The Holographic Principle

Mathematical relationship or physical reality? Does the difference even matter?

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A particular buzzword that flies around physics articles these days is holography. When most people think of holograph it conjures images of science-fiction movies with shimmering images.  In the movies, the hologram is a 3 dimensional illusion projected from a 2 dimensional screen. The physics concept is a bit different.

In general, holography refers to some higher-dimensional phenomena (a 3D image in this case) that is ‘encoded’ somehow into a lower-dimensional form (a 2D projector).  In physics, this general idea is applied to reality itself with the holographic principle.

Black Holes at the Surface

During the early studies of black holes, it became clear that we could need to understand how they work not only in terms of their gravity, but in terms of their thermodynamics. In particular, it was important to determine the entropy of a black hole. Entropy is often referred to as the disorder of a system but it also reflects the density of information. Jacob Bekenstein developed a model where black holes contained the maximum possible entropy for anything with the same volume.

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S here represents the entropy for the entire volume.  k and l are constants. A is the surface area of the black hole.

In the course of understanding this they found that this maximum entropy was directly proportional to the surface area of the black hole. In other words, you only needed information about the surface of a black hole in order to calculate the entropy of the entire volume. Since higher dimensional information (the entropy of a black hole) is completely defined by a lower-dimension system (the surface), this is a form of holography. This observation has inspired physicists to ask why it is holographic? what does it mean? And are other physical systems holographic?

What does holography mean in Physics?

Since the early studies of black hole physics, new theories have adopted what we call the holography principle, the idea that a volume of space can be described by a lower dimensional boundary. Black holes were are early example, but now the concept is applied to string theories, quantum gravity and more. Some theories suggest that entire universe itself is actually caused by some behavior on a two dimensional boundary.

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Where holography is concerned, The behavior on the boundary is directly linked to what occurs in the volume.

Let’s unpack this a little bit because it isn’t entirely clear what all this means. A major dispute in discussions of relationships of holography is how to interpret it. The literal interpretation is that the higher dimensional reality doesn’t actually exist and that it is illusion created by interactions on some lower-dimensional structure. This goes against our intuition that 3 dimensional structures exist as whole entities that are more than just their bounding surface.

Another interpretation is that the holography principle doesn’t tell us what dimensions really exist or not, but instead it tells us about the relationships and structure of these system. Saying that we can describe something using a bounding surface doesn’t imply that the volume isn’t real, it is just that the information in that volume is bound by certain relationships that cause the total behavior to be describable by the surface. This approach has some interesting implications for how information itself flows in our universe.

Regardless of how you interpret it, the holography principle has many applications to physics. One of the major areas of research is in what we call AdS/CFT correspondence. This is a approach to quantum gravity where two theories are able to be connected through holography. Both theories produce the same results, however one theory acts on the volume (AdS) and one theory acts on the boundary (CFT).

These kind of correspondences is called a duality. That is where different theories and approaches end up being simply two ways of describing the same thing. It seems fitting that all the varied approaches to studying the universe might lead to different perspectives on the same reality.
VIDEO: The World as a Hologram,  Leonard Susskind
A different approach that focuses on the black hole information paradox when discussing holography. A very interesting watch.

PDF: The Holographic Universe, Jean-Pierre Luminet
A very readable and accessible introduction to the approaches to holography and their implications.
https://arxiv.org/ftp/arxiv/papers/1602/1602.07258.pdf

COURSE: String Theory and Holographic Duality,  Prof. Hong Liu
A free graduate course that introduces string theory and the AdS / CFT correspondence in detail.  Includes pdf readings and videos.
https://ocw.mit.edu/courses/physics/8-821-string-theory-and-holographic-duality-fall-2014/index.htm

Quick Intro: Group Theory

A group is an abstract description of how a system can transform.

Now and again in higher-level Physics you might run into terms like this: SO(3), Lie Group, U(2), etc… These sound like strange and disconnected terms but they are all part of how we categorize systems into groups.

What is a group you might ask? A group is an abstract description of how that system can transform. Specifically, a group consists of a set of elements that create our group space. With this group there is an operator that allows us to combine any two elements into a third element. The operator is usually represented by a plus + symbol so that if A, B and C are group elements, a statement like A + B = C makes sense. You might notice that this looks a lot like simple algebra, and groups do form their own algebra in a sense, but they aren’t limited to the math operators that we learned in primary school.

Since a group is completely abstract, it can have different ‘representations’ that express the group. A cyclic group can be represented by the way we rotate an object, so that if you add two rotations, you might end up rotating the object 360 degrees and returning to the original position. Another example of a group representation would be how we can combine simple twists of string to form complex braids (the group in this case is called the braid group).

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One example of a group is the ways that we can flip a mattress. This helpful article explores group theory in that scenario: https://opinionator.blogs.nytimes.com/2010/05/02/group-think/

While there are infinite representations of groups, we try to reduce each group to it’s simplest form, known as the irreducible representation. This allows us to compare different systems in terms of their groups.

Breaking systems down into their fundamental groups allows us to predict transitions that appear in the group before we can even observe them. Group theory can also be used to understand symmetry, this perspective can not only be applied to the physical sciences, but the study of art.

As far as physics goes, here are a few common groups that come up a lot:

SO(3): The special orthogonal group in three dimensions. This group consists of simple rotations in 3 dimension space around a fixed point.

U(2): The unitary group of degree 2. This group consists of all 2×2 matrices that are unitary.

SU(2): The special unitary group of degree 2. Like the unitary group, but the determinant of each matrix is equal to 1.

There are a few ways of discussing different types of groups. Many groups are characterized by some kind of symmetry, which is defined by a parameter of the system that doesn’t change during a transformation. For example matrices in the special unitary group always have a determinant equal to 1, or the special orthogonal group doesn’t change the length of a given radius as it rotates.

A Lie group is any group that is defined by a continuous transformation. A particle might have discrete transformation between one energy state of another, but rotations can be made continuously.

There are so many different ways of looking at groups. Even if it doesn’t become your field of study, it can make for a very interesting way of looking at different systems. Here are a few sources where you can learn a little bit more about groups:

PDF/BOOK: Introduction to Group Theory for Physicists – Marina von Steinkirch
This book breaks gives us the mathematical formulation of group theory and looks at a number of groups that have special significance for Physics.
http://www.astro.sunysb.edu/steinkirch/books/group.pdf

VIDEO: Group Theory – Robert de Mello Koch
The first in a series of lectures that cover the study of group theory and representations. A great place to start if you have some experience with Quantum Mechanics.
https://www.youtube.com/watch?v=3wNPrSwbtQ8

The Principle of Least Action and Lagrangian Mechanics

The principle of least action is a idea that helped transition physics away from Newtonian methods and toward a more general description of physical systems.

Lately I’ve been studying methods of calculating path integrals, and while I would very much like to write about that, I think I should start with a more foundational topic. The principle of least action is a idea that helped transition physics away from Newtonian methods and toward a more general description of physical systems. It has been so important that it is still used across many different areas of physics today.

Principle of least action

To start, we should discuss what the principle of least action actually is. In the 1600s, Pierre de Fermat discovered that light always takes a path that takes the least amount of time to traverse. Knowing the initial point, and the final point of light, you would know what path it took by finding that path that minimized the time taken by the light. This would become known as Fermat’s principle or the principle of least time.

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The light ray moves slower when passing through a material. As per Fermat’s principle, the ray is deflected in such a way that minimizes the light takes to pass from beginning to end.

This was a novel idea, since older approaches of physics first considered the present state of a system and then apply laws to determine what further point it will transition to. In Fermat’s principle, you consider the endpoints of the system (beginning and end of the light path), and then you could derive the laws based on some kind of constraint (ie: minimizing the time that light travels).

Lagrangian Mechanics

As it turns out, light isn’t the only place you can apply this kind of approach. In Lagrangian mechanics, you could define the starting point of a system, and the end point of that system, then you’d try to minimize and maximize a particular quantity known as the action (S). The action is an abstract value that depends on the path the system takes; if find the path that minimizes or maximizes the action, that path corresponds to the real path that the system takes. This sounds pretty abstract but there is a well defined system for calculating the action for a path, and it comes from formula you can build for your system called the Lagrangian function or just the Lagrangian (L).

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 The Lagrangian is defined at each point in time. Integrating along the path of the system provides the action for that path.

So by deriving the Lagrangian function and applying the principle of least action, you can describe almost any physical system. This has been a very successful approach and was later adapted to a similar technique in what we call Hamiltonian mechanics.

There are some conceptual oddities of Lagrangian mechanics. Particularly, the idea of both the past and the future influencing the behaviour of the present can be a little discordant with our idea of causality. Regardless, it has been proven as an effective approach to understanding the behaviour of a large variety of scenarios in physics.

What does it mean?

The principle of least action is an extremely handy concept that lets us tackle some problems that would otherwise be extremely difficult in physics. A commonly used example is the problem of coupled oscillators. In this problem, a series of masses are connected by springs and strung between two walls. Deriving it using tradition newtonian methods can be extremely tedious for a large number of masses. On the flip-side using lagrangian mechanics can vastly simplify the problem. If you are interested in lagrangian methods for harmonic systems, this problem can illustrate can illustrate the power of lagrangian methods:

Coupled Harmonic Oscillators by Irina Yakimenko

https://people.ifm.liu.se/irina/teaching/sem6.pdf

Keep in mind that this approach can offer more than just mathematical power, it is conceptual change in how we look at interactions in the universe. Instead seeing the universe as of a series of sequential events, we can see a system of relationships across time and space that emerge from the optimization of a few constraints.

The principle of least action also provides interesting implications for our perspective of time. As mentioned earlier, since the lagrangian derives our equations of motion from past and future constraints, it violates our intuition about causality. Some have argued that it implies an eternalist universe where the past, present and future all exist simultaneously and are separated only by the dimension of time. This is largely philosophical and as such, is difficult to demonstrate with any certainty.

 

Here are a few resources for getting a handle on Lagrangian Mechanics and the principle of least action:

PDF: Introduction to Lagrangian and Hamiltonian Mechanics, Melanie Ganz

A short pdf that introduces the main concepts and equations of Lagrangian Mechanics

http://image.diku.dk/ganz/Lectures/Lagrange.pdf

BOOK: Mechanics: Volume 1, Landau and Lifshitz

The go-to book on classical mechanics. Uses Lagrangian methods from the beginning and covers many different methods applying them to mechanical situations.

https://www.amazon.ca/Mechanics-1-L-D-Landau/dp/0750628960

VIDEO: A Special Lecture: Principle of Least Action, Kenneth Young

A fantastic introduction to the material and great speaker.

https://www.youtube.com/watch?v=IhlSqwZBW1M