scussions in Section 8.

Enhanced Symmetries
===================

In this section, we discuss the enhanced symmetries in two dimensional (2D) field theory with boost symmetry and anisotropic scalings, using the method developed in [@Polchinski:1987dy] and [@Hofman:2011zj]. Usually for a theory with global symmetries, we can defines the corresponding conserved Noether currents and their conserved charges. However there could be ambiguities in defining the currents. In 2D quantum field theory with scaling symmetry and boost symmetry, under the assumption that there exist a complete basis of local operators as the eigenvectors of the dilation operator with a discrete spectrum, the conserved currents can be organized in a form such that they have the canonical commutation relations with the generators. But the currents can be shifted by certain local operators without changing the commutation and conservation relations. Analyzing the behavior of the local operators leads to special relations of the currents, which in turn tell us that there may be infinite conserved charges.

Global symmetries
-----------------

The global symmetries of 2D QFT we consider in this work include the translations along two directions $x$ and $y$ $$x\rightarrow x'=x+\delta x,
\hs{3ex}
y\rightarrow y'=y+\delta y,$$ the dilations $$x\rightarrow x'=\lambda ^c x,\hs{3ex}y\rightarrow y'=\lambda ^d y,$$ where $c,d$ are non-negative. And the Galilean boost symmetry behaves as, $$y\rightarrow y'=y+v x.$$ It is worth noting that the dilation scales two directions at the same time, but could be with different weights $c$ and $d$. We use a slightly different notation from the one in [@Hofman:2014loa]. The generators of the above symmetry transformations are denoted as $H, \bar{H}, D$ and $B$ respectively. They annihilate the vacuum, and satisfy the commutation relations =0,&\[D,H\]=-c H,&\[D,|[H]{}\]=-d |[H]{},=-|[H]{},&\[B,|[H]{}\]=0,&\[B,D\]=(d-c)B.

We assume that the dilation operator has a discrete spectrum and the theory has a complete basis o