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What are zero modes in quantum field theory, and what are they used for? Or, where can I read about them? I was never able to find a good introduction on the subject.

I am particularly interested in zero modes that appear in the context of Yang-Mills, QCD and QCD-like theories, at zero or finite temperature. My doubt consists on the nature of these zero modes in these theories: are they special field configurations, or quantum states, or something else? Basically, I'd like to know what's their physical significance.

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The name "zero mode" just refers to a state of zero energy. They can be interesting in QFT applied to say condensed matter as they can be Majorana states, or signals of nontrivial topological order. In what context did you hear of this term? –  j.c. Jan 15 '11 at 16:42

2 Answers 2

Moshe is of course completely right, but let me be a bit less abstract for a while and list a few examples that cover most of the contexts where the term "zero mode" is used.

When one discusses the motion of an extended object, such as the string, the position of a point along the string - the point is parameterized by $\sigma$ - is $X^\mu(\sigma)$. The "zero mode" of $X$ is then nothing else than the average over the string - the center-of-mass coordinate.

The dual quantity is the "zero mode" of the momentum - the integral of $P_\mu(\sigma)$: that's nothing else than the total momentum. These two zero modes, the center-of-mass position and the total momentum, are the quantities we would normally associate with a point-like particle. And indeed, in some approximation, the string may be considered to be a particle, and the zero modes are the usual properties of the particle.

If one solves differential equations on compact manifolds, such as the Calabi-Yau manifolds, one may find zero modes, too. For example, we may have a Dirac equation $$D^\mu \gamma_\mu \psi(x_1, \dots, x_6) = 0$$ The solutions $\psi$ to this massless Dirac equations are just zero modes. The equation above only included functions of the compact coordinates. But if you consider both 6 compact and the 3+1 non-compact coordinates, one may consider spinors $\psi$ whose dependence on the 6 compact coordinates is given by the "zero mode" - the solution of the equation above. Such a field will then behave as a massless particle in 4 dimensions - it will solve the 3+1-dimensional equation with a vanishing mass.

In both cases above, there was an operator - either $-d^2/d\sigma^2$, or the Dirac operator $D^\mu \gamma_\mu$, that annihilated the solution and that's why the solution was ultimately called a "zero mode". In this context, it's useful to mention what are the "non-zero modes" or "normal modes". They are eigenvalues of the same operators with different eigenvalues.

For example, the operator $-d^2/d\sigma^2$ on the string may act on functions such as $\exp(in\sigma)$ and it will produce $n^2$ - the same with sines and cosines. These non-zero modes are eigenstates of an operator, so if you include all possible eigenvalues and the right degeneracy, you may reconstruct any function - by Fourier expansions.

The same is true for the Dirac example. All the spinors $\psi$ on the 6-dimensional manifold may be written as a combination of the "modes", some of which are the zero modes but most of them are non-zero modes. When one discusses supersymmetry, the non-zero modes typically come in pairs and it can be proved, but the zero modes - massless particles in 3+1 dimensions, as I mentioned - may come non-paired. So the number of zero modes of the Dirac equation (more precisely the number of solutions minus the number of "anti-solutions" with some opposite charge or chirality etc.) is a special number known as the "index". It is invariant under all continuous transformations. In heterotic string theory, it is interpreted as the number of generations (minus the number of antigenerations) of leptons and quarks we obtain in 3+1 dimensions.

We may continue with solutions to Dirac and similar equations. But consider a different background, e.g. an instanton in gauge theory. And include the gauge field term to the Dirac operator. Then this total Dirac operator will again have some zero modes on $R^{3,1}$; the zero mode will be nonzero and nontrivial especially in the region where the instanton is located. Any fermionic particles in 3+1 dimensions that has a zero mode will actually appear as a factor in a complicated product of many fermionic fields - and this multi-product interaction term is actually induced by the instanton.

The zero mode is a $c$-number-valued solutions of the right differential equation with the right operator. But its real meaning is that the whole quantum field should be expanded into modes - zero modes and non-zero modes. Each term in the sum has a $c$-number valued function of the "space" or "spacetime" and it is multiplied by an operator corresponding to the mode. The operators that multiply the zero modes in the expansion play a special role.

Instead of the Dirac operator, you may consider the differential operators for other fields, including bosonic fields such as the gauge field itself. The "zero modes" of the differential equation for the bosonic fields - essentially various forms of (covariantized) wave equations - describe deformations of the background. They're "collective degrees of freedom" of the instanton. It's called the 't Hooft interaction.

Note that the description "collective degree of freedom" applies to the string as well. The center-of-mass position and the total momentum are "collective degrees of freedom" for all the string "bits" along the string; they share these collective degrees of freedom. Some of the zero modes tell you how the whole object may move in space - like in the example of the string. But the fermionic zero modes can also tell you how the whole object may move in the superspace. And the zero modes of the instanton also include some modes whose variation determines changes of the size (and/or shape) of the instanton - or any other object.

This text was only meant for the reader to have some more concrete ideas about the words by Moshe which are of course totally valid.

Best wishes Lubos

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Thank you, Lubos! It's a very clear exposition, and I really liked it. I know that zero modes are defined as zero-eigenvalue solutions to some operator equations, but I wanted to grasp their physical significance. And your description of them as being degrees of freedom that describe the important/relevant motions of some object, extended or not, helps a lot. Thanks. :) –  hjcv Jan 15 '11 at 19:10
    
It was a pleasure, hjcv. The motion of an object - identified with a localized solution etc. - is the simplest example. But for example, for the instantons that you apparently plan to study, the zero modes may also describe its size. An instanton is a solution of a typical radius, and gauge theory, which is classically scale-invariant, allows you to rescale the radius arbitrarily. The difference of the gauge field configurations for an instanton of radii $R$ and $R+\epsilon$ - divided by $\epsilon$ to get a finite function - is also a zero mode $A_\nu^0(x_\mu)$ - one responsible for the size. –  Luboš Motl Jan 15 '11 at 20:15
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Also, in superstring theory, there can be fermionic modes on the string. In the Ramond sectors, there are some periodic fermions along the strings - so one can also integrate them over the whole string to get a zero mode. By quantizing these $N$ fermions, one gets a degeneracy with $2^N$ states. If they're spacetime vectors - like the fermions in the Ramond sector - the quantization produces spacetime spinors (or, if this is done both for left-movers and right-movers, differential forms). That's where the spin of all the supergraviton massless states from from in superstring theory. –  Luboš Motl Jan 15 '11 at 21:23
    
There are also periodic fermions on strings that are spacetime scalars - for example in the heterotic string, one can write down the world sheet with 32 extra fermions. One must allow strings with periodic as well as antiperiodic fermions as well. In the periodic sector, there exist zero modes of the fermions. By quantizing them, one either gets (massive) states in the spinor rep of SO(32), or, if they're divided to 16+16 fermions and only one group is periodic, one gets 256/2=128 states that complete 120 of SO(16) to the 248-dimensional rep of E8 (two of them). –  Luboš Motl Jan 15 '11 at 21:25

Zero modes are solution to an eigenvalue equation with zero eigenvalue. They appear many times in mathematics and physics, so it is hard to know where to point you to.

Roughly, bosonic zero modes indicate degeneracy in the system, for example such that results from symmetry. If you have a configuration of some classical field, think about it as some localized lump of energy, it can oscillate with different frequencies, called the normal modes of the system. They are classified by their frequency $\omega$ (or energy, $\hbar \omega$).

Some of those normal modes are special in that they have zero frequency $\omega=0$, and then they are not oscillations at all, instead they are rigid translations of the whole system in some direction. This means there is a family of solutions, for example localized in different positions, and the zero mode corresponds to translating the configuration to a new location. This is a zero mode because translation to a new direction does not cost you energy (unlike real oscillations).

Fermionic zero modes are even more interesting, via index theorems they are related to topology and anomalies, but you probably need more background in QFT (or topology) to appreciate those.

(Lubos has more quantitative discussion, you can read both to have a more complete picture).

For reference I'd recommend Rajaraman's "Solitons and Instantons". The simple example of kink in two dimension, and its translational zero mode, should make the general pattern clear. I think there is also good discussion of fermionic zero modes, e.g. in the presence of an instanton, and their relation to topology via the Atiyah-Singer index theorem.

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I am actually interested in zero modes in Yang-Mills, QCD and QCD-like theories, at zero or finite temperature. My doubt consists on the nature of these zero modes in these theories, are they special field configurations, or quantum states, or something else? Basically, I'd like to know what's their physical significance. –  hjcv Jan 15 '11 at 18:11
    
Bosonic of Fermionic? –  user566 Jan 15 '11 at 18:23
    
Either. Because they are both called 'zero modes', I thought they could have a common physical interpretation, apart from the common definition as zero-eigenvalue solutions of some operator equation. I was aware of this definition, but that didn't tell me much about the physical significance of such modes. For example, what would be the difference, physically, between zero, normal or other modes? –  hjcv Jan 15 '11 at 18:58
    
Thank you for your answer, Moshe. :) –  hjcv Jan 15 '11 at 19:47
    
@Moshe Can you give some references to get started about the topological significance of fermionic zero modes? Also can you elaborate on the comment that bosonic zero modes indicate degeneracy. I have been asking about similar things here, physics.stackexchange.com/questions/4320/… –  user6818 Feb 1 '11 at 10:55

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