Scalar fields describe particles with zero spin (1 component) and transform under unitary irreducible representations of Poincare group.

A scalar field \(\phi(x)\) is a infinite dimensional vector with component at each \(x^{\mu }\). So fields are objects with infinite number of degree of freedom.

\[ \psi (x)\to e^{\frac{1}{2}\omega_{\mu \nu }M^{\mu \nu } + a^{\mu }p_{\mu } }\phi (x)=\phi (x) \]

Since scalar fields have only one component, it will transform back to same, or we can just say that it does not transform under unitary irreducible representations of Poincare group (ISO(3,1)). Translational invariance is mostly present if we look for lagrangian density of kind \(\mathcal{L}=\mathcal{L}(\phi ,\partial _{\mu }\phi )\), so we will only look into lorentz invariance. For lagrangian explicitly depending upon \(x^{\mu }\) (\(\mathcal{L}=\mathcal{L}(\phi ,\partial _{\mu }\phi ,x)\)), it changes under translational operation and hence does not respect translational symmetry.

The inspiration for real scalar fields came from 4 component vector field \(A_{\mu }(x)\) which is (\(\phi,\vec{A}\))
First we guess the similar lorentz covariant (since scalar fields remain invariant we just look for lorentz invariant EOM) K.G equation for scalar fields \((\partial ^2+m^2)\phi(x)=0\), then we guess the lorentz invariant(or Poincare invariant) lagrangian density (with L=T-V form) for free real scalar fields.

Free scalar fields are described by the following free lagrangian density \(\mathcal{L}\):
for \(\eta_{\mu \nu }=(+,-,-,-)\)

\[ \begin{align*} \mathcal{L} &=\frac{1}{2}\partial _{\mu }\phi \partial ^{\mu }\phi -\frac{1}{2}m^2\phi ^2 \\ &= -\phi (\partial ^2+m^2)\phi + \partial _{\mu }(\phi \partial ^{\mu }\phi ) \\ \end{align*} \]

for \(\eta_{\mu \nu }=(-,+,+,+)\)

\[ \begin{align*} \mathcal{L} &=-\frac{1}{2}\partial _{\mu }\phi \partial ^{\mu }\phi -\frac{1}{2}m^2\phi ^2 \\ &= \phi (\partial ^2+m^2)\phi + \partial _{\mu }(\phi \partial ^{\mu }\phi ) \\ \end{align*} \]

Here the lagrangian density retains the form (L=T-V) which can be seen from \((\partial _{\mu}\phi \partial ^{\mu }\phi = \frac{1}{2}\dot\phi ^2-\frac{1}{2}\delta ^2-\frac{1}{2}m^2\phi ^2)\).

The distinction is due to requirement that Hamiltonian shall be bounded from below (zero). i.e. Kinetic energy shall be positive.

Free complex scalar fields have the following lagrangian:

\[ \begin{align*} \mathcal{L} &= \partial_{\mu }\phi ^{*}\partial ^{\mu }\phi - m^2\phi ^2 \\ &= \frac{1}{2}\partial _{\mu }\phi_{1} \partial ^{\mu }\phi_{1} -\frac{1}{2}m^2\phi_{1} ^2 + \frac{1}{2}\partial _{\mu }\phi_{2} \partial ^{\mu }\phi_{2} -\frac{1}{2}m^2\phi_{2} ^2 \\ \end{align*} \]

The second step implies that complex scalar lagrangian is sum of two real scalar lagrangian densities. This can be seen if we write \((\phi = \frac{1}{\sqrt{2} }(\phi _{1} + i\phi _{2}))\).

Imagine that we have real scalar field in a cubical box of side 'l', boundary condition imposition implies that,

\[ \phi (x+l,y,z)=\phi (x,y+l,z)=\phi (x,y,z+l)=\phi (x,y,z)\\ \]

We know that there are fields like vector potential fields \(A_{\mu }\) which have the dynamics given by :

\[ \begin{align*} \partial _{\mu }F^{\mu \nu }=0\\ \partial _{\mu }(\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu })=0\\ \partial ^2A^{\nu }-\partial _{\nu}(\partial _{\mu }A^{\mu })=0\\ \end{align*} \]

which in lorentz gauge it takes the form of :

\[ \partial ^2A^{\nu }=0\\ \]

Similerly we can try the same equation of motion for scalar fields but with mass term.
Our scalar field obey :

\[ \begin{align*} (\partial ^2+m^2)\phi (\vec{x},t)=0 \\ (\partial ^2+m^2)\phi (x)=0 \\ \end{align*} \]

Using our plane wave solutions \(e^{i\vec{k}.\vec{x}}\), we get,