, I hope this paragraph is expounded upon elsewhere. I’m sure there’s more to it than that.

      2. To Dr. Strassler: Thank you for your interesting answers/explanations.
        You said “This potential ambiguity of which one is which is the key to the story, and that’s why the interesting properties of bosons and of fermions are only important for two bosons of the same type and for two fermions of the same type.” But why is this applicable to fermions and bosons of the same types but not to the identical ordinary objects like balls (assuming they have identical structures at atomic level and molecule by molecule)? because no matter how carefully made, an objects/system made of multiple fundamental particles cannot be exactly the same (cannot have exactly the same state)? or is it something to do with the size or mass of the object? Only fundamental particles (without internal/sub/ structure?) can (behave) be indistinguishably identical in physical phenomena? A proton has internal structure but is it always possible to distinguish proton A and B before/after their collision event etc.?

        P.S. sometime ago, I read you would write articles on the current “standard model (=partial grand unified theory(GUT)?)” Are some of the articles already posted in this site? (if so, could you point the links.)

  5. How does Pauli exclusion principle apply to a pair of an electron and a positron just about to collide? Do they keep feeling repulsive force due to Pauli exclusion principle until just before they collide and turn into a pair of two photons? First of all how can they merge(?) if Pauli exclusion principle prohibits two fermions to occupy the same position/state? and what happens to the angular momentum(=spin) they have? Do an electron and a positron always collide spin up or down alignment (head-to-head? or tail-to-tail? collision) to cancel out(conserve) total spin?

    1. The Pauli exclusion principle does not apply to an electron and a positron. It only applies to two electrons, or to two positrons — generally, to any two fermions of exactly the same type.

    1. Hey, there’s no harm in treating a theoretical particle as if it were confirmed already, it did wonders for the higgs and neutrino.

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