Yang-Mills equations

Yang-Mills

Description
Equation	$\,D_{\alpha }F^{\alpha \beta }=0\!$
Fields	$A_{\alpha }:\mathbb {R} ^{1+d}\to {\mathfrak {g}}$
Data class	$A_{\alpha }[0]\in H^{s}(\mathbb {R} ^{d})\times H^{s-1}(\mathbb {R} ^{d})$
Basic characteristics
Structure	Hamiltonian
Nonlinearity	semilinear with derivatives
Linear component	Wave
Critical regularity	${\dot {H}}^{d/2-1}(\mathbb {R} ^{d})$
Criticality	energy critical for d=4
Covariance	Lorentzian, gauge, conformal
Theoretical results
LWP	varies
GWP	varies
Related equations
Parent class	DNLW
Special cases	Yang-Mills on R2, R3, R4
Other related	MKG, Cubic NLW, Yang-Mills-Higgs

The Yang-Mills equations

Classical equations

Let $A$ be a connection on $R^{d+1}$ which takes values in the Lie algebra g of a compact Lie group G. Formally, the connection A is said to obey the Yang-Mills equation if it is a critical point for the Lagrangian functional

$\int F^{\alpha \beta }F_{\alpha \beta }$

where $F:=dA+[A,A]$ is the curvature of the connection $A$. The Euler-Lagrange equations for this functional have the schematic form

$\Box A+\nabla (\nabla _{x,t}A)=[A,\nabla A]+[A,[A,A]]$

where $\nabla _{x,t}A=\partial _{a}A^{a}$ is the spacetime divergence of $A$. A more succinct (but less tractable) formulation of this equation is

$\,D_{\alpha }F^{\alpha \beta }=0\!$.

It is often convenient to split $A$ into temporal and spatial components as $\,A=(A_{0},A_{i})\!$.

As written, the Yang-Mills equation is under-determined because of the gauge invariance

$\,A\rightarrow U^{-1}dU+U^{-1}AU\!$ $\,F\rightarrow U^{-1}FU\!$

in the equation, where U is an arbitrary function taking values in $G$. In order to correctly formulate a Cauchy problem, one must impose a further constraint on the gauge. There are three standard ones:

Temporal gauge: $\,A^{0}=0\!$ Coulomb gauge: $\partial _{i}A_{i}=0$ Lorenz gauge: $\nabla _{x,t}A=0$

There are also several other useful gauges, such as the Cronstrom gauge Cs1980 centered around a point in spacetime.

The Lorenz gauge has the advantage of being invariant under conformal transformations, but it appears that the Yang-Mills equation is not well-behaved in this gauge for rough data. (For smooth data one can obtain local well-posedness in this gauge by energy estimates). The Coulomb gauge is the simplest to work with technically, and in this gauge the bilinear expression $[A,\nabla A]$ acquires a null structure KlMa1995 which allows for a satisfactory analysis of the equation. Unfortunately there are often difficulties in creating a global Coulomb gauge, and one often has to rely instead on local Coulomb gauges pieced together using finite speed of propagation; see KlMa1995. The Temporal gauge is fairly close to the Coulomb gauge, and one can develop a parallel theory for this gauge. The temporal gauge has the advantage of being easy to establish globally, but the null form structure is less obvious (one needs to partition the connection into divergence-free and curl-free components). See e.g. Ta2003.

In the Coulomb or Temporal gauges, one can create a model equation for the Yang-Mills system by ignoring cubic terms and any contribution from the "elliptic" portion of the gauge ($A_{0}$ in the Coulomb gauge, or the curl-free portion of $A_{i}$ in the Temporal gauge). The resulting model equation is

$\Box A=\nabla ^{-1}Q(A,A)+Q(\nabla ^{-1}A,A)$

where $Q(A,A')$ is some null form such as

$Q(A,A'):=\partial _{i}A\partial _{j}A'-\partial _{j}A\partial _{i}A'$.

The results known for the model equation are slightly better than those known for the actual Yang-Mills or Maxwell-Klein-Gordon equations.

The Yang-Mills equations come with a positive definite conserved Hamiltonian

$\int |F_{0,i}|^{2}+|F_{i,j}|^{2}dx$

which mostly controls the $H^{1}$ norm of $A$ and the $L^{2}$ norm of $A_{t}$. However, there are some portions of the $H^{1}\times L^{2}$ norm which are not controlled by the Hamiltonian (in the Coulomb gauge, it is $\partial _{t}A_{0}$; in the Temporal gauge, it is the $H^{1}$ norm of the curl-free part of $A_{i}$). This causes some technical difficulties in the global well-posedness theory.

The Yang-Mills equations can also be coupled with a g-valued scalar field $f$, with the Lagrangian functional of the form

$\int F^{\alpha \beta }F_{\alpha \beta }+D_{\alpha }f\cdot D^{\alpha }f+V(f)$

where $D_{\alpha }:=\partial _{\alpha }+[A_{\alpha },.]$ are covariant derivatives and $V$ is some potential function (e.g. $V(f)=|f|^{k+1})$. The corresponding Euler-Lagrange equations have the schematic form

$\Box A+\nabla (\nabla _{x,t}A)=[A,\nabla A]+[A,[A,A]]+[f,Df],D_{\alpha }D^{\alpha }f=V'(f)$

and are generally known as the Yang-Mills-Higgs system of equations. This system may be thought of as a Yang-Mills equation coupled with a semi-linear wave equation. The Maxwell-Klein-Gordon system is a special case of Yang-Mills-Higgs.

The theory of Yang-Mills connections is considerably more advanced in the elliptic case (when the Minkowski metric is replaced by a Riemannian one), especially in the critical case of four dimensions, but a discussion of this topic is beyond our expertise.

Attention has mostly focused on the three and four dimensional cases; the one-dimensional case is trivial (e.g. in the temporal gauge it collapses to $A_{tt}=0$). In higher dimensions n=5,7,9 singularities can develop from large smooth radial data CaSaTv1998 (see also Biz-p). Numerics suggest this phenomenon is generic, and also one appears to have blowup also at the critical dimension BizTb2001, Biz-p.

The Yang-Mills equations can also be coupled with a spinor field. In the $U(1)$ case this becomes the Maxwell-Dirac equation.

The Yang-Mills equations in dimension n have many formal similarities with the wave maps equation at dimension d-2 (see e.g. CaSaTv1998 for a discussion).

Yang-Mills on $R^{2}$

• Scaling is $s_{c}=0$.
• One can use the method of descent and finite speed of propagation to infer R² results from the R^3 results. Thus, for instance, one has LWP for s > 3/4 in the temporal gauge and GWP in the temporal gauge for $s\geq 1$. These results are almost certainly non-optimal, however, and probably have much simpler proofs (for instance, one can obtain the LWP result from the general theory of DNLW without using any null form structure).

Yang-Mills on R³

• Scaling is s_c = 1/2.
• LWP for s > 3/4 in the Temporal gauge if the norm is sufficiently small Ta2003. The main tools are bilinear estimates involving both $X^{s,\theta }$ spaces and product Sobolev spaces.
  • Presumably the small data assumption can be removed, but the usual methods to do this fail because there are too many time derivatives in the non-linearity in the temporal gauge.
  • For $s\geq 1$in the Temporal or Coulomb gauges LWP for large data was shown in KlMa1995.
  • For s > 1 LWP for the Temporal, Coulomb, or Lorenz gauges follows from Strichartz estimates PoSi1993.
  • For s > 3/2 LWP for the Temporal, Coulomb, or Lorenz gauges follows from energy estimates EaMc1982.
  • There is a tentative conjecture that one in fact has ill-posedness in the energy class for the Lorenz gauge.
  • For the model equation LWP fails for s < 3/4 MaStz-p
  • The endpoint s = 1/2 looks extremely difficult, even for Besov space variants.
• GWP is known for data with finite Hamiltonian (morally, this is for $s\geq 1$) in the Coloumb or Temporal gauges KlMa1995.
  • For smooth data this was proven in EaMc1982.
    • This result was extended to curved space in CcSa1997
  • It seems likely that one can improve this to something like s>7/8, in analogy with the theory for the Maxwell-Klein-Gordon equation.

MKG and Yang-Mills in R⁴

• Scaling is s_c = 1.
• For the MKG equations in the Coulomb gauge, LWP is known for s > 1 Sb-p5. This is still not known for Yang-Mills.
  • For the model equations this is in KlTt1999
    • For general quadratic DNLW this is only known for s > 5/4 (e.g. by the estimates in FcKl2000). Strichartz estimates need s > 3/2 PoSi1993, while energy estimates need s > 2.
  • The latter two results (Strichartz and energy) easily extend to the actual MKG and YM equations in all three standard gauges.
• It is conjectured that one has global well-posedness results for small energy, but this is open.
  • For small smooth compactly supported data, one can obtain global existence from the general theory of quasi-linear equations.
  • For large data Yang-Mills, numerics suggest that blowup does occur, with the solution resembling a rescaled instanton at each time BizTb2001, Biz-p.
    • Further numerics suggests that the radius of the instanton in fact decays like $Ct/{\sqrt {\log t}}$ BizOvSi-p.
  • GWP for small $B^{1,1}$ data (with an additional angular derivative of regularity) in the Lorenz gauge is in Stz-p2.

MKG and Yang-Mills in R^d, d>4

• Scaling is s_c = d/2 - 1.
• LWP is almost certainly true for MKG-CG for s > s_c by adapting the results in Sb-p5. The corresponding question for Yang-Mills is still open.
  • For the model equations one can probably achieve this by adapting the results in Tt1999
• For dimensions $d\geq 6$, GWP for small H^{d/2} data in MKG-CG is in RoTa-p. The corresponding question for Yang-Mills is still open, but a Besov result follows (in the Lorenz gauge) from Stz-p3.

Yang-Mills-Higgs on R³

• Suppose the potential energy V( f ) behaves like $|f|^{p+1}$ (i.e. defocussing p^th power non-linearity). When $p\leq 3$, the Higgs term is negligible, and the theory mimics that of the ordinary Yang-Mills equation. The most interesting case is p=5, since the Higgs component is then H^1-critical.
• There is no perfect scale-invariance to this equation (unless p=3); the critical regularity is $s_{c}=max(1/2,3/2-2/(p-1))$.
• In the sub-critical case p<5 one has GWP for smooth data EaMc1982, GiVl1982b. This can be pushed to H^1 by the results in Ke1997. The local theory might be pushed even further.
• In the critical case p=5 one has GWP for $s\geq 1$ Ke1997.
• In the supercritical case p>5 one probably has LWP for $s\geq s_{c}$(because this is true for the Yang-Mills and NLW equations separately), but this has not been rigorously shown. No large data global results are known, but this is also true for the supposedly simpler supercritical NLW. It seems possible however that one could obtain small-data GWP results.

Quantum equations

Differently from the classical case, the corresponding quantum formulation has not proved to exist yet. The main difficulties rely on the gauge invariance that makes already an Euclidean formulation, eventually manageable through Wiener path integrals, at best problematic. From a physical standpoint, one ignores this kind of difficulties and puts down a quantum field theory exploiting it through a small perturbation theory in the coupling, limiting in this way the analysis at higher momenta (smaller distances). This approach has been proved frutiful in understanding the phenomenolgy observed at laboratory facilities and using lattice computations with computers. On the other side, at small momenta (larger distances), difficulties are overwhelming great making best suitable a numerical approach on the lattice. So, also non rigorous methods seem to fail to give a clever understanding of the situation for this limit. Then, in what follows, we just give a formal presentation of material having in view the idea that, through this, fundamental questions like the existence of the mass gap or existence of the theory itself could finally be proved. We will use as a reference FaSl1980.

Hamiltonian formulation

Yang-Mills equations can be stated into a Hamiltonian form. The canonical variables are $(E^a_k,A^b_l)$ with $a,b$ running on the Lie group index and $k,l$ enumerating the spatial coordinates. One has

$$ E_k=F_{k0} $$ $$ B_k=-\frac{1}{2}\epsilon_{ijk}F_{ij} $$ $$ C = DE $$

and the Hamiltonian can easily be written down as

$$ H=\int d^{D-1}x{\rm Tr}(E^2+B^2) $$

The dynamics has a constraint and we need a gauge condition to get the equations of motion. This can also be seen in the corresponding Lagrangian formulation, after a Legendre transform, where a Lagrange multiplier indeed appears.

Now, one can write down the corresponding Poisson brackets obtaining

$$\{E_k^a(x),A_l^b(y)\}=\delta_{kl}\delta_{ab}\delta(x-y) $$ $$\{C^a(x),C^b(y)\}=f^{abc}C^c(x)\delta(x-y) $$

being $f^{abc}$ the structure constants of the Lie group.

Heisenberg equations

In order to obtain the quantum dynamics we can use Heisenberg equations of motion

$$\partial_0E_k^a=i[H,E_k^a]$$ $$\partial_0A_l^b=i[H,A_l^b]$$

and the set of commuting relations obtained from the Poisson brackets through Dirac quantization

$$[E_k^a(x),A_l^b(y)]=\delta_{kl}\delta_{ab}\delta(x-y)$$

$$[E_k^a(x),E_l^b(y)]=0,\ [A_k^a(x),A_l^b(y)]=0$$

$$[C^a(x),C^b(y)]=f^{abc}C^c(x)\delta(x-y).$$

A further condition to fix the gauge is also needed.

We note that these are operatorial equations and so we need to build a proper state space to give them a meaning. Presently, a rigorous proof of existence of all this construction does not exist yet and would be a proof of existence for the Yang-Mills theory itself. The best one can do is to perform a small perturbation theory of these equations on a Fock space built on the solutions of the leading order equations. Also this construction has not a rigorous proof but is common practice between physics community with a considerable success. But, in order to perform such kind of computations, a different approach is used that heavily relies on path integration. No sound foundation for path integrals in Minkowski spaces exists yet.