Martin Bauer

And if you consider the limit of x_1 -> x_2 you can see how the Pauli principle follows for fermions

14/14

The number of different ways to exchange particles and ending up in exactly the same configuration corresponds to different ways a WF transforms under particle exchange. There are only two classes in 3D, so only fermions and bosons. In 2D instead there're anyons (that can have any phase)

13/

In 2D this can be any complex number with a different phase factor for each winding number. In 3D+ it can be only +1 or -1

12/

In Quantum mechanics the question is how does the wavefunction describing two identical particles transform under this exchange. This corresponds to an operator O(\lambda) with is a 1-dimensional unitary representation of the fundamental group acting on the WF

11/

So the fundamental group of the configuration space in 3D+ is the group with exactly 2 elements

10/

Here is a representation of a non-contractible loop that can't be continuously transformed into a point

9/

Here is a representation of how the contractible loop can be continuously transformed into a point (=no exchange of the two particles)

Again, the bar on the right part is the line connecting the two particles

8/14

And closing the q2 loop twice is equivalent to a contractible loop

7/14

Here, there're only 2 different classes of loops that can't be transformed into each other. Contractible loops (q1) and non-contractible loops (q2)

6/14

Now in 3D (or higher) the situation is different. Here, the configuration space is a half sphere, or a disk with opposite points identified

5/14

The different classes of such loops that can't be transformed into each other by stretching and moving them around the cone (w/o a tip) form the first fundamental group of the configuration space

While R^2 is simply connected (every loop can be contracted to a dot), this space is not

4/14

Any exchange of these two particles corresponds to a loop on this cone

Two loops that can be continually transformed into each other are equivalent. But because the tip of the cone is missing, any number of rotations around the cone can't be transformed into one with fewer or more windings

3/14

For 2 indistinguishable particles, the configuration in which particle 1 is at r and particle 2 is at -r is equivalent to the configuration where they swap positions

So in 2D, the configuration space is R^2 with opposite points identified (excluding the origin)

This space is a cone w/o a tip

2/14

In 3 (or more) dimensions, all fundamental particles are either fermions and bosons. But why?

This is a direct consequence of the properties of the configuration space for identical particles

🧵 1/14

Here is it: The worlds best measurement of the anomalous magnetic moment of the muon

In great agreement with previous experimental results and in agreement with the Standard Model prediction using the LO HVP contribution from lattice input

YouTube video by Fermilab Final Muon g-2 Measurement at Fermilab

Livestream of the final result of the anomalous magnetic moment of the muon measurement at Fermilab
right now:

www.youtube.com/watch?v=huLv...

Mary Gaillard's calculations for this and similar processes made it possible to establish an upper limit on the charm quark mass at < 2 GeV

John Rosner's list of predictions turned out to be very accurate

11/11

The GIM Mechanism: origin, predictions and recent uses The GIM Mechanism was introduced by Sheldon L. Glashow, John Iliopoulos and Luciano Maiani in 1970, to explain the suppression of Delta S=1, 2 neutral current processes and is an important element of ...

This was first proposed by The Glashow, Iliopoulos and Maiani. The cancellation between the different quark lines was therefore called GIM mechanism

arxiv.org/abs/1303.6154

10/11

Therefore any possible contribution to this process must be proportional to the quark mass differences, which allows to determine the mass of the charm quark by measuring the decay of a bound state without any charm quarks into a final state without any charm quarks

9/11

But since it is a basis transformation, the CKM matrix must be unitary. If the function f(m_i) was the same for all quarks, the result would be exactly 0

8/11

With this information one can read off the form of the amplitude from the diagram without calculation.

Here, f(m_i) is an unknown loop function that we'd get by explicitly calculating this diagram

7/11

We can follow the argument here:

The interactions of the W boson are proportional to the CKM matrix, they measure the difference between the quark mass eigenbasis and interaction eigenbasis

6/11

However, the calculations for these diagrams were divergent. They only give a finite result assuming a fourth quark, that shared all quantum numbers with the up-quark yet was so heavy that it eluded particle accelerator searches at the time

5/11

But observations showed decays of *neutral* bound states involving strange quarks into *neutral* final states without any quarks.

Quantum effects in the electroweak model could explain these as at the cost of internal fermion lines

4/11

In the electroweak theory the only way to change from one type of quark to another is via charged interactions. This explained beta decay, in which a down-quark turns into an up-quark

The neutral Z boson and the photon can't change a strange quark into a down quark

3/11

In the 70s there was evidence for 3 quarks, up- and down-quarks, that make up the nuclei of atoms, and strange quarks, which share all quantum numbers with down quarks but are 20 times heavier

2/11

Before the charm quark was discovered in bound states, it was predicted via indirect effects. An early success story of the electroweak model and QFT!

Mary Gaillard and John Rosner, who both passed last week, contributed to calculating the charm quark mass

🧵1/11

Yes, fair point.

Attracting smart young people is the most valuable resource any nation can have today. In the future it will become even more critical. Scientific research is one of the strongest magnets for talent. You can ignore it, but the US is dismantling one of its most powerful engines of innovation

Over 50% of international students in the U.S. are in STEM fields. Do you think this will continue if their research funding collapses by >70% and they can be kicked out at any time because the current government picks a fight with their University?