Real Scalar Field

A real scalar field is the simplest relativistic quantum field: it creates neutral particles with spin zero. Its quadratic action makes every vacuum correlation function calculable from one two-point function. It is a baseline for scattering theory, perturbation theory, and comparisons with interacting models, rather than a description of all scalar matter.

Real scalar dynamics

In $d$-dimensional flat spacetime, take a real field $\phi$ with mass $m>0$ and action

$$S=\int \mathrm d^dx\, \left[\frac12\partial_\mu\phi\,\partial^\mu\phi-\frac12m^2\phi^2\right].$$

The canonical momentum is $\pi=\partial_t\phi$. Varying the action gives $(\Box+m^2)\phi=0$, so a mode $e^{-ip\cdot x}$ obeys $p^2=m^2$ and has energy $E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}$. The field has engineering dimension $[\phi]=(d-2)/2$. The symmetry $\phi\mapsto-\phi$ is discrete; it does not supply a continuous conserved particle-number charge.

Quantization imposes $[\phi(t,\mathbf x),\pi(t,\mathbf y)]=i\delta^{(d-1)}(\mathbf x-\mathbf y)$. Positive-frequency modes multiply annihilation operators and negative-frequency modes their adjoints. A real field needs only one species: the particle is its own antiparticle. The mode construction and energy interpretation are developed in Srednicki 2007, §3.

The propagator and a Gaussian check

For the Minkowski vacuum, the time-ordered two-point function is

$$\langle0|T\phi(x)\phi(0)|0\rangle =\int\frac{\mathrm d^dp}{(2\pi)^d} \frac{i\,e^{-ip\cdot x}}{p^2-m^2+i0}.$$

The $i0$ prescription specifies time ordering; it is not a retarded response prescription. Since the theory is Gaussian, the four-point function is the sum of its three pairings:

$$\langle T\phi_1\phi_2\phi_3\phi_4\rangle =G_{12}G_{34}+G_{13}G_{24}+G_{14}G_{23},$$

where $G_{ij}=\langle T\phi(x_i)\phi(x_j)\rangle$. Therefore its connected four-point function vanishes. This is a useful numerical and algebraic test: a nonzero connected scattering amplitude cannot come from this free action.

Uses, observables, and limits

The mass is visible in the propagator pole and in the dispersion relation. Local commutators vanish at spacelike separation, although the two-point function need not vanish there. Vacuum fluctuations are consequently compatible with causal propagation.

Adding $\phi^4$ interactions changes the Gaussian factorization and creates scattering. Curved backgrounds change the mode problem and can remove a preferred vacuum. The massless limit in low dimensions can also introduce infrared problems; it should not be inferred simply by setting $m=0$ in every integrated expression.

Develop the calculation

Read The Klein–Gordon Field and Its Modes for the solution space, then Quantizing the Real Scalar Field for its Fock space. Compare the charged species in the Complex Scalar Field and the first interaction in Scalar φ⁴ Theory.

References

• Srednicki, Mark. Quantum Field Theory. Cambridge University Press, 2007, §§3 and 8. DOI.