New Directions in the Foundations of Physics 2014

This is the last post about the conference because the remaining of the talks gradually drifted outside of my domain of expertise. Next time I’ll resume the geometry series. For today I want to talk a bit about the Many Worlds Interpretation (MWI) which was showcased in its talk by David Wallace http://arxiv.org/abs/1111.2187v1 and about Causal Sets (presented by Rafael Sorkin and David Rideout).

Since I am working in an approach different than MWI which also contends that quantum evolution is pure unitary I am very critical of MWI claims. I talked in the past about MWI but I want to clearly present why I believe MWI is fundamentally misguided.

When I was first introduced to quantum mechanics in college, the lessons followed the standard approach: Hilbert spaces, kets, bras, Schrodinger equation. Then at the end of that class, mixed states were introduced as an afterthought, and they were introduced in a misleading way. Suppose we prepare many copies of a quantum state A, and many copies of a quantum state B. We can then create a mixed state by the following preparation process: randomly choose p% of the time the quantum state A and (1-p)% of the time quantum state B. For example, suppose some photons are vertically polarized (state A), and some photons are horizontally polarized (state B). Then if we randomly select 50% of the time horizontally polarized photons and 50% of the time vertical polarized photons we end up with a random mixture of the two. Big deal I said at that time. Why do we want to do that? To make our lives harder in computing the answers to problems? And in thinking that, I completely missed the point about mixed states. Here is how:

The key point of mixed states IN QUANTUM MECHANICS is that the decomposition of MIXED STATES into PURE STATES  is not unique.  Unlike classical mechanics, an ignorance interpretation of mixed states in not tenable. For example, suppose that in the process above, “A” corresponds instead to photons which are left-circularly polarized, and “B” corresponds to photons which are right-circularly polarized. Mix them up 50%-50% and the end state is the same as mixing vertical and horizontal polarized photons: no experiment is able to distinguish between the two preparation procedures.

So now back to MWI. MWI asserts that the wavefunction evolves ONLY unitary, and this means that until observation, Schrodinger’s cat is both dead and alive. However, experiments reveal only a dead or an alive cat, and MWI’s explanation is that the world splits into two copies at the time of measurement: in one branch the cat is dead, while in another the cat is alive. (Isn’t this nice? It means we are all immortals: there is always a branch in which we never die).

But here is the catch: if the quantum state is in a mixed state, there is an ambiguity on which decomposition is used by the measurement process. For a photon in the mixed state from above, is the world going to split into a branch where a photon polarized vertically and a branch where it is polarized horizontally, or is the universe going to split into two branches where the photon is left or right circularly polarized? Somehow there must be a preferred base for this split. On this MWI is silent, but decoherence comes to the rescue. Indeed, decoherence solves the basis ambiguity problem, but in the process we miss a fundamental tension between MWI and decoherence:

• On one hand decoherence is trivial and natural because is rooted in unavoidable interaction with the environment.
• On the other hand, the universe splitting process is nothing short of extraordinary: a humble photon can split the entire universe into two branches.

Why is the split happening only after decoherence? What is so SPECIAL about an un-special, mundane, common-place process like decoherence? This is MWI’s big pink elephant in the room: merging two ideas opposite in spirit to solve the measurement problem. MWI without decoherence does not work due to basis ambiguity; decoherence without MWI does not work because the outcome is not unique. Together they work, but it is like mixing fire with water. No wonder some people deride MWI and call it: the Many Words Interpretation. I fully agree with Zurek when he says MWI has the feeling of a cheap shot in solving a complex problem.

Now onto Causal Sets. I never listened before to a talk about this approach and I was fortunate to hear Rafael Sorkin’s cogent presentation. To me, my first impression was about the unreasonable effectiveness of the approach. The key result of this approach is the correct order of magnitude prediction for the Cosmological Constant: 10^-120.

So how do the competing physical theories stack up to the challenge of predicting the value of the cosmological constant?

• In quantum field theory, vacuum is a very violent place and if you count vacuum fluctuations as contributing to dark energy then you make a prediction which is wrong by about 120 orders of magnitude!!!
• In supersymmetric field theory, the cosmological constant must be exactly zero!
• String theory predicts a small value, but of the wrong sign, and models of the right sign were called Rube Goldberg models.

Where all the smart and complex solution failed, causal sets succeeded. Here how the magic happens:

The diameter of the known universe in Plank’s units is 10⁶⁰ It’s volume is then approximately ~10²⁴⁰ and the cosmological constant Lambda is computed by the model to be = 1/sqrt(Volume)~ 10^-120 (applause please).

This is because a causal set is a set of elements with an order relation: transitive + acyclicity + local finiteness (discreetness). In causal sets “Geometry =order + number” and causal set are uniformly embeddable in a manifold. Hence Number=Volume.

Causal sets can be employed also outside physics, for example in the growth of the web pages on the internet! But the predictions become bizarre and obviously wrong in this case: just as the universe can expand and recollapse in causal set cosmological models, so too the Internet can vanish one day by pure chance of birth and death of new web pages.

This is why my impression of causal sets is that of unreasonable effectiveness. I tend to think the approach is incorrect, but still, I cannot simply discard the correct cosmological value prediction where no other theory succeeded. Either something is fundamentally right in this approach, or the model is simply incredibly lucky – don’t know which one is the correct explanation.

A flea on Schrödinger’s Cat

New Directions in the Foundations of Physics  2014

Continuing the conference talks presentations, in “Asymptotic Theory Reduction and the Measurement Problem”, Klaas Landsman introduced a fresh start on the measurement problem.

Landsman started by defining low-level and high-level theories:

L = lower-level theory = fundamental theory (physics) = reducing theory (philosophy)

H = higher-level theory = phenomenological theory (physics) = reduced theory (philosophy)

Here are some examples:

L = Quantum Physics                            (ħ→0)                         H = Classical Physics

L = Statistical Mechanics                      (N→∞)                         H = Thermodynamics

L = Molecular Dynamics                                                           H = Hydrodynamics

L = Wave Optics                                  (wavelength→0)           H = Geometric Optics

Then there are a couple of observations:

• H is defined and understood by itself
• H has ‘novel’ feature(s) not present in L (classical physics has counterfactual definiteness, thermodynamics allows irreversibility, etc)

Also a quick observation is that technically ħ→0 is apparently impossible because ħ is a constant. However ħ→0 is actually shorthand for changing a dimensionless combination. For example ħ²/2m→0 is the same as m→∞

Now back to the measurement problem. Here there is no consensus among physicists: some claim it is not a problem, some that it is a pseudo-problem, some that it is a very serious problem.  Basically the problem is that quantum mechanics fails to predict that measurements have outcomes:

-theoretically, Schrödinger’s Cat states of L yield mixed limit states of H

-experimentally, outcomes are sharp, hence pure states in H

Regardless of physicist’s consensus (or lack of it), this can be stated in a mathematical precise way:

H is ħ→0 limit of L, but limit L→H induces the wrong classical states.

The proposed solution: asymptotic reduction similar with spontaneous symmetry breaking. Here is how it works in a completely soluble example:

Start with a symmetric double well potential in classical physics which has reflection symmetry. A test classical particle at rest can reside either at the left or right potential well bottom, but not in the middle (between the two minima) because it is an unstable equilibrium point. Hence a symmetric invariant state (a mixed state) in unphysical.

Now solve the same problem in quantum mechanics and observe that the ground state (the lowest energy state) in symmetric! Then we can take the limit ħ→0 and obtain two sharp localization peaks.

But then here comes a flea!

It can be shown analytically in several problems (double well potential, quantum Ising model, quantum Curie-Weisz model), that a tiny perturbation induces an exponential splitting of the lowest pair of energy levels. And this shows that the ground state of perturbed Hamiltonian shifts to a localized state and the density matrix not only decoheres but gets single peak!

In turn this implies that asymptotic emergence does not exist because tiny perturbations have exponentially large effects! In short, reduction is real, emergence is not.

The “Old Measurement Problem” solutions also involve reduction. The “Old Measurement Problem” demands to show that a pure state evolves to a mixt state. The proposed solutions were:

-Classical description of apparatus (Borh) for ħ→0 or N→∞

-Superselection sector (Hepp, Emch, Wigner) for N→∞

-Decoherence (Joos, Zeh, Zurek) for t→∞

Then selection of one term in mixture would completely solve the measurement problem.

The proposed solution shows how pure unitary time evolution is compatible with wavefunction  collapse. So the Everett interpretation is NOT the only game in town when it comes to unitary evolution explanation of the collapse.

There are several open problems related to this approach:

• Where does the “flea” come from?
• Are the perturbations deterministic or stochastic?
• Is the mechanism dynamically viable?
• Is the mechanism experimentally testable?
• Is the mechanism universal?

For references, see this paper and this paper.

Decoherence in a box

New Directions in the Foundations of Physics 2014

Schrodinger considered superposition the characteristic trait of quantum mechanics:

"When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives [the quantum states] have become entangled.”

So why we don’t see then in our macroscopic world a cat both dead and alive? In other words, how does classicality emerge from quantum mechanics? The answer comes (in part) from decoherence which shows that interaction with environment makes the density matrix diagonal, free of interference terms. But is this a general enough mechanism?

In his talk “A Closed System Perspective for Decoherence”, Sebastian Fortin presented a framework which generalizes the current approach for both open and closed systems.

First let’s present the problem with closed systems. Unitary time evolution in quantum mechanics can prevent canceling the non-diagonal parts of the density matrix in a closed system. Decoherence works for open systems by talking advantage of the very large numbers of degrees of freedom of the environment which can absorb the “unwanted” information. There are two time characteristics in an open quantum system: decoherence time and relaxation time.  You can have decoherence without equilibrium (an obvious example are the planets of the solar system) but dissipation always implies decoherence. (This means that a quantum system reaches decoherence faster than equilibrium.)

But how can we talk about irreversible processes in quantum mechanics? Doesn’t this contradict unitary time evolution? Non-unitary time evolution can arise when we sum over (trace over) the degrees of freedom of the environment. This leads to a (non-unitary) master equation.

In a closed systems, equilibrium or standard decoherence may not occur, but we can still get emergent classical behavior. So here is Fortin’s proposed solution: non-unitary time evolution is achieved by course-graining:

• split the information in relevant and irrelevant parts
• compute time evolution of relevant observables
• determine if you reach equilibrium (achieve relaxation)
• if so, compute the decay times

This is not at all unusual: for example in the kinetic theory of gasses individual molecules never rest and reach (static) equilibrium. However, course graining can extract the relevant macroscopic information: gas density, pressure, etc.

In a closed quantum system the non-diagonal elements do not go to zero, but they may cancel each other out.

This mechanism also works in open systems in addition to the standard way of evolving the non-diagonal elements to zero. Hence, the framework is universal. As an application: Fortin talked about self-induced decoherence:

• Operators belong to relevant observables O
• States belong to dual of O
• Computed expectation values are real quantities

A good example of relevant observables is van Hove observables. To understand more of the proposed framework, I recommend this paper:  “The problem of identifying the system and the environment in the phenomenon of decoherence”.

The key strength of the proposed unified classicality emergence framework is that it can be put to the test against real physical systems, and its predictions can be confirmed or not by experiments. 

Two tales of time: Carlo Rovelli vs. Lee Smolin

New Direction in the Foundations of Physics 2014

Following the New Directions in the Foundations of Physics conference, two interesting consecutive talks made the case for opposite points of view and it is best to present them at the same time. On one hand, Carlo Rovelli made the case for the emergence of space and time and talked about the possibility of having fundamental physical theories without talking about space and time at all. On the other hand Lee Smolin made the point for the reality of time and the possibility of change for the physical laws. Because the problem of time is very hard and there is no universally accepted solution, this debate cannot be settled for now.

Let’s start with Carlo Rovelli’s position. From general quantum gravity considerations, it is not that strange to consider the possibility that space-time is not continuous. But how can you recover time? There are intuitive arguments and mathematical rigorous arguments available.

For example we can ask a silly question: “why things fall in Newtonian physics?” There is no notion of up and down in Newton’s equations. The answer is obviously due to the existence of our planet that “up” and “down” are defined. So this is a relational definition. Another question: what does it mean that entropy increases? We know that entropy increases in time but this can be turned around and used to define time in a relational way too: time is the variable in conjunction to which entropy grows. Let’s try to make this more precise. Here is the usual picture in physics: from averages in time to thermal phenomena to equilibrium states. If we flip this: start with an equilibrium state, extract the dynamics (the Hamiltonian) and arrive at time. This is the idea of the Thermal Time. At this point you may object to this seemingly naïve picture and you will have very good reasons for this. What if the system is not in equilibrium? Does time exist? What if you have only one particle for which the notion of thermodynamics is meaningless? Does time exist? However the thermal time hypothesis is not at all naïve and has a very solid mathematical and physical foundation in advanced quantum mechanics theory. This deserves a post of its own and after I will be done presenting the conference talks I will resume the mathematical explanations and build the pre-requisites to properly explain and discuss the thermal time hypothesis.

Now for the opposite point of view, that of reality of time, Lee Smolin started by presenting some ideas about time:

           

• Timeless naturalism: related to block universe idea in relativity
• Temporal naturalism: no timeless objects, future is not real, past is real
• Barbour’s instantaneous naturalism: only moments exists which are all timeless

Smolin argued for the need to start over in thinking about time and discussed some fallacies:

• Cosmological fallacy: we look at the universe as a whole from a “bird eye’s view”. There is no such thing as viewing the universe form the outside.
• Naturalism trap: “our senses impressions are illusions, and behind them is a natural world, which really is X”.
• For example X = Tegmark’s mathematical objects which he called a “metaphysical fantasy”

Two paradigms were discussed:

• Newtonian paradigm (or the usual paradigm in physics. By the way, this includes quantum mechanics and the name is somewhat misleading):
• Laws and initial conditions are the key ingredients
• A state space (invariant under time) is constructed.
• The history in time is represented by a timeless mathematical object
• It is fallacious to infer from this that nature is a mathematical object or that is really timeless.
• Falls apart when we apply to universe as a whole
• Temporal naturalism and the laws of nature paradigm (a proposed paradigm for answering: “why those laws?”)
• No timeless laws
• State-laws distinction breaks down
• Time is prior to laws of nature
• Meta-law dilemma: how the laws change?

A solution for the meta-law dilemma was proposed by a “principle of precedence in quantum theory” described in arxiv:1205.3707

Of course temporal naturalism is highly speculative and attempts to win acceptance by making falsifiable predictions with the hope that the predictions will turn up to be true. For example an older idea by Smolin was that if our universe is typical then it is likely to be tuned to maximize the production of black holes. Then if the universe expands and re-collapses, at each bounce the laws of physics mutate slightly. Big if, but maximization of black hole production demands:

• Small amounts of carbon and oxygen  for star formation dynamics
• Supernovas require tuning weak interactions
• Gravity must be weak.

Are those predictions enough? Not at all, but it is a start. How promising? Remains to be seen.

The Bronstein Hypercube of Quantum Gravity

New Direction in the Foundations of Physics 2014

As advertised, I will stop the mathematical presentations for a little while and I will present interesting talks from New Direction in the Foundations of Physics 2014 conference in Washington DC.

The first talk was by Daniele Oriti and was entitled “The Bronstein Hypercube of Quantum Gravity”*. To introduce the topic I wave to say a bit about the Bronstein cube. Matvei Bronstein imagined a cube specified by the fundamental constants of nature and placed at the corners various limiting theories.

This was popularized by Sabine Hossenfelder at her blog:  and the point of Daniele Oriti’s talk is that we need an extra dimension added to this picture. First how can we visualize such an object (which is called a Tesseract)? The shadow on a 2-dimensional surface (the computer screen) looks like this:

and in each node all lines are suppose to make 90 degrees angles (impossible to build in 3D), but you get the picture.

The point is that in addition to gravitational constant, the speed of light, and Planck’s constant we need to consider N-the number of degrees of freedom- because the limits for large N is qualitatively different.

For example, it is evident that fluids represent an emergent behavior of a large number of molecules under certain conditions. Here is a picture from the talk illustrating emergence of new physics in case of a large number of degrees of freedom.

But what is the basis for this, particularly in the case of gravity? Is space-time intrinsically discrete and what we experience is only an emergent property?

In Quantum Gravity there are hints of the “disappearance of space-time”:

• In General relativity there is no absolute space
• Planck distance is very possibly a minimal length
• Existence of singularities
• Black hole thermodynamics shows that entropy is finite meaning discreetness may be present

Welcome to the world of quantum gravity where radical conceptual shifts are considered in the hope is to obtain the continuous space-time limit from discrete ingredients. Here are some examples:

• Loop Quantum Gravity: based on spin networks (graphs labeled by algebraic data). It has discrete spectra and uses path integral and Regge calculus.
• Group Field Theory (GFT): is using SU(2) and the building blocks are tetrahedrons (more precisely GFT is the second quantization version of LQG and is using spin networks dual to simplicial complexes labeled by data from the SU(2) group)
• Tensor models: removes algebraic data and keeps only the combinatorics.

But how is smooth spacetime and geometry emerge from many discrete building blocks? This is the key question in quantum gravity. In the original Bronstein cube, in the limit of large N, classical and quantum mechanics will generate solid state physics and condensed matter-fluids/hydrodynamics.

A crucial tool in the large N limit is the renormalization group. Then one can perform for example GFT renormalization in standard quantum field theory.

There are major conceptual and technical issues however. For example the fundamental degrees of freedom are ambiguous because they depend on the vacuum state (which corresponds in algebraic field theory to the representation of the algebra). Also you need to prove the emergence of robust novel properties.

As a standard example of emergence one can consider the Bose-Einstein condensate where you have a single collective wavefunction responsible for superfluidity.

The emergence problem for quantum gravity is a major open problem in physics and there are some tempting speculations that we can understand the Big Bang as a phase transition for example. In everyday life we experience phase transitions every Winter when liquid water tuns into snow and ice. Maybe the universe started its existence by condensing some primordial "quantum gravity atoms" into space-time. And maybe before the inflation period “geometrogenesis” occurred.

When we will have a good model for the emergence of space-time we can investigate if other phase transitions can be physically realized. So the prospects are exciting and amazing if we will be able to work out the problems.

Until all the puzzles of quantum gravity are solved, here are some interesting results showing the way for the emergence problem using GFT renormalization: arXiv:1303.3576 (PRL 111 (2013) 031301) and arXiv:1311.1238 Here (states of) macroscopic homogeneous universes are build as condensate states in group field theory. This is similar with coherent (condensate) states in Bose-Einstein condensates, and the effective dynamic is extracted directly from the underlying quantum theory.

For the benefit of the reader, let me make a few additional points not directly relevant to the talk but which help frame the general context: is it really necessary to quantize gravity? The answer is yes because it can be shown that there are no consistent mixed theories of quantum and classical physics. And with quantum mechanics we naturally encounter discrete spectra which give rise to quantized areas and volumes. String theory takes another route and presupposes a continuous space-time background, but the supporters of loop quantum gravity and related approaches counter that with the main lesson from general relativity that there is no such thing as an absolute background. String theorists reply by pointing out that a discrete space-time would have measurable consequences, like violations of Lorenz symmetry for light. Why? The same reason the sky is blue: light gets scattered by the molecules of air and different colors (wavelengths) scatter differently (in particular the blue color gets scattered the most). Similarly light can get scattered by the granular structure of space-time. The effect is small, but it can be amplified over galactic distances. This was tested with supernova explosion observations which checked the arrival time of various wavelengths and no delay was observed. So the background independent theories should also solve the Lorenz symmetry problem, but it is hoped that space-time emergence will take care of this issue.

* I thank Daniele Oriti for giving the permission to post his completye talk. Please use the link to view the document. I also thank him for suggestions to make this post better. I am not an expert in quantum gravity and I am the sole person responsible for any inaccuracies in this post.

Homotopy and Homology

Let us continue the discussion about homotopy and introduce some definitions. Let π₀(X) be the set of paths in X and let π₀(X,p) be the set of paths containing the point p. Remembering from last time, we want to make closed loops and consider a unit circle which parameterizes a loop. Then for each point p of X we can introduce a loop space “Ω_p X” which is the space of maps from the unit circle to X and for which p is the starting point of the loop.

With those preliminaries, we can now introduce a fancy abstract redefinition of the first homotopy group (the fundamental group) as follows:

π₁(X, p) = π₀(Ω_p X, p)

The advantage of this abstraction is that it naturally generalizes to higher levels:

π_k+1(X, p) = π_k(Ω_p X, p)

So for example π₂ represents the first homotopy group of the loop space meaning we cover X with a surface and we try to continuously deform it to a loop. In higher dimensions, this becomes a nightmare of visualization and we need a way to compute those groups algebraically (we need a mechanical method relatively free of visual intuition). And why do we want to compute those groups in the first place? The point is that spaces with different groups are distinct.

As a group, π₁ is non-abelian in general, and this points the way towards a key simplification: let’s construct an abelian (commutative) group out of π_1. This will be the first homology group, but we are not yet ready to discuss it. Instead let’s try to understand what abelianization means. It means that we don’t care about the point p and we replace the closed loops with cut and stitching operations, the same way a tailor is making clothes. 

For example take an inner tube which is equivalent with a doughnut, or torus.  If we cut the inner tube along its two circles we obtain a flat surface. Equivalently, if we stitch (glue, zip) a rectangular piece of cloth along the opposite sides matching the stitching direction, we obtain a torus. As a nontrivial example of a torus, in early computer games, because of memory limitation, sometimes an object exits through the right edge of the screen and reappears from the left (the same behavior at top and bottom).

Probably the best way to understand cut and stitch is to try to understand the technique of the proof of a problem mathematicians solved around 1900: what is the complete classification of two dimensional compact surfaces?

Two dimensional surfaces are easiest to visualize and it turns out that orientability and Euler’s characteristic are the only ingredients needed for a complete classification. For the record, the answer to the classification problem is that the surface is characterized by the numbers of “holes” (or genus) and the numbers of “cross caps”.

Euler’s characteristic was discussed two posts ago, and non-orientability is easy to understand using Mobius band.

Now it is (relatively) straightforward to solve the classification theorem for 2 dimensional compact surfaces. All we have to do is to cut the surface until we obtain small flat pieces, remembering the stitching back instructions. Then we need to find some equivalent way of writing stitching instructions in a “standard normalized way”. In the end we tally all “standard normalized ways” and obtain the proof.

For a beautiful presentation of the proof, please see those lectures by Norm Wildberger:

https://www.youtube.com/watch?v=qei9LarWwXk

and

https://www.youtube.com/watch?v=wzSf9yXH5Cc

If you are new to algebraic topology, I highly encourage you to invest the time and energy to watching and understanding those lectures.

The technique of the theorem’s proof will naturally introduce us to a way of thinking about homology. Homology is related to the boundary of a space. The link with physics is that integration on the boundary can tell us essential things about the bulk (does Gauss theorem ring a bell?) Moreover when we jump from homology to cohomology, we’ll see that differential forms can also reveal information about space. This is the realm of the beautiful Hodge and de Rham theories which are essential to a modern understanding of physics. 

Homotopy and Aridne’s thread

Last time we introduced Euler’s characteristic and noted that it is an intrinsic geometric invariant. We can now take the geometry story in two different directions and we will follow it both in due time.

Let us start with a game: try to cover the Earth’s surface in triangles (which can be as small as we need to follow the contour precisely). Also when we add a new triangle, color the surface green (substitute here you favorite color) not to loose count of faces, edges, and vertices. We can cover the Earth all the way around until only a last triangle is left to be colored. Up to that point, F+V-S=1 (because when we add a new vertex and we color the new triangle the number does not change), and when we completely color the surface of the Earth, at the very last step: F+V-S=2. Now imagine increasing the gravitational attraction which smoothes the surface to a perfect sphere. Nothing changes in F+V-S however. This is the key idea of homotopy: one can continuously deform a shape in such a way that no tears occur. Then the initial and final shapes are equivalent and retain some core characteristics. For example one can deform a coffee mug into a doughnut and this is called homeomorphism. Mathematically a homeomorphism is a continuous map with a continuous inverse. It is important to note that the continuous property for the inverse map is essential otherwise a line segment can be equivalent with a circle, and this contradicts the no tearing property.

Now back to "intrinsic" geometrical games: suppose you are an ant on a doughnut. How can you uncover the type of shape you are on? One way is to play the triangle game from before, but what if the space is infinite, or very large? (remember that Euler’s characteristic can change at the very last step). Fortunately help comes from ancient mythology in the form of Ariadne’s thread.

but with a twist: after you randomly journey through the surface of your space, make sure to always return to the starting point. Then try to reel the thread back on both ends at the same time. If your space has no intrinsic holes, and is simply connected, you will succeed to get all your thread back. However, if you go around a hole N times, your thread will become stuck. This may look like trivial observations, but it has extremely deep connections with electromagnetism and high energy physics as I will show over time. 

What we need to realize is that this thread game has secretly a hidden mathematical structure: a group. A group has an associative operation, an identity element, and each operation has an inverse. The group elements in our case are the trips, and the associative operation is concatenating two trips one after another. When we do that, we tie the end of the thread for the first trip to the beginning of the thread of the second trip, and try to reel in the combined thread. The inverse operation of a trip is a reversed trip, and the identity element is the “stand still” no trip.  What we just introduced is the first homotopy group (or the fundamental group). Child’s play? For the first homotopy group yes, but things will quickly get out of hand. Next time we’ll generalize Ariadne’s thread game, observe that is no longer intuitive, and seek a simplification which will lead us to the wonderful craft of tailoring clothes or as mathematicians like to call it in fancy abstract and impenetrable obscure language: homology. Who knew that fashion design and making clothes is deeply relevant to understanding Nikola Tesla’s alternative current generators?