Scale Invariant Classical Field Theory

In summary, the given action for a real scalar field is invariant under the scaling of all lengths by ##x^{\mu} \rightarrow (x')^{\mu}=\lambda x^{\mu}## and ##\phi(x) \rightarrow \phi'(x) = \lambda^{-D}\phi(\lambda^{-1}x)##, where ##D## is the scaling dimension of the field. The scaling dimension ##D## such that the derivative terms remain invariant is found to be ##D=-2##. This conclusion holds for any values of ##m## and ##p##, and for a scalar field living in any ##(n+1)##-dimensional spacetime. Using Noether's theorem,
  • #1
1,344
33

Homework Statement



A class of interesting theories are invariant under the scaling of all lengths by ##x^{\mu} \rightarrow (x')^{\mu}=\lambda x^{\mu}## and ##\phi(x) \rightarrow \phi'(x) = \lambda^{-D}\phi(\lambda^{-1}x)##.

Here ##D## is called the scaling dimension of the field.

Consider the action for a real scalar field given by ##S = \int d^{4}x\ \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}-g\phi^{p}##.

Find the scaling dimension ##D## such that the derivative terms remain invariant.

For what values of ##m## and ##p## is the scaling ##x^{\mu} \rightarrow (x')^{\mu}=\lambda x^{\mu}## and ##\phi(x) \rightarrow \phi'(x) = \lambda^{-D}\phi(\lambda^{-1}x)## a symmetry of the theory? How do these conclusions change for a scalar field living in an ##(n+1)##-dimensional spacetime instead of a ##3+1##-dimensional spacetime?

In ##3+1## dimensions, use Noether's theorem to construct the conserved current ##D^{\mu}## associated to scaling invariance.

Homework Equations



The Attempt at a Solution



Under the scaling of all lengths by ##x^{\mu} \rightarrow (x')^{\mu}=\lambda x^{\mu}## and ##\phi(x) \rightarrow \phi'(x) = \lambda^{-D}\phi(\lambda^{-1}x)##, where ##D## is called the scaling dimension of the field, the given action for the real scalar field transforms as follows:

##S = \int d^{4}x\ \frac{1}{2}\partial_{\mu}\phi(x)\partial^{\mu}\phi(x)-\frac{1}{2}m^{2}\phi^{2}(x)-g\phi^{p}(x)##

##\rightarrow \int d^{4}x\ \Big|\frac{1}{\lambda^{4}}\Big| \Big[ \frac{1}{2}({\Lambda^{\rho}}_{\mu}\lambda^{-D}\partial_{\rho}\phi)(\lambda^{-1}x)({\Lambda_{\sigma}}^{\mu}\lambda^{-D}\partial^{\sigma}\phi)(\lambda^{-1}x)-\frac{1}{2}m^{2}(\lambda^{-D}\phi)^{2}(\lambda^{-1}x)-g(\lambda^{-D}\phi)^{p}(\lambda^{-1}x)\Big]##, where the factor of ##\Big|\frac{1}{\lambda^{4}}\Big|## is the Jacobian for the coordinate transformation

Therefore, the scaling dimension ##D## such that the derivative terms remain invariant is found as follows:

##\Big|\frac{1}{\lambda^{4}}\Big| \Big[ \frac{1}{2}({\Lambda^{\rho}}_{\mu}\lambda^{-D}\partial_{\rho}\phi)(\lambda^{-1}x)({\Lambda_{\sigma}}^{\mu}\lambda^{-D}\partial^{\sigma}\phi)(\lambda^{-1}x)\Big]=\frac{1}{2}(\partial_{\nu}\phi)(\lambda^{-1}x)(\partial^{\nu}\phi)(\lambda^{-1}x)##

##\Big(\frac{1}{\lambda^{4}}\Big)(\lambda^{-D})(\lambda^{-D}) \Big[ \frac{1}{2}({\Lambda^{\rho}}_{\mu}{\Lambda_{\sigma}}^{\mu})(\partial_{\rho}\phi)(\lambda^{-1}x)(\partial^{\sigma}\phi)(\lambda^{-1}x) \Big]=\frac{1}{2}(\partial_{\nu}\phi)(\lambda^{-1}x)(\partial^{\nu}\phi)(\lambda^{-1}x)##

##\lambda^{-4-2D} \Big[ \frac{1}{2}{\Lambda^{\rho}}_{\mu}{(\Lambda^{-1})^{\mu}}_{\sigma}(\partial_{\rho}\phi)(\lambda^{-1}x)(\partial^{\sigma}\phi)(\lambda^{-1}x) \Big]=\frac{1}{2}(\partial_{\nu}\phi)(\lambda^{-1}x)(\partial^{\nu}\phi)(\lambda^{-1}x)##

##\lambda^{-4-2D} \Big[ \frac{1}{2}{\eta^{\rho}}_{\sigma}(\partial_{\rho}\phi)(\lambda^{-1}x)(\partial^{\sigma}\phi)(\lambda^{-1}x) \Big]=\frac{1}{2}(\partial_{\nu}\phi)(\lambda^{-1}x)(\partial^{\nu}\phi)(\lambda^{-1}x)##

##\lambda^{-4-2D} \Big[ \frac{1}{2}(\partial_{\rho}\phi)(\lambda^{-1}x)(\partial^{\rho}\phi)(\lambda^{-1}x) \Big]=\frac{1}{2}(\partial_{\nu}\phi)(\lambda^{-1}x)(\partial^{\nu}\phi)(\lambda^{-1}x)##

so that

##\lambda^{-4-2D}=\lambda^{0}##
##D=-2##

Am I correct so far?
 
Physics news on Phys.org
  • #2
Please help.

Yes, you are correct. The scaling dimension for the field is -2 in this case. To determine the values of m and p for which the scaling is a symmetry of the theory, we can look at the terms involving the field in the action. Since the scaling of the field is given by ##\phi(x) \rightarrow \phi'(x) = \lambda^{-D}\phi(\lambda^{-1}x)##, the term ##m^2\phi^2## scales as ##m^2\phi^2 \rightarrow \lambda^{-2D}m^2\phi^2##. For the scaling to be a symmetry, the term must remain invariant, so we have ##\lambda^{-2D}m^2 = m^2##, which implies ##D=0##. Therefore, for the scaling to be a symmetry, we must have ##D=0##, which means that the scaling dimension of the field must be 0. This is only possible if ##m=0##, since ##D=-2## for this particular theory. So, the scaling is a symmetry of the theory if ##m=0## and ##D=0##, which implies that the field has no mass and its scaling dimension is 0.

If the scalar field lives in an (n+1)-dimensional spacetime instead of a 3+1-dimensional spacetime, the result is still the same. The scaling dimension of the field must be 0 for the scaling to be a symmetry of the theory, which means that the field has no mass. The only difference is that the Jacobian for the coordinate transformation will be different, but the result will still be the same.

To construct the conserved current associated to scaling invariance using Noether's theorem, we start by considering the infinitesimal transformation of the field:

##\phi(x) \rightarrow \phi'(x) = \phi(x) + \delta \phi(x)##

where ##\delta \phi(x)## is the infinitesimal change in the field. This transformation should leave the action invariant, so we have:

##S = \int d^{4}x\ \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}-g\phi^{p}##

##\rightarrow S' = \int d
 

1. What is Scale Invariant Classical Field Theory?

Scale Invariant Classical Field Theory is a theoretical framework used to describe the behavior of fields in physics. It is based on the principle that the equations governing the fields should remain the same regardless of the scale at which they are observed.

2. How is Scale Invariant Classical Field Theory different from other field theories?

Scale Invariant Classical Field Theory is unique in that it does not rely on any specific length or mass scales. This means that the equations remain the same regardless of the size or energy of the system, making it applicable to a wide range of physical phenomena.

3. What are the applications of Scale Invariant Classical Field Theory?

Scale Invariant Classical Field Theory has many applications in various fields of physics, including particle physics, cosmology, and condensed matter physics. It has been used to study phenomena such as phase transitions, symmetry breaking, and the behavior of particles at high energies.

4. How is Scale Invariant Classical Field Theory tested and validated?

Scale Invariant Classical Field Theory is tested and validated through experiments and observations. By comparing the theoretical predictions to real-world data, scientists can determine the accuracy and validity of the theory.

5. Can Scale Invariant Classical Field Theory be combined with other theories?

Yes, Scale Invariant Classical Field Theory can be combined with other theories, such as General Relativity and Quantum Field Theory, to create a more comprehensive understanding of physical phenomena. This allows for a more complete description of the behavior of fields at different scales.

Suggested for: Scale Invariant Classical Field Theory

Back
Top