# Internal and spacetime symmetries?

If the probability distribution has any target space symmetries, i.e. symmetries in the space of the stochastic variable (also called internal symmetries), then the correlation matrix will have induced symmetries. If there are symmetries of the space (or time) in which the random variables exist (also called spacetime symmetries) then the correlation matrix will have special properties.

And then it follows with two examples about symmetries. However I didn't quite understand the definition nor I know any intuitive example for each case?

Can somebody help me regarding this; first an explanation of definition and secondly some examples. What does it mean to say "target space" symmetries in space of stochastic variable and what is the difference of that to "symmetries of the space (or time) in which the random variables exist"?

• The first are related to changes in the space time coordiantes (e.g. a translation), the other ones to internal degree of freedom (e.g. gauge symmetry). It depend on how you're into these argument (field theory) on how deep the answer should be. – Antonio Ragagnin Apr 2 '14 at 18:22

Let $M$ be a manifold, let $V$ be a finite-dimensional vector space, and let $\Omega$ be a sample space (in the sense of probability). For each point $p\in M$, let $X(p):\Omega\to V$ be a random variable.

A mapping $T_V:V\to V$ is called a target space transformation and a mapping $T_M:M\to M$ is called a space transformation (or spacetime transformation if $M$ can be thought of as a spacetime.)

Let $\rho:\Omega\to \mathbb R$ be a probability distribution on $\Omega$. We call a function $T_\Omega:\Omega\to\Omega$ a sample space transformation. We say that the measure $d\omega \rho(\omega)$ is invariant under this transformation provided $dT(\omega)\,\rho(T(\omega)) = d\omega\,\rho(\omega)$ for all $\omega\in \Omega$. In this case $T_\omega$ is called a symmetry. We define the correlation function $C^{ij}_X(p,q)$ as \begin{align} C^{ij}_X(p,q) = \int_\Omega d\omega\,\rho(\omega) \,X^i(p)(\omega)X^j(q)(\omega) \end{align} where $X^i(p)$ is the $i^\mathrm{th}$ component of $X(p)$ when it is expanded in some basis of $V$.

Example. Conformal field theory on $\mathbb R^2$.

Let $M = \mathbb R^2$, $V = \mathbb R$, and $\Omega$ be a set of admissible functions $\phi:M\to V$. This is the situation one would have in a quantum field theory of a real scalar field on the plane. For each $\lambda\neq 0$ and each real $\Delta$, we define a scale transformation $T_\Lambda$ on $\Omega$ as \begin{align} T_{\lambda,\Delta}(\phi)(\mathbf x)= \lambda^{-\Delta} \phi(\lambda^{-1}x) \end{align} which is simply the space transformation $\mathbf x\to \lambda^{-1}\mathbf x$ composed with the target space transformation $\phi(\mathbf x)\to \lambda^{-\Delta}\phi(\mathbf x)$. Formally, there are certain distributions on $\Omega$ that exhibit so-called scale-invariance, namely invariance under $T_{\lambda, \Delta}$. Let $X_\phi(\mathbf x)$ be the random variable \begin{align} X(\mathbf x)(\phi) = \phi(\mathbf x) \end{align} then given scale-invariance, $T_\omega$ induces a scale transformation on the correlator which I leave it to you to verify. \begin{align} C_X(\mathbf x, \mathbf y) \to \lambda^{-2\Delta}C_X(\lambda^{-1}\mathbf x, \lambda^{-1}\mathbf y). \end{align}

• Thank you very very much for this detailed answer. I am actually coming from mathematical background. Still there are two things that I don't understand: One is that when you apply the transformation how that will affect the association between points in M and V(actually X) say, when transformation is not 1-1. And secondly I was asking examples not for transformations but for symmetries?; according to your answer an example for that will be correlation function in target space(although I would be quite grateful if you give a real example for that or guide me to some text for it) – Cupitor Apr 2 '14 at 23:22
• And what about symmetries in domain?(on M) – Cupitor Apr 2 '14 at 23:23
• @Cupitor I edited the answer to make it both more terminologically consistent with your question, and to include an example of the type you requested. I'll try to include another example with $V$ having dimension greater than one as soon as I have more time. – joshphysics Apr 3 '14 at 1:35