## Appendix A Local Symmetry  
Symmetry in physical laws means that every physical law is invariant under certain operations. In the present context, we can discuss symmetry under coordinate transformations. Consider a spring is placed along the \(x\) axis of an \(xy\) coordinate system with its one end fixed at the origin (Figure 11). You stretch the spring by pulling the other end along the \(x\) axis in the positive \(x\) direction. According to [PERSON]'s law, we can express the force-stretch relation as \(\mathbf{f}=-k\mathbf{s}\) where \(\mathbf{f}\), \(k\) and \(\mathbf{s}\) are the force vector, spring constant and stretch vector, respectively. With the unit vector notation (\(\hat{x}\) and \(\hat{y}\)), \(\mathbf{f}=f\hat{x}+0\hat{y}\) and \(\mathbf{s}=s\hat{x}+0\hat{y}\). Now consider that you rotate the coordinate system by \(\pi/4\) from the positive \(x\) axis toward the positive \(y\) axis and call the new axes \(x^{\prime}\) and \(y^{\prime}\) axes. During this operation, the stretched spring is intact. With the new coordinate system, the force and stretch vectors become \(\mathbf{f}^{\prime}=(f/\sqrt{2})\hat{x}^{\prime}-(f/\sqrt{2})\hat{y}^{\prime}\) and \(\mathbf{s}^{\prime}=(s/\sqrt{2})\hat{x}^{\prime}-(s/\sqrt{2})\hat{y}^{\prime}\). This indicates that the coordinate rotation does not keep the components of the force or stretch vector invariant, but keeps vectors themselves unchanged. This is because vectors are physical quantities but their components are not (because they can change as we define the coordinate system, which is not a physical quantity). Since [PERSON]'s law relates the two vectors, it is also invariant under this coordinate transformation. We say that [PERSON]'s law is symmetric under the coordinate axis rotation. This type of symmetry is referred to as global symmetry because the coordinate transformation is performed globally. If the entire spring is divided into \(n\) segments and a local coordinate system is defined for each segment, the situation is fundamentally different under coordinate axis rotations. Unless all the local coordinate systems undergo the same axis rotation so that they are aligned with one another (i.e., \(x_{1}^{\prime}\) is parallel to \(x_{2}^{\prime}\cdots x_{i}^{\prime}\cdots x_{n}^{\prime}\) ), even if the physical phenomenon (the stretch of the spring) remains the same during the coordinate rotations, [PERSON]'s law cannot be expressed by a common coordinate system. The same problem arises if segments of the spring undergoes physical rotations during the stretching action. (This time segments of the spring under go rotation, not coordinate axes.) In this case the entire spring does not keep the linear shape. It takes a zigzag shape after the stretch. In this case, if we set up a local coordinate system for each segment of the zigzag shape, [PERSON]'s law can still be expressed in each individual local coordinate systems for the corresponding segments (because the local coordinate system sees only the linear stretching of the corresponding segment). At the global level, however, it is impossible to express the liner relation between the force and total stretch. (We can say that this is because [PERSON]'s law describes linear force law; the stretch vector and the force vector is always parallel.) We say that the physical system loses local symmetry because the stretching operation makes the spring in a zigzag shape. The concept of gauge theories fixes the problem by introducing a force that somehow reshapes the spring linear (this is a fake force because physically the spring is zigzag). In this way, we can formulaically express the force stretch relation with [PERSON]'s law with the global coordinate system. This type of fake force is referred to as a compensation (gauge) field. We say that with the introduction of a gauge field, the system regains the local symmetry. This fake effect is important because it contains the information of the effect that makes the spring zigzag.