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An overview of the kaon physics program of the RBC and UKQCD collaborations will be presented with a focus on the lattice calculation of K → ππ decay and the direct CP violation parameter ε ′. We will describe substantial improvements to our earlier 2015 K → ππ calculation, including the use of three independent ππ interpolating operators and the results we obtain for I = 0 ππ scattering for energies at and below the kaon mass. While the new result for ϵ ′ is not yet complete, the enhanced statistics and improved analysis that underlies our expanded calculation will be presented.

We report the first lattice QCD calculation of the complex kaon decay amplitude A_{0} with physical kinematics, using a 32³×64 lattice volume and a single lattice spacing a, with 1/a=1.3784(68) GeV. We find Re(A_{0})=4.66(1.00)(1.26)×10(-7) GeV and Im(A_{0})=-1.90(1.23)(1.08)×10(-11) GeV, where the first error is statistical and the second systematic. The first value is in approximate agreement with the experimental result: Re(A_{0})=3.3201(18)×10(-7) GeV, while the second can be used to compute the direct CP-violating ratio Re(ϵ^{'}/ϵ)=1.38(5.15)(4.59)×10^{-4}, which is 2.1σ below the experimental value 16.6(2.3)×10(-4). The real part of A_{0} is CP conserving and serves as a test of our method while the result for Re(ϵ^{'}/ϵ) provides a new test of the standard model theory of CP violation, one which can be made more accurate with increasing computer capability.

We present results for several light hadronic quantities ($f_\pi$, $f_K$, $B_K$, $m_{ud}$, $m_s$, $t_0^{1/2}$, $w_0$) obtained from simulations of 2+1 flavor domain wall lattice QCD with large physical volumes and nearly-physical pion masses at two lattice spacings. We perform a short, O(3)%, extrapolation in pion mass to the physical values by combining our new data in a simultaneous chiral/continuum `global fit' with a number of other ensembles with heavier pion masses. We use the physical values of $m_\pi$, $m_K$ and $m_\Omega$ to determine the two quark masses and the scale - all other quantities are outputs from our simulations. We obtain results with sub-percent statistical errors and negligible chiral and finite-volume systematics for these light hadronic quantities, including: $f_\pi$ = 130.2(9) MeV; $f_K$ = 155.5(8) MeV; the average up/down quark mass and strange quark mass in the $\bar {\rm MS}$ scheme at 3 GeV, 2.997(49) and 81.64(1.17) MeV respectively; and the neutral kaon mixing parameter, $B_K$, in the RGI scheme, 0.750(15) and the $\bar{\rm MS}$ scheme at 3 GeV, 0.530(11)., Comment: 131 pages, 30 figures. Updated to match published version

We report on the first complete calculation of the K_{L}-K_{S} mass difference, ΔM_{K}, using lattice QCD. The calculation is performed on a 2+1 flavor, domain wall fermion ensemble with a 330 MeV pion mass and a 575 MeV kaon mass. We use a quenched charm quark with a 949 MeV mass to implement Glashow-Iliopoulos-Maiani cancellation. For these heavier-than-physical particle masses, we obtain ΔM_{K}=3.19(41)(96)×10^{-12} MeV, quite similar to the experimental value. Here the first error is statistical, and the second is an estimate of the systematic discretization error. An interesting aspect of this calculation is the importance of the disconnected diagrams, a dramatic failure of the Okubo-Zweig-Iizuka rule.

We develop and demonstrate techniques needed to compute the long distance contribution to the ${K}_{L}\ensuremath{-}{K}_{S}$ mass difference, $\ensuremath{\Delta}{M}_{K}$, in lattice QCD and carry out a first, exploratory calculation of this fundamental quantity. The calculation is performed on $2+1$ flavor, domain wall fermion, ${16}^{3}\ifmmode\times\else\texttimes\fi{}32$ configurations with a 421 MeV pion mass and an inverse lattice spacing $1/a=1.73\text{ }\text{ }\mathrm{GeV}$. We include only current-current operators and drop all disconnected and double penguin diagrams. The short distance part of the mass difference in a $2+1$ flavor calculation contains a quadratic divergence cut off by the lattice spacing. Here, this quadratic divergence is eliminated through the Glashow-Iliopoulos-Maiani mechanism by introducing a valence charm quark. The inclusion of the charm quark makes the complete calculation accessible to lattice methods provided the discretization errors associated with the charm quark can be controlled. The long distance effects are discussed for each parity channel separately. While we can see a clear signal in the parity odd channel, the signal to noise ratio in the parity even channel is exponentially decreasing as the separation between the two weak operators increases. We obtain a mass difference $\ensuremath{\Delta}{M}_{K}$ which ranges from $6.58(30)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}\text{ }\text{ }\mathrm{MeV}$ to $11.89(81)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}\text{ }\text{ }\mathrm{MeV}$ for kaon masses varying from 563 to 839 MeV. Extensions of these methods are proposed which promise accurate results for both $\ensuremath{\Delta}{M}_{K}$ and ${ϵ}_{K}$, including long distance effects.

There has been much speculation as to the origin of the ΔI=1/2 rule (ReA0/ReA2≃22.5). We find that the two dominant contributions to the ΔI=3/2, K→ππ correlation functions have opposite signs, leading to a significant cancelation. This partial cancelation occurs in our computation of ReA2 with physical quark masses and kinematics (where we reproduce the experimental value of A2) and also for heavier pions at threshold. For ReA0, although we do not have results at physical kinematics, we do have results for pions at zero momentum with mπ≃420 MeV [ReA0/ReA2=9.1(2.1)] and mπ≃330 MeV [ReA0/ReA2=12.0(1.7)]. The contributions which partially cancel in ReA2 are also the largest ones in ReA0, but now they have the same sign and so enhance this amplitude. The emerging explanation of the ΔI=1/2 rule is a combination of the perturbative running to scales of O(2 GeV), a relative suppression of ReA2 through the cancelation of the two dominant contributions, and the corresponding enhancement of ReA0. QCD and electroweak penguin operators make only very small contributions at such scales.

We describe the computation of the amplitude ${A}_{2}$ for a kaon to decay into two pions with isospin $I=2$. The results presented in [T. Blum et al., Phys. Rev. Lett. 108, 141601 (2012)] from an analysis of 63 gluon configurations are updated to 146 configurations giving $\mathrm{Re}{A}_{2}=1.381(46{)}_{\mathrm{stat}}(258{)}_{\mathrm{syst}}{10}^{\ensuremath{-}8}\text{ }\text{ }\mathrm{GeV}$ and $\mathrm{Im}{A}_{2}=\ensuremath{-}6.54(46{)}_{\mathrm{stat}}(120{)}_{\mathrm{syst}}{10}^{\ensuremath{-}13}\text{ }\text{ }\mathrm{GeV}$. $\mathrm{Re}{A}_{2}$ is in good agreement with the experimental result, whereas the value of $\mathrm{Im}{A}_{2}$ was hitherto unknown. We are also working toward a direct computation of the $K\ensuremath{\rightarrow}(\ensuremath{\pi}\ensuremath{\pi}{)}_{I=0}$ amplitude ${A}_{0}$ but, within the Standard Model, our result for $\mathrm{Im}{A}_{2}$ can be combined with the experimental results for $\mathrm{Re}{A}_{0}$, $\mathrm{Re}{A}_{2}$ and ${\ensuremath{\epsilon}}^{\ensuremath{'}}/\ensuremath{\epsilon}$ to give $\mathrm{Im}{A}_{0}/\mathrm{Re}{A}_{0}=\ensuremath{-}1.61(28)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$. Our result for $\mathrm{Im}{A}_{2}$ implies that the electroweak penguin (EWP) contribution to ${\ensuremath{\epsilon}}^{\ensuremath{'}}/\ensuremath{\epsilon}$ is $\mathrm{Re}({\ensuremath{\epsilon}}^{\ensuremath{'}}/\ensuremath{\epsilon}{)}_{\mathrm{EWP}}=\ensuremath{-}(6.25\ifmmode\pm\else\textpm\fi{}{0.44}_{\mathrm{stat}}\ifmmode\pm\else\textpm\fi{}{1.19}_{\mathrm{syst}})\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$.

We calculate the masses of bottom mesons using an improved relativistic action for the b-quarks and the RBC/UKQCD Iwasaki gauge configurations with 2+1 flavors of dynamical domain-wall light quarks. We analyze configurations with two lattice spacings: a^{-1} = 1.729 GeV (a ~ 0.11 fm) and a^{-1} = 2.281 GeV (a ~ 0.086 fm). We use an anisotropic, clover-improved Wilson action for the b-quark, and tune the three parameters of the action nonperturbatively such that they reproduce the experimental values of the B_s and B_s* heavy-light meson states. The masses and mass-splittings of the low-lying bottomonium states (such as the eta_b and Upsilon) can then be computed with no additional inputs, and comparison between these predictions and experiment provides a test of the validity of our method. We obtain bottomonium masses with total uncertainties of ~0.5-0.6% and fine-structure splittings with uncertainties of ~35-45%; for all cases we find good agreement with experiment. The parameters of the relativistic heavy-quark action tuned for b-quarks presented in this work can be used for precise calculations of weak matrix elements such as B-meson decay constants and mixing parameters with lattice discretization errors that are of the same size as in light pseudoscalar meson quantities. This general method can also be used for charmed meson masses and matrix elements if the parameters of the heavy-quark action are appropriately tuned., 30 pages, 14 figures, 14 tables

We report on the first realistic ab initio calculation of a hadronic weak decay, that of the amplitude ${A}_{2}$ for a kaon to decay into two $\ensuremath{\pi}$ mesons with isospin 2. We find $\mathrm{Re}{A}_{2}=(1.436\ifmmode\pm\else\textpm\fi{}{0.063}_{\mathrm{stat}}\ifmmode\pm\else\textpm\fi{}{0.258}_{\mathrm{syst}}){10}^{\ensuremath{-}8}\text{ }\text{ }\mathrm{GeV}$ in good agreement with the experimental result and for the hitherto unknown imaginary part we find $\mathrm{Im}{A}_{2}=\ensuremath{-}(6.83\ifmmode\pm\else\textpm\fi{}{0.51}_{\mathrm{stat}}\ifmmode\pm\else\textpm\fi{}{1.30}_{\mathrm{syst}}){10}^{\ensuremath{-}13}\text{ }\text{ }\mathrm{GeV}$. Moreover combining our result for $\mathrm{Im}{A}_{2}$ with experimental values of $\mathrm{Re}{A}_{2}$, $\mathrm{Re}{A}_{0}$, and ${ϵ}^{\ensuremath{'}}/ϵ$, we obtain the following value for the unknown ratio $\mathrm{Im}{A}_{0}/\mathrm{Re}{A}_{0}$ within the standard model: $\mathrm{Im}{A}_{0}/\mathrm{Re}{A}_{0}=\ensuremath{-}1.63(19{)}_{\mathrm{stat}}(20{)}_{\mathrm{syst}}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$. One consequence of these results is that the contribution from $\mathrm{Im}{A}_{2}$ to the direct CP violation parameter ${ϵ}^{\ensuremath{'}}$ (the so-called Electroweak Penguin contribution) is $\mathrm{Re}({ϵ}^{\ensuremath{'}}/ϵ{)}_{\mathrm{EWP}}=\ensuremath{-}(6.52\ifmmode\pm\else\textpm\fi{}{0.49}_{\mathrm{stat}}\ifmmode\pm\else\textpm\fi{}{1.24}_{\mathrm{syst}})\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$. We explain why this calculation of ${A}_{2}$ represents a major milestone for lattice QCD and discuss the exciting prospects for a full quantitative understanding of $CP$ violation in kaon decays.

We report a direct lattice calculation of the K to ?? decay matrix elements for both the ?I=1/2 and 3/2 amplitudes A0 and A2 on 2+1 flavor, domain wall fermion, 163×32×16 lattices. This is a complete calculation in which all contractions for the required ten, four-quark operators are evaluated, including the disconnected graphs in which no quark line connects the initial kaon and final two-pion states. These lattice operators are nonperturbatively renormalized using the Rome-Southampton method and the quadratic divergences are studied and removed. This is an important but notoriously difficult calculation, requiring high statistics on a large volume. In this paper, we take a major step toward the computation of the physical K??? amplitudes by performing a complete calculation at unphysical kinematics with pions of mass 422 MeV at rest in the kaon rest frame. With this simplification, we are able to resolve Re(A0) from zero for the first time, with a 25% statistical error and can develop and evaluate methods for computing the complete, complex amplitude A0, a calculation central to understanding the ?=1/2 rule and testing the standard model of CP violation in the kaon system.

We report on the first realistic ab initio calculation of a hadronic weak decay, that of the amplitude A(2) for a kaon to decay into two π mesons with isospin 2. We find ReA(2)=(1.436±0.063(stat)±0.258(syst))10(-8) GeV in good agreement with the experimental result and for the hitherto unknown imaginary part we find ImA(2)=-(6.83±0.51(stat)±1.30(syst))10(-13) GeV. Moreover combining our result for ImA(2) with experimental values of ReA(2), ReA(0), and ε'/ε, we obtain the following value for the unknown ratio ImA(0)/ReA(0) within the standard model: ImA(0)/ReA(0)=-1.63(19)(stat)(20(syst)×10(-4). One consequence of these results is that the contribution from ImA(2) to the direct CP violation parameter ε' (the so-called Electroweak Penguin contribution) is Re(ε'/ε)(EWP)=-(6.52±0.49(stat)±1.24(syst))×10(-4). We explain why this calculation of A(2) represents a major milestone for lattice QCD and discuss the exciting prospects for a full quantitative understanding of CP violation in kaon decays.

We determine the neutral kaon mixing matrix element BK in the continuum limit with 2+1 flavors of domain wall fermions, using the Iwasaki gauge action at two different lattice spacings. These lattice fermions have near exact chiral symmetry and therefore avoid artificial lattice operator mixing. We introduce a significant improvement to the conventional nonperturbative renormalization (NPR) method in which the bare matrix elements are renormalized nonperturbatively in the regularization invariant momentum scheme (RI-MOM) and are then converted into the MS? scheme using continuum perturbation theory. In addition to RI-MOM, we introduce and implement four nonexceptional intermediate momentum schemes that suppress infrared nonperturbative uncertainties in the renormalization procedure. We compute the conversion factors relating the matrix elements in this family of regularization invariant symmetric momentum schemes (RI-SMOM) and MS? at one-loop order. Comparison of the results obtained using these different intermediate schemes allows for a more reliable estimate of the unknown higher-order contributions and hence for a correspondingly more robust estimate of the systematic error. We also apply a recently proposed approach in which twisted boundary conditions are used to control the Symanzik expansion for off-shell vertex functions leading to a better control of the renormalization in the continuum limit. We control chiral extrapolation errors by considering both the next-to-leading order SU(2) chiral effective theory, and an analytic mass expansion. We obtain BKMS? (3??GeV)=0.529(5)stat(15)?(2)FV(11)NPR. This corresponds to B?KRGI? =0.749(7)stat(21)?(3)FV(15)NPR. Adding all sources of error in quadrature, we obtain B?KRGI? =0.749(27)combined, with an overall combined error of 3.6%.

We present physical results obtained from simulations using 2+1 flavors of domain wall quarks and the Iwasaki gauge action at two values of the lattice spacing $a$, ($a^{-1}$=\,1.73\,(3)\,GeV and $a^{-1}$=\,2.28\,(3)\,GeV). On the coarser lattice, with $24^3\times 64\times 16$ points, the analysis of ref.[1] is extended to approximately twice the number of configurations. The ensembles on the finer $32^3\times 64\times 16$ lattice are new. We explain how we use lattice data obtained at several values of the lattice spacing and for a range of quark masses in combined continuum-chiral fits in order to obtain results in the continuum limit and at physical quark masses. We implement this procedure at two lattice spacings, with unitary pion masses in the approximate range 290--420\,MeV (225--420\,MeV for partially quenched pions). We use the masses of the $\pi$ and $K$ mesons and the $\Omega$ baryon to determine the physical quark masses and the values of the lattice spacing. While our data are consistent with the predictions of NLO SU(2) chiral perturbation theory, they are also consistent with a simple analytic ansatz leading to an inherent uncertainty in how best to perform the chiral extrapolation that we are reluctant to reduce with model-dependent assumptions about higher order corrections. Our main results include $f_\pi=124(2)_{\rm stat}(5)_{\rm syst}$\,MeV, $f_K/f_\pi=1.204(7)(25)$ where $f_K$ is the kaon decay constant, $m_s^{\bar{\textrm{MS}}}(2\,\textrm{GeV})=(96.2\pm 2.7)$\,MeV and $m_{ud}^{\bar{\textrm{MS}}}(2\,\textrm{GeV})=(3.59\pm 0.21)$\,MeV\, ($m_s/m_{ud}=26.8\pm 1.4$) where $m_s$ and $m_{ud}$ are the mass of the strange-quark and the average of the up and down quark masses respectively, $[\Sigma^{\msbar}(2 {\rm GeV})]^{1/3} = 256(6)\; {\rm MeV}$, where $\Sigma$ is the chiral condensate, the Sommer scale $r_0=0.487(9)$\,fm and $r_1=0.333(9)$\,fm., Comment: 129 pages, 59 figures, Published version containing an extended discussion of reweighting and including a new appendix (Appendix C)

The large mass of the ninth pseudoscalar meson, the ${\ensuremath{\eta}}^{\ensuremath{'}}$, is believed to arise from the combined effects of the axial anomaly and the gauge field topology present in QCD. We report a realistic, $2+1$-flavor, lattice QCD calculation of the $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$ masses and mixing which confirms this picture. The physical eigenstates show small octet-singlet mixing with a mixing angle of $\ensuremath{\theta}=\ensuremath{-}14.1(2.8)\ifmmode^\circ\else\textdegree\fi{}$. Extrapolation to the physical light quark mass gives, with statistical errors only, ${m}_{\ensuremath{\eta}}=573(6)\text{ }\text{ }\mathrm{MeV}$ and ${m}_{{\ensuremath{\eta}}^{\ensuremath{'}}}=947(142)\text{ }\text{ }\mathrm{MeV}$, consistent with the experimental values of 548 and 958 MeV.

The large mass of the ninth pseudoscalar meson, the η', is believed to arise from the combined effects of the axial anomaly and the gauge field topology present in QCD. We report a realistic, 2+1-flavor, lattice QCD calculation of the η and η' masses and mixing which confirms this picture. The physical eigenstates show small octet-singlet mixing with a mixing angle of θ=-14.1(2.8)°. Extrapolation to the physical light quark mass gives, with statistical errors only, mη=573(6) MeV and mη'=947(142) MeV, consistent with the experimental values of 548 and 958 MeV.

We present results for the dependence of the residual mass of domain wall fermions (DWF) on the size of the fifth dimension and its relation to the density and localization properties of low-lying eigenvectors of the corresponding hermitian Wilson Dirac operator relevant to simulations of 2+1 flavor domain wall QCD. Using the DBW2 and Iwasaki gauge actions, we generate ensembles of configurations with a $$16^3\times 32$$ space-time volume and an extent of 8 in the fifth dimension for the sea quarks. We demonstrate the existence of a regime where the degree of locality, the size of chiral symmetry breaking and the rate of topology change can be acceptable for inverse lattice spacings $$a^{-1} \ge 1.6$$ GeV.

We present a calculation of the renormalization coefficients of the quark bilinear operators and the K-Kbar mixing parameter B_K. The coefficients relating the bare lattice operators to those in the RI/MOM scheme are computed non-perturbatively and then matched perturbatively to the MSbar scheme. The coefficients are calculated on the RBC/UKQCD 2+1 flavor dynamical lattice configurations. Specifically we use a 16^3 x 32 lattice volume, the Iwasaki gauge action at beta=2.13 and domain wall fermions with L_s=16., 80 pages and 33 figures

We summarize the latest results from performing Lattice QCD calculations with two different discretization schemes, asqtad and p4fat3, on the LLNL BG/L supercomputer, and present a plan for continued running to complete the calculation of the QCD transition temperature and equation of state.

We present the first results for neutral kaon mixing using 2+1 flavors of domain wall fermions. A new approach is used to extrapolate to the physical up and down quark masses from our numerical studies with pion masses in the range 240 -- 420 MeV; only $SU(2)_L \times SU(2)_R$ chiral symmetry is assumed and the kaon is not assumed to be light. Our main result is $B_K^{\bar{\rm MS}}(2 \mathrm{GeV}) = 0.524(10)(28)$ where the first error is statistical and the second incorporates estimates for all systematic errors., Comment: 4 pages, 4 figures. RBC and UKQCD collaborations

We present numerical results for the kaon B-parameter, B_K, determined in the quenched approximation of lattice QCD. Our simulations are performed using domain-wall fermions and the renormalization group improved, DBW2 gauge action which combine to give quarks with good chiral symmetry at finite lattice spacing. Operators are renormalized non-perturbatively using the RI/MOM scheme. We study scaling by performing the simulation on two different lattices with a^{-1} = 1.982(30) and 2.914(54) GeV. We combine this quenched scaling study with an earlier calculation of B_K using two flavors of dynamical, domain-wall quarks at a single lattice spacing to obtain B_K(MS,NDR,mu=2GeV)=0.563(21)(39)(30), were the first error is statistical, the second systematic (without quenching errors) and the third estimates the error due to quenching., 77 pages, 44 figures, to be published in Phys. Rev. D

QCDOC is a supercomputer designed for high scalability at a low cost per node. We discuss the status of the project and provide performance estimates for large machines obtained from cycle accurate simulation of the QCDOC ASIC., 3 pages 1 figure. Lattice2002(machines)

We find the renormalisation coefficients of the quark field and the flavour non-singlet fermion bilinear operators for the domain wall fermion action, in the regularisation independent (RI) renormalisation scheme. Our results are from a quenched simulation, on a 16^3x32 lattice, with beta=6.0 and an extent in the fifth dimension of 16. We also discuss the expected effects of the residual chiral symmetry breaking inherent in a domain wall fermion simulation with a finite fifth dimension, and study the evidence for both explicit and spontaneous chiral symmetry breaking effects in our numerical results. We find that the relations between different renormalisation factors predicted by chiral symmetry are, to a good approximation, satisfied by our results and that systematic effects due to the (low energy) spontaneous chiral symmetry breaking and zero-modes can be controlled. Our results are compared against the perturbative predictions for both their absolute value and renormalisation scale dependence., Comment: 53 pages, 21 figures, revtex

We study the chiral properties of quenched domain wall fermions with several gauge actions. We demonstrate that the residual chiral symmetry breaking, which is present for a finite number of lattice sites in the fifth dimension ($L_s$), can be substantially suppressed using improved gauge actions. In particular the Symanzik action, the Iwasaki action, and a renormalization group improved gauge action, called doubly blocked Wilson (DBW2), are studied and compared to the Wilson action. All improved gauge actions studied show a reduction in the additive residual quark mass, $\mres$. Remarkably, in the DBW2 case $\mres$ is roughly two orders of magnitude smaller than the Wilson gauge action at $a^{-1}=2$ GeV and $L_s=16$. Significant reduction in $\mres$ is also realized at stronger gauge coupling corresponding to $a^{-1}=1.3$ GeV. As our numerical investigation indicates, this reduction is achieved by reducing the number of topological lattice dislocations present in the gauge field configurations. We also present detailed results for the quenched light hadron spectrum and the pion decay constant using the DBW2 gauge action., 50 pages, 25 figures, RevTeX formated, minor corrections

Results for the temperature of the color deconfinement phase transition in pure SU(3) lattice gauge theory are described. These were obtained on a specially built 16-node, 256 Megaflop computer using the Metropolis algorithm. The architecture, performance, and expansion plans for this machine are also discussed.

The usual lattice gauge theory developed by Wilson1 and discussed by many of the previous speakers has two important advantages over the continuum theory: i) The number of degrees of freedom per-unit-volume is finite, so that the theory is regularized and numerical calculation is possible, ii) Both the weak- and strong-coupling regimes can be evaluated analytically. The long distance behavior of the weak-coupling limit agrees with continuum perturbation theory while the strong-coupling limit shows confinement.

This talk is divided into two parts. In the first part, our present results on the deconfining phase transition for pure SO(3) theory will be discussed. The second part contains a brief description of our computers: the architecture of the three machines now in varying stages of operation/construction, their performance and the expected schedual for their completion.

The question of operator ordering in the quantum Yang-Mills Hamiltonian is discussed for Coulomb gauge. Unconventional terms, first proposed by Schwinger, are found and their effect on the Coulomb gauge Feynman rules is analysed. The central role played by gauge theories in our understanding of hadronic physics provides the motivation for a detailed study of the quantum Yang-Mills theory in a variety of gauges. The work which I will describe, done in collaboration with Lee [ l] , pays particular attention to the question of operator ordering in non-covariant gauges defined by conditions, x( vi) = 0, imposed on the spacial components of the vector potential vi. Let us begin by considering the simplest gauge for canonical quantization, the Vo = 0 gauge. The conventional Hamiltonian operator is given by where g l i z i B = Eijk ajv; + f z a b ~ a ~ : E is a group index running from 1 to N 2 1 for SU(N) and f l a b the structure constants. K'(r) and n:(r) are the (unconstrained) field variables and their conjugate momenta obeying [F'(r), np(r ') l = itiZmsijs3(r r') (2) Since the equation of motion obtained by varying the action with respect to Vo is not among the Hamilton equations of motion we must impose Gauss' law as a constraint on the "physical" states ( Q T ~ ) ~ I $ ) = (air; + g p k m & k n T ) l $ ) = o (3) Let us adopt the view that this is the correct formulation of the V, = 0 gauge Yang-Mills theory and attempt to transform it into the non-covariant gauge, x ( A i ) = 0. For simplicity I will specialize to the case x ( A i ) = aiAi(r), Coulomb gauge. First write i g fi = u ~ ~ ~ 1 + ~ a ~ ~ 1 (4) where A i is in Coulomb gauge, aiAi = 0, andA and Vare represented by N x N Hermitian matrices, e.g., = K'T' for N x N , traceless, Hermitian SU(N) generators T'. Next parametrize the SU(N) matrix U by N 2 1 angles G f f (for example the three Euler angles in the case of SU(2)). Thus, we have two sets of variables Vi'(r) and &(r), Af(r) with which to describe a particular Yang-Mills configuration. This view that Coulomb gauge can be obtained by a particular choice of Vo = 0 gauge Physica Scripta 23 variables has been developed previously by Gribov, Gervais and Sakita; and Creutz, Muzinich and Tudron [ 21. We must now rewrite H and Dini in terms of the new variables and Ai . To do this, we treat n!(r) as a Schrodinger differential operator