Large and small-scale phenomena
- Vesna Lukic
- May 31, 2021
- 6 min read
In the previous posts we explored applying deep learning techniques to classify large-scale structures such as radio galaxies, as well as using a variation on these techniques to find the sources.
In the current post we will discuss phenomena that occur across different length scales, as well as explore how large-scale phenomena emerges from laws at the microscopic level.
On one extreme of the length scale, on the order of minuscule fractions of a metre, we enter the world of fundamental particles. The image to the left shows a simulation of a collision that would produce a microscopic evaporating black hole, as seen by the ATLAS detector. Image credit:
"CERN / ATLAS Particle Collision" by Ars Electronica is licensed under CC BY-NC-ND 2.0.
At the other end of the length scale spectrum, we observe galaxies, such as the radio galaxies which we have worked on classifying, which span cosmological distances. To the right is shown a Hubble Ultra Deep Field image of 10,000 galaxies, dated from a few hundred millions years after the Big Bang. Image credit: "Hubble Ultra Deep Field 2014" by NASA Hubble is licensed under CC BY 2.0
The behaviour of a system depends on the length scale that it is viewed, so different laws operate on different scales. For example, on length scales above 10^-31 - 10^-9 m, we observe subatomic particles such as protons, neutrons and electrons, and their constituents, such as gluons and quarks.
The laws of quantum mechanics and quantum field theory dictate the behaviour on the sub-atomic scale level. Shown to the left is the internal structure of the proton. It is made up of two up quarks and a down quark. The springs represent the gluons, which carry the strong force. Source: Wikimedia Commons. Author: Bakeed
In quantum mechanics, particles take on a wave-like nature. When presented with several different paths to go from point A to point B, the particles can sample the paths simultaneously. In the subatomic world of quantum mechanics, superpositions of states, particles going back in time, and virtual particles appearing and disappearing are observed. The behaviour of quantum systems is very different to the classical world we have direct experience with.
The diagram to the right illustrates the principle of superposition in action. What is shown are the phase-amplitude plots of several Laguerre-Gauss modes. The phase and amplitude are represented by the color and brightness respectively. The bottom right plot shows that the system can be in a superposition of the m=+1 and m=-1 states at the same time, where m is the orbital angular momentum quantum number. Source: Wikimedia commons. Author: Tyharvey313
In quantum field theory, Feynman diagrams are very useful in computing the interactions between fundamental particles. The left and right sides of individual Feynman diagrams represent the initial and final states respectively. The parts in between represent the manner of particle interactions, over all the possible time-orderings that the interactions can occur.
For example, to the left is shown the cross section of the interaction a+b->c+d, which is made up of two possible time orderings. The interaction is mediated by the exchange of a virtual particle. Source: Modern Particle Physics by Mark Thomson
As we increase in length above around the nanometre scale, the interaction and coupling of particles with the environment causes the loss of coherence - also known as decoherence. This is what leads to a transition from quantum to classical behaviour.
The plot below shows the time evolution of a Gaussian wave packet. The initial state is given in the left-most plot. The width of the Gaussian in the off-diagonal direction becomes increasingly reduced due to the presence of a scattering environment, resulting in a probability distribution of positions centered around the diagonal.
Therefore, decoherence explains why quantum behaviour does not persist above a certain length scale.
A useful way to illustrate how the macroscopic world arises from small-scale interactions is to look at Effective Field Theory. As an example, one can consider the macroscopic behaviour of a system consisting of say 10^30 particles. The properties can be determined by treating the system quantum-mechanically, where the interactions between all combinations of particles and the environment are taken into account, however this is very computationally inefficient and time consuming. Effective Field Theory allows one to get a reasonable approximation to the overall behaviour of the system, and to see how large-scale properties such as temperature and pressure emerge.
Shown below is another example of the transition between the microscopic to the macroscopic world. If we consider the force of pressure of individual fluid particles on increasingly smaller surfaces, the measurement would show chaotic fluctuations with time. However if we consider the pressure on increasingly larger surfaces, it will become more stable due to the contribution of increasing numbers of particles.
If the laws of physics stay the same despite changing the length or energy scale, they are scale invariant. In order to investigate the changes of a physical system at different length scales, one can use the renormalisation group equations. If we consider varying the energy or length scale at which different physical phenomena occur in particle physics, it is possible to observe the changes in the underlying force law.
For a theory to be renormalisable, the system at one scale should be made up of self-similar copies at a smaller scale, with a change to the parameters that describe the parts of the system.
We can think of self-similarity in a different way, for example by observing the internal structure of a leaf, where the larger veins have a similar pattern to the smaller veins. Source: "Fractal pattern"by @Doug88888 is licensed under CC BY-NC-SA 2.0
In particle physics, renormalisation was introduced to address the infinities in a quantum field theory, by developing a theory to renormalise mass and charge.
The original idea for renormalisation stemmed from condensed matter physics, where a block-spin renormalisation group was proposed. In this system, multiple length scales exist simultaneously.
The idea was to consider collections of components from smaller length scales as defining the components of the theory at larger scales.
It is assumed that only nearest-neighbour interactions exist, where the strength of the interaction is denoted by C. The second assumption is that the system is at a given temperature T. The system can be described using a Hamiltonian H(T,C), which is an equation describing the total energy of the system.
To get the average behaviour of the block, one can divide it into smaller blocks of 2*2 atoms. A similar Hamiltonian can describe the behaviour of these smaller blocks, but with different values of T and C. The original problem is too hard to solve, because there are too many atoms to consider. However, if we keep dividing the system into increasingly smaller blocks, there are fewer atoms and interactions to consider, so the system becomes solvable.
In order to get the long-range behaviour of the entire block, we keep iterating over the smaller blocks until we get left with the single block. In doing so we find that the system ends up with some fixed points.
One fixed point corresponds to T=0, infinite coupling C, where there is no disorder due to the coupling. Another fixed point corresponds to C=0, infinite temperature T, where the system is completely disordered. The third fixed point corresponds to a critical temperature T and coupling C, here changing the scale does not change the physics, and the system is in a fractal state, as shown in the figure.
In summary, we discussed how large-scale phenomena emerges from laws at the microscopic level, and behaviours that can be observed across different length scales, from the subatomic to cosmological scales. We discussed how as systems increase in size from the subatomic scale, there is more interaction with the environment, which causes them to decohere and transition to classical rather than quantum behaviour. We also covered Effective Field Theory, which is used as an approximation to describe large-scale behaviour without having to compute interactions between large numbers of individual particles. The last area discussed was renormalisation theory, which was invented to address the infinities encountered in quantum field theory. The idea for renormalisation originated from considering a block-spin model, of particles having a coupling C and an overall temperature T. One can solve for the overall behaviour of the system by continually dividing it into smaller-sized blocks, where eventually several fixed points are found, in considering the extremes of the coupling and temperature values.
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