Is the collision in LHC quantum mechanically so we cannot predict the trajectory?

Question

I'm puzzled as always. In the LHC 2 streams of particles are accelerated in opposite direction and allowed to smash against each other in a controlled accident, then is it classical or quantum mechanically? For classical we can know where the products will end up but for quantum mechanical then we must examine all the detectors see which one has picked up the signals.

Answer 1

The particles in the accelerator are described within the framework of quantum field theory, a relativistic quantum theory that treats particles as excitations of interacting operator-valued fields. This theory allows for the prediction of a "differential cross section" that explains the experimental output. The output field is described in terms of a Hilbert space known as a "Fock space" which is related to a similar input Fock space via a linear transformation called the "scattering" matrix, or "S" matrix for short. The "S" matrix encodes the details of the interaction and allows one to relate the input and output field operators with a similarity transformation, i.e. $\phi_{in}=S\phi_{out} S^{\dagger}$. From there, one can calculate a differential cross section, a simple example (electrons colliding with positrons) of which is given below: $${d\sigma\over d\Omega}={\alpha^2\over4E^2}(1+cos^2\theta).$$ The important thing to take away is that the differential cross section is a real number derived from quantum field theory that can be measured in a collider experiment. They are analogous to the differential cross sections of ordinary classical dynamics, but are derived from principles of modern physics.

Thus, one can mathematically predict the outcome, the differential cross section, of collider experiments with quantum field theory. The detector experiments allow one to test the theoretical constructs to verify whether or not they have real predictive power and are thus good theories. On the other hand, studying the experimental results can guide theoretical research in the correct direction so that theory and experiment are a mutually reciprocating cooperative. For example, the LHC allowed researchers to confirm what theory had long since predicted, viz. that there should exists a large mass bosonic quantum, the "Higgs boson". If anomalies are detected in the scattering results, this indicates that researchers must adjust the theory so that it produces the correct physics.