Mass gap

Short description: Energy difference between ground state and lightest excited state(s)

Quantum field theory

Feynman diagram

History

Background

* Field theory
* Electromagnetism
* Weak force
* Strong force
* Quantum mechanics
* Special relativity
* General relativity
* Gauge theory

Symmetries

* Symmetry in quantum mechanics
* C-symmetry
* P-symmetry
* T-symmetry
* Space translation symmetry
* Time translation symmetry
* Rotation symmetry
* Lorentz symmetry
* Poincaré symmetry
* Gauge symmetry
* Explicit symmetry breaking
* Spontaneous symmetry breaking
* Yang–Mills theory
* Noether charge
* Topological charge

Tools

* Anomaly
* Crossing
* Effective field theory
* Expectation value
* Ghost fields
* Faddeev–Popov ghosts
* Feynman diagram
* Lattice gauge theory
* LSZ reduction formula
* Partition function
* Propagator
* Quantization
* Regularization
* Renormalization
* Vacuum state
* Wick's theorem
* Wightman axioms
* Feynman parametrization

Equations

* Dirac equation
* Klein–Gordon equation
* Proca equations
* Wheeler–DeWitt equation
* Bargmann–Wigner equations
* Joos–Weinberg equation
* Weyl equation

Standard Model

* Quantum electrodynamics
* Electroweak interaction
* Quantum chromodynamics
* Higgs mechanism

Incomplete theories

* Topological quantum field theory
* String theory
* Superstring theory
* M-Theory
* Supersymmetry
* Supergravity
* Technicolor
* Theory of everything
* Quantum gravity

Scientists

* C. D. Anderson
* P. W. Anderson
* Bethe
* Bjorken
* Bogoliubov
* Brout
* Callan
* Coleman
* DeWitt
* Dirac
* Dyson
* Englert
* Fermi
* Feynman
* Fierz
* Fock
* Fröhlich
* Glashow
* Gell-Mann
* Gross
* Guralnik
* Heisenberg
* Higgs
* Haag
* Hagen
* 't Hooft
* Jordan
* Kendall
* Kibble
* Lamb
* Landau
* Lee
* Majorana
* Mills
* Nambu
* Nishijima
* Parisi
* Polyakov
* Salam
* Schwinger
* Skyrme
* Sudarshan
* Tomonaga
* Veltman
* Ward
* Weinberg
* Weisskopf
* Weyl
* Wilczek
* Wilson
* Yang
* Yukawa

* v
* t
* e

For the mass gaps of stellar-mass black holes, see Stellar_black_hole § Mass_gaps.

In quantum field theory, the mass gap is the difference in energy between the lowest energy state, the vacuum, and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle.

Since the energies of exact (i.e. nonperturbative) energy eigenstates are spread out and therefore are not technically eigenstates, a more precise definition is that the mass gap is the greatest lower bound of the energy of any state which is orthogonal to the vacuum.

The analog of a mass gap in many-body physics on a discrete lattice arises from a gapped Hamiltonian.

## Contents

* 1 Mathematical definitions
* 2 Examples from classical theories
* 3 Yang–Mills theory
* 4 Källén–Lehmann representation
* 5 See also
* 6 References
* 7 External links

## Mathematical definitions

For a given real-valued quantum field ϕ(x), where x=(x,t), we can say that the theory has a mass gap if the two-point function has the property

⟨ϕ(0,t)ϕ(0,0)⟩∼∑nAnexp⁡(−Δnt)

with Δ0>0 being the lowest energy value in the spectrum of the Hamiltonian and thus the mass gap. This quantity, easy to generalize to other fields, is what is generally measured in lattice computations. It was proved in this way that Yang–Mills theory develops a mass gap on a lattice.[1][2] The corresponding time-ordered value, the propagator, will have the property

limp->0Δ(p)=constant

with the constant being finite. A typical example is offered by a free massive particle and, in this case, the constant has the value 1/m2. In the same limit, the propagator for a massless particle is singular.

## Examples from classical theories

An example of mass gap arising for massless theories, already at the classical level, can be seen in spontaneous breaking of symmetry or the Higgs mechanism. In the former case, one has to cope with the appearance of massless excitations, Goldstone bosons, that are removed in the latter case due to gauge freedom. Quantization preserves this gauge freedom property.

A quartic massless scalar field theory develops a mass gap already at classical level[clarification needed]. Consider the equation

◻ϕ+λϕ3=0.

This equation has the exact solution

ϕ(x)=μ(2λ)14sn(p⋅x+θ,−1) —where μ and θ are integration constants, and sn is a Jacobi elliptic function—provided

p2=μ2λ2.

At the classical level, a mass gap appears while, at quantum level, one has a tower of excitations, and this property of the theory is preserved after quantization in the limit of momenta going to zero.[3]

## Yang–Mills theory

Main articles: Yang–Mills theory and Yang–Mills existence and mass gap

While lattice computations have suggested that Yang–Mills theory indeed has a mass gap and a tower of excitations, a theoretical proof is still missing. This is one of the Clay Institute Millennium problems and it remains an open problem. Such states for Yang–Mills theory should be physical states, named glueballs, and should be observable in the laboratory.

## Källén–Lehmann representation

If Källén–Lehmann spectral representation holds, at this stage we exclude gauge theories, the spectral density function can take a very simple form with a discrete spectrum starting with a mass gap

ρ(μ2)=∑n=1NZnδ(μ2−mn2)+ρc(μ2)

being ρc(μ2) the contribution from multi-particle part of the spectrum. In this case, the propagator will take the simple form

Δ(p)=∑n=1NZnp2−mn2+iϵ+∫4mN2∞dμ2ρc(μ2)1p2−μ2+iϵ

being 4mN2 approximatively the starting point of the multi-particle sector. Now, using the fact that

∫0∞dμ2ρ(μ2)=1

we arrive at the following conclusion for the constants in the spectral density

1=∑n=1NZn+∫0∞dμ2ρc(μ2).

This could not be true in a gauge theory. Rather it must be proved that a Källén–Lehmann representation for the propagator holds also for this case. Absence of multi-particle contributions implies that the theory is trivial, as no bound states appear in the theory and so there is no interaction, even if the theory has a mass gap. In this case we have immediately the propagator just setting ρc(μ2)=0 in the formulas above.

## See also

* Coleman–Mandula theorem
* Scalar field theory

## References

1. ↑ Lucini, Biagio; Teper, Michael; Wenger, Urs (2004). "Glueballs and k-strings in SU(N) gauge theories: calculations with improved operators". Journal of High Energy Physics 0406 (6): 012. doi:10.1088/1126-6708/2004/06/012. Bibcode: 2004JHEP...06..012L. .
2. ↑ Chen, Y.; Alexandru, A.; Dong, S. J.; Draper, T.; Horvath, I.; Lee, F. X.; Liu, K. F.; Mathur, N. et al. (2006). "Glueball Spectrum and Matrix Elements on Anisotropic Lattices". Physical Review D 73 (1). doi:10.1103/PhysRevD.73.014516. Bibcode: 2006PhRvD..73a4516C. .
3. ↑ Frasca, Marco (2006). "Strongly coupled quantum field theory". Physical Review D 73 (2). doi:10.1103/PhysRevD.73.027701. Bibcode: 2006PhRvD..73b7701F.

## External links

* Sadun, Lorenzo. Yang-Mills and the Mass Gap. Video lecture outlining the nature of the mass gap problem within the Yang-Mills formulation.
* Mass gaps for scalar field theories on Dispersive Wiki
* "Witten talk on the mass gap problem in 3D quantum Yang-Mills theory". Francis Villatoro. August 1, 2012. https://www.youtube.com/watch?v=KHNvuSNV09Q.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Mass gap. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Physics:Mass_gap&oldid=4184707"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template