A Note on the scale symmetry and Noether current

Usually we consider the symmetry of action as the symmetry of the theory, however, in the Keplar problem the scaling symmetry existing in equation of motion is not the ones for action. It changes the multiplicative constant of action and the time boundary. In such a case that the scale transformation does not leave the action invariant but keeping the equation invariant, the following statement is proved. The time integration of Lagrangian is explicitly performed and the action can be expressed by the difference of formal (non-conserved) Noether charges at time boundaries. In field theory the action can be expressed by the boundary integration of the formal Noether current. 1 Scale invariance and Keplar motion The scale transformation of dynamical systems is widely used and discussed for a long time. The well known phenomena in classical mechanics is the Keplar’s third theorem and the law of similitude for flow. In the high energy physics the Bjorken’s scaling law is well established fact [1], and further discussions depending on the renormalization group equations are essentially coming from the scale transformation. The essential property of fractals is also the scale invariance. When the theory is called scale invariant, there are two different meanings. One is the invariance of equation of motion, another one is the invariance of action. Even if the action changes its value by multiplicative constant, the equation of motion is invariant. Therefore there are the transformations which make equation invariant and changes the action as discussed above [2]. One example is the scale transformation of Keplar problem which will be seen later clearly. E-mail: [email protected], [email protected]