1. Introduction
2. Ordered exponentials
3. Additional properties
Let $x,y\in U$, and $1$ denote the unit function. Then, under the conditions described above, we have
It is obvious that the initial condition ${{\rm{\Phi }}}^{ab}(y,y)={\delta }^{ab}$ holds, because ${\sigma }^{\rho }(x,y)$ and ${\sigma }^{\rho }\left(x,y\right)$ tend to zero when x → y. Let us check that the right hand side of (
Let $x,y\in U$, and $1$ denote the unit function. Also, ${N}_{r}(x,y)$ and ${N}_{l}(x,y)$ denote matrix-valued operators ${\sigma }^{\rho }(x,y){\mathop{D}\limits^{\longrightarrow}}_{\rho }^{ab}$ and ${\mathop{D}\limits^{\longrightarrow}}_{\rho ^{\prime} }^{ab}{\sigma }^{\rho ^{\prime} }\left(x,y\right),$ respectively. Then, under the conditions described above, we have
For simplicity, we work with matrix-valued operators. Then, using property (
Let $x,y\in U.$ Then, under the conditions described above, we have
Let us apply the operator ${\sigma }^{\rho }(x,y){\partial }_{\rho }$ to the left hand side of formula (
Let $x,y,z\in U.$ Then, under the conditions described above, we have the following relation
Let us introduce two gauge transformed derivatives according to the formulae
4. Application to the Yang–Mills theory
• | Does the effective action contain the infrared singularity in coordinate representation, depending on the ‘out' part? For the positive answer, it is sufficient to provide an explicit expression for some ${W}_{n}(B,{\rm{\Lambda }}).$ We draw attention that the two-loop contribution ${W}_{1}(B,{\rm{\Lambda }})$ includes only singularities, related to the ‘in' part because in this case the quantum correction is split into a sum of ‘in' and ‘out' parts, without mixing. Hence, without any fundamental justifications and assumptions, we are interested in the three-loop calculation. Moreover, it would be meaningful to describe the structure of such contributions for a cutoff regularization. |
• | Does the effective action contain the infrared singularity in coordinate representation, depending on the non-local part ${{\mathscr{N}}}^{ab}$? This is one more non-obvious issue. The fact is that in the two-loop contribution, a special combination of non-local terms occurs, but it is converted to the local part with the use of the Seeley–DeWitt coefficients. The appearance or absence of the non-local part in the multi-loop terms is quite an interesting challenge because it leads to the possibility of using simplifications in the calculations. The last two questions are related to the form of the coefficients, their locality and dependence on the boundary conditions. However, we have a problem, corresponding to a structure of singularities itself. Indeed, in the quantum corrections after the regularization introduced, we have some dimensional parameters: ${\rm{\Lambda }}$ is the parameter of regularization, $\mu $ is the parameter from ( |
• | Let $i,j\in {\mathbb{N}}\cup \{0\}$ and $n\gt 0.$ What combinations of ${({\rm{\Lambda }}/m)}^{i}{\mathrm{ln}}^{j}({\rm{\Lambda }}/m)$ does the correction ${W}_{n}(B,{\rm{\Lambda }})$ contain, where $m\in \{\mu ,{\mu }_{1}\}$ is a dimension parameter? For example, it was shown [27, 32] that the two-loop correction includes only $\mathrm{ln}({\rm{\Lambda }}/\mu ).$ And it is expected that for other orders we have $i=0$ and $j\in \{1,\ldots ,n\}.$ |