A whirlwind tour through Quantum Field Theory to Conformal Field Theory

This is designed as a very casual introduction, there are many aspects which are brushed over and ignored. This is because either it isn’t specifically relevant or I didn’t want to bog down the post with unnecessary minutia when the motivation is just to give one an idea of the subject. There are no references here, if you want one check any textbook worth its salt. This is merely a retelling geared towards my specific goals for future posts.

Quantum mechanics (not yet field theory) is an inherently probabilistic theory. Solving the Schrödinger equation gives a wavefunction ${\psi}$ which dictates how a state will evolve from some initial set of conditions. In order to make measurements in quantum theory, self-adjoint operators within a Hilbert space ${\mathcal{H}}$ must be used. The Hilbert space is required to ensure that the operators used are well defined. With a wavefunction ${\psi}$ , it is possible to measure its position using the position operator ${X}$ then

$\displaystyle \left(X\psi\right)(x)=x\psi(x) \ \ \ \ \ (1)$

with ${\psi\in L^{2}(\mathbb{R})}$ defined as square integrable function on ${\mathbb{R}}$ . Now a spectral projection is defined to any interval ${I\subset\mathbb{R}}$

$\displaystyle \left(P\left(I\right)\psi\right)(x)=\chi_{I}(x)\psi(x) \ \ \ \ \ (2)$

with

$\displaystyle \chi_{I}(x)=\left\{\begin{array}{ll}{1} & {x \in I} \\ {0} & {x \notin I}\end{array}\right. \ \ \ \ \ (3)$

and ${P^{2}=P=P^{*}}$ . The map ${I\rightarrow P(I)}$ extends to a projection-valued measure and to each square integrable ${\psi}$ that is normalised ${I\rightarrow \langle\psi|P(I)\psi\rangle}$ extends to a probability measure. So if one were interested in the probability of finding ${\psi}$ in the range ${(a,b)}$ the calculation would then be

$\displaystyle \langle\psi|P\left(\left[a,b\right]\right)\psi\rangle=\int_{\mathbb{R}}\overline{\psi(x)}\chi_{\left[a,b\right]}(x)\psi(x)dx \ \ \ \ \ (4)$

which reduces to the standard expression in undergraduate texts!

$\displaystyle \int^{b}_{a}|\psi(x)|^{2}dx \ \ \ \ \ (5)$

To every self-adjoint operator ${A}$ on ${\mathcal{H}}$ there is a projection valued measure ${I\rightarrow P_{A}(I)}$ and to each state ${\psi\in \mathcal{H}}$ that is normalised ${||\psi||=1}$ , ${I\rightarrow\langle \psi|P_{A}(I)\psi\rangle}$ gives a probability measure. We can define

$\displaystyle Prob\left(A\in I|\psi\right)= \langle\psi|P_{A}(I)\psi\rangle \ \ \ \ \ (6)$

and have the following result for some function ${g}$

$\displaystyle \langle\psi|g(A)\psi\rangle=\int g(\lambda)d\langle\psi|P_{A}(-\infty,\lambda)\psi\rangle \ \ \ \ \ (7)$

If one has a discrete spectrum of eigenvalues ${\lambda_{n}}$ with basis ${ | e_{n}\rangle}$ one can find the related formula

$\displaystyle \langle \psi|g(A)\psi\rangle=\sum_{n}g(\lambda_{n})|\langle e_{n} | \psi\rangle|^{2} \ \ \ \ \ (8)$

In the original theory of quantum mechanics the objects solved for by the governing equations of Schrödinger, Klein & Gordon and Dirac were wavefunctions. It was found however upon moving from the original quantum mechanics to quantum field theory (and by continuation to conformal quantum field theory) that the field ${\phi(x)}$ is not well defined. Instead it is necessary to use operator valued distributions which will become operators upon smearing against a smooth test function. An example of smearing ${\phi}$ against a test function ${f}$ is given as

$\displaystyle \phi(f)=\int\phi(x)f(x)d^{d}x \ \ \ \ \ (9)$

These test functions are a fundamental part of the theory and the choice of them can have a profound impact on what is seen in terms of the probability distribution.

The stress energy tensor is used within both quantum field theory and general relativity. It contains the energy density and related quantities in the theories. Within this project the goal is to get to the probability distribution of the stress energy tensor smeared against a function ${f\in\mathbb{R}}$ defined as

$\displaystyle T(f)=\int_\mathbb{R}T(u)f(u)du \ \ \ \ \ (10)$

Conformal quantum field theory is defined to be a quantum field theory which obeys conformal invariance or has an invariance under transformations from the conformal group. One can think of conformal transformations as those which map from the space to itself & preserve angles within the spacetime. The main interest for my work results in working on a ${1+1}$ Minkowski spacetime which means using only one spatial and one temporal variable on a flat spacetime. In ${1+1}$ dimensions, space can be either a line or a circle. The ability to map between these two representations allows the use of a larger class of techniques within the theory as a whole. I frequently make use of the Cayley map for this reason, it allows one to draw an equivalence between the stress energy tensor on the line and on the circle and use problem solving techniques in either scenario for progress.

This scale invariance property means that a conformal transformation is a local rescaling of the metric. This often allows for simplification in calculation and CFTs can allow for exact calculations within quantum field theories which in and of itself incredibly valuable.

CFT is also used throughout String theory due to the description of the worldsheet of a string’s excitations, admittedly I’m less interested in this area but it is of note, and before I can be accused of playing around in abstraction I would note to the reader that other uses of CFTs include statistical physics and condensed matter. Admittedly I’m not quite at the stage of doing lab work but here we are… I will in future posts use conformal methods to motivate work without this brief tour through its preliminaries.

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