I have the question on inherent connection between Lagrangian Equation and invariance norm, asking chatGPT,

The relationship between the Lagrangian formalism and the invariant norm, usually understood as the spacetime interval in special relativity, is a foundational part of modern physics, particularly in the realm of relativistic quantum field theory.

In special relativity, the invariant interval (or “norm”) is a physical quantity that remains constant regardless of the frame of reference in which it’s measured. It’s given by the four-dimensional distance between two events in spacetime, and it’s expressed in terms of differences in space and time coordinates between the two events.

In the Lagrangian formalism, we derive equations of motion from the principle of stationary action (or least action). This principle posits that the path that a system will follow is one that minimizes (makes stationary) the action, which is the integral of the Lagrangian over time.

These two concepts are fundamentally related when we build a Lagrangian for a system that obeys the principles of special relativity, or, more generally, a Lorentz invariant field theory. In this context, the space and time dependence of the fields (and not just particles) is integrated into a single spacetime dependence. The Lagrangian (including the kinetic and potential pieces) depends on the fields and their spacetime derivatives in a way that it’s invariant under Lorentz transformations, i.e., transformations that preserve the spacetime interval.

For example, the Lagrangian for a free (non-interacting) scalar field, which can describe spinless particles, is constructed as a difference between a kinetic term (related to the spacetime derivative of the field) and a mass term (related to the field itself), both in a way that ensures Lorentz invariance.

In summary, this relationship arises from the requirement that the laws of physics—and thus, the description of physical systems using the Lagrangian formalism— must not change under a change of inertial frames, a principle known as the principle of relativity.

And then question about Gauge Theory, what on earth is Gauge?

In physics, a gauge refers to the freedom to make certain changes in the mathematical formalism used to describe physical systems without affecting the predicted observable quantities.

More specifically, in quantum field theory and the standard model of particle physics, gauge theories are used to describe fundamental interactions. Here, “gauge” refers to a type of symmetry of the system – known as gauge symmetry – that the laws of physics exhibit.

For example, in electromagnetism, gauge symmetry is the freedom to choose the electromagnetic potentials arbitrarily without affecting the physical fields (the electric and magnetic fields).

In quantum field theory, each of the fundamental forces (except for gravity) is associated with a certain type of gauge symmetry and force particles (gauge bosons):

1. Electromagnetic force is associated with U(1) gauge symmetry and the photon.
2. Weak nuclear force is associated with SU(2) gauge symmetry and the W and Z bosons.
3. Strong nuclear force is associated with SU(3) gauge symmetry and gluons.

The “gauge” in gauge theories refers to our ability to transform the properties of the field at each point in spacetime independently, in a way that leaves the fundamental equations unchanged. This local symmetry forms the basis for the fundamental forces, and the way the fields respond to these transformations leads to the concept of force-carrying particles.

When was asked to use U(1) to illustrate how it explain photon mediates mageneticelectronic force, chatGPt is not able to really answer

“

he U(1) symmetry surfaces in quantum electrodynamics (QED), a quantum field theory that describes the electromagnetic force as mediated by photons. This idea of “a quantum field theory with U(1) gauge symmetry” is basically the technical way of saying “the theory of how photons and electrically charged particles interact”.

In broad terms (and simplifying considerably), the U(1) symmetry conveys an invariance under transformations that globally shift the phase of the wave function. The “1” in U(1) demonstrates that one complex value keeps the quantum-mechanical amplitude of the system constant. Essentially, it symbolizes a freedom – we can choose a phase at any point in spacetime and it doesn’t affect what we observe in the physics of the situation.

Despite the freedom to choose a different phase at each spacetime point, we still wish the physics of the system to be the same. This requirement of local U(1) gauge invariance obligates us to introduce a new field (termed the gauge potential) to the equations, which will allow us to handle these position-dependent phase changes.

The minimally coupled U(1) gauge potential and matter fields (like electrons) result in interactions between the new field and the initial particles that look just like the electromagnetic interactions. Upon quantizing this field, we obtain photons – hence, the unification of the U(1) gauge potential and the electromagnetic potential.

In essence, the U(1) gauge symmetry, when used with QED, obliges interactions to conserve electric charge. These interactions implicitly need a force mediator (since electrically charged particles can’t just interact instantly across spacetime), and that forces us to introduce a new field which, after being quantized, gives us photons as the carriers of the electromagnetic force.”