Classical field

Definition

See Classical Field Theory .
See gauge theory#Coming from Classical field theories.

A classical field theory is the following data:

• A $n$-dimensional manifold (space-time) $M$,
• A target space $V\cong {\mathbb{R}}^m$, to construct the space of fields $\mathcal{F}=\text{Maps}(M,V)$.
• A Lagrangian, which is a polynomial $L(\varphi ,{\partial }_I\varphi )$, for $|I|\le s\in \mathbb{N}$, where $\varphi \in \mathcal{F}$.
  Then, they are generalized to gauge theories, which are the same stuff but instead of $\mathcal{F}=\text{Maps}(M,V)$ we use sections of a fiber bundle $E\to M$ as matter fields.

Types of fields

• Scalar fields. Section of a rank 1 vector bundle over spacetime. Related to spin 0 particles.
• Vector fields. Section of a rank $n+1$ vector bundle over a $(n+1)$-dimensional spacetime. Related to spin 1 particles.
• Spinor fields (classical spinor field?). When they say a particle has spin 1/2 what they mean is that the particle is the quantum version of a classical spinor field, that is, for every point $p$ in the (3+1)-spacetime we have a vector $v_p$ in such a way that if we consider a change of local frame in $p$ through an element $\varphi \in SO(1,3)$, then $\varphi$ acts on $v_p$ by means of a spin representation of $SO(1,3)$.

Transformations of fields

Given a field, in the sense of an element $\varphi \in \mathcal{F}$, we can consider a diffeomorphism $F:M\to M$ and define a new field $\tilde{\varphi }=F^{\ast }\varphi$, i.e.,

In the picture it looks like if the transformation $F$ is pushing the points of $M$ two units right, but the corresponding transformation of $\varphi$ consists of pushing the field two units left.

Usually, $F$ is an element of a local group of transformations of $M$, depending on parameters.

Examples: Consider $M=\mathbb{R}$, and the family of diffeomorphisms $F_{ϵ}:M\to M$ given by $F_{ϵ}(x)=x+ϵv$, where $v\in \mathbb{R}$. Then, the transformation of a field $\varphi$ is given by

TO BE ENDED. SEE xournal 275