Explicit evaluation of group‐invariant measure as by‐product of path integration over Yang–Mills fields

1984 ◽  
Vol 25 (4) ◽  
pp. 1093-1101 ◽  
Author(s):  
Claudio Teitelboim
Keyword(s):  
Invariant Measure ◽  
Path Integration ◽  
Yang Mills ◽  
Group Invariant ◽  
Explicit Evaluation ◽  
Group Invariant Measure

AN EXCEPTIONAL E8 GAUGE THEORY OF GRAVITY IN D = 8, CLIFFORD SPACES AND GRAND UNIFICATION

2009 ◽  
Vol 06 (06) ◽  
pp. 911-930 ◽  
Author(s):  
CARLOS CASTRO
Keyword(s):  
Gauge Theory ◽  
Gravitational Theory ◽  
Grand Unification ◽  
Quantum Level ◽  
Yang Mills ◽  
Group Invariant ◽  
Gravitational Model ◽  
Theory Of Gravity ◽  
Chern Simons ◽  
Gauge Theory Of Gravity

A candidate action for an Exceptional E8 gauge theory of gravity in 8D is constructed. It is obtained by recasting the E8 group as the semi-direct product of GL(8,R) with a deformed Weyl–Heisenberg group associated with canonical-conjugate pairs of vectorial and antisymmetric tensorial generators of rank two and three. Other actions are proposed, like the quarticE8 group-invariant action in 8D associated with the Chern–Simons E8 gauge theory defined on the 7-dim boundary of a 8D bulk. To finalize, it is shown how the E8 gauge theory of gravity can be embedded into a more general extended gravitational theory in Clifford spaces associated with the Cl(16) algebra and providing a solid geometrical program of a grand unification of gravity with Yang–Mills theories. The key question remains if this novel gravitational model based on gauging the E8 group may still be renormalizable without spoiling unitarity at the quantum level.

GEOMETRICAL DERIVATION OF THE FADDEEV-POPOV ANSATZ

1989 ◽  
Vol 04 (24) ◽  
pp. 2397-2407 ◽  
Author(s):  
P. ELLICOTT ◽  
G. KUNSTATTER ◽  
D.J. TOMS
Keyword(s):  
Invariant Measure ◽  
Sigma Models ◽  
Configuration Space ◽  
Effective Action ◽  
General Class ◽  
Gauge Theories ◽  
Yang Mills ◽  
Linear Sigma Models ◽  
Supersymmetric Theories

A geometrical derivation of the Faddeev-Popov measure is presented. This derivation is valid in any gauge for a general class of gauge theories, including Yang-Mills theory, gravitation and non-linear sigma models, and can easily be generalized to include supersymmetric theories. We stress the role of a non-trivial, finite contribution to the effective action from the invariant measure on the orbit over each point in the physical configuration space.

scholarly journals FROM PEIERLS BRACKETS TO A GENERALIZED MOYAL BRACKET FOR TYPE-I GAUGE THEORIES

2007 ◽  
Vol 04 (03) ◽  
pp. 349-360 ◽  
Author(s):  
GIAMPIERO ESPOSITO ◽  
COSIMO STORNAIOLO
Keyword(s):  
Vector Fields ◽  
Field Operator ◽  
Functional Integral ◽  
Gauge Fields ◽  
Order Differential Operator ◽  
Type I ◽  
Gauge Invariant ◽  
Yang Mills ◽  
Group Invariant ◽  
Gauge Fixing

In the space-of-histories approach to gauge fields and their quantization, the Maxwell, Yang–Mills and gravitational field are well known to share the property of being type-I theories, i.e. Lie brackets of the vector fields which leave the action functional invariant are linear combinations of such vector fields, with coefficients of linear combination given by structure constants. The corresponding gauge-field operator in the functional integral for the in-out amplitude is an invertible second-order differential operator. For such an operator, we consider advanced and retarded Green functions giving rise to a Peierls bracket among group-invariant functionals. Our Peierls bracket is a Poisson bracket on the space of all group-invariant functionals in two cases only: either the gauge-fixing is arbitrary but the gauge fields lie on the dynamical sub-space; or the gauge-fixing is a linear functional of gauge fields, which are generic points of the space of histories. In both cases, the resulting Peierls bracket is proved to be gauge-invariant by exploiting the manifestly covariant formalism. Moreover, on quantization, a gauge-invariant Moyal bracket is defined that reduces to iħ times the Peierls bracket to lowest order in ħ.

Integral Kernels with Reflection Group Invariance

1991 ◽  
Vol 43 (6) ◽  
pp. 1213-1227 ◽  
Author(s):  
Charles F. Dunkl
Keyword(s):  
Coxeter Groups ◽  
Poisson Kernel ◽  
Multivariable Analysis ◽  
Root Systems ◽  
Reflection Group ◽  
Laplacian Operator ◽  
Harmonic Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Group Invariant ◽  
Group Invariant Measure

Root systems and Coxeter groups are important tools in multivariable analysis. This paper is concerned with differential-difference and integral operators, and orthogonality structures for polynomials associated to Coxeter groups. For each such group, the structures allow as many parameters as the number of conjugacy classes of reflections. The classical orthogonal polynomials of Gegenbauer and Jacobi type appear in this theory as two-dimensional cases. For each Coxeter group and admissible choice of parameters there is a structure analogous to spherical harmonics which relies on the connection between a Laplacian operator and orthogonality on the unit sphere with respect to a group-invariant measure. The theory has been developed in several papers of the author [4,5,6,7]. In this paper, the emphasis is on the study of an intertwining operator which allows the transfer of certain results about ordinary harmonic polynomials to those associated to Coxeter groups. In particular, a formula and a bound are obtained for the Poisson kernel.

scholarly journals MORE ON GENERALIZED SIMPLICIAL CHIRAL MODELS

2001 ◽  
Vol 16 (06) ◽  
pp. 1161-1171
Author(s):  
M. ALIMOHAMMADI ◽  
KH. SAAIDI
Keyword(s):  
Phase Transition ◽  
Partition Function ◽  
Density Function ◽  
Path Integration ◽  
Two Dimensional ◽  
Third Order ◽  
Dimensional Simplex ◽  
Yang Mills ◽  
Eigenvalue Density ◽  
Large N

By generalizing the auxiliary field term in the Lagrangian of simplicial chiral models on a (d-1)-dimensional simplex, the generalized simplicial chiral models has been introduced in Ref. 1. These models can be solved analytically only in d=0 and d=2 cases at large-N limit. In the d=0 case, we calculate the eigenvalue density function in strong regime and show that the partition function computed from this density function is consistent with one calculated by path integration directly. In the d=2 case, it is shown that all V= Tr (AA†)n models have a third order phase transition, the same as the two-dimensional Yang–Mills theory.

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