Quantum Mechanics

Sabine Hossenfelder’s “Lost in Math”

December 27, 2020 Incoherent ramblings , , , , , , ,

“Lost in Math” is a book that I’ve been curious to read, as I’ve been a subscriber to Sabine’s blog and youtube channel for quite a while.  On her blog and channel, she provides overviews of many topics in physics that are well articulated, as well as what appear to be very well reasoned and researched criticisms of a number of topics (mostly physics related.)  Within the small population of people interested in theoretical physics, I think that she is also very well known for her completely fearlessness, as she appears to have none of the usual social resistance to offending somebody should her statements not be aligned with popular consensus.

This book has a few aspects:

  • Interviews with a number of interesting and prominent physicists
  • A brutal take on the failures of string theory, supersymmetry, theories of everything, and other research programs that have consumed significant research budgets, but are detached from experimental and observational constraints.
  • An argument against the use of beauty, naturalness, and fine tuning avoidance in the constructions of physical theory.  Through the many interviews, we get a glimpse of the specific meanings of these words in the context of modern high level physical theories.
  • Some arguments against bigger colliders, given that the current ones have not delivered on their promises of producing new physics.
  • A considerable history of modern physics, and background for those wondering what the problems that string theory and supersymmetry have been trying to solve in the first place.
  • Some going-forward recommendations.

While there were no equations in this book, it is not a terribly easy read.  I felt that reading this requires considerable physics sophistication.  To level set, while I haven’t studied particle physics or the standard model, I have studied special relativity, electromagnetism, quantum mechanics, and even some introductory QFT, but still found this book fairly difficult (and I admit to nodding off a few times as a result.)  I don’t think this is really a book that aimed at the general public.

If you do have the background to attempt this book, you will probably learn a fair amount, on topics that include, for example: the standard model, general relativity, symmetry breaking, coupling constants, and the cosmological constant.  An example was her nice illustration of symmetry breaking.  We remember touching on this briefly in QFT I, but it was presented in an algebraic and abstract fashion.  At the time I didn’t get the high level view of what this meant (something with higher energy can have symmetries that are impossible at lower energies.)  In this book, this concept is illustrated by a spinning top, which when spinning fast is stable and has rotational symmetry, but once frictional effects start to slow it down, it will start to precess and wobble, and the symmetry that is evident at higher spin rates weakens.  This was a particularly apt justification for the title of the book, as her description of symmetry breaking did not require any mathematics!

Deep in the book, it was pointed out that the equations of the standard model cannot generally be solved, but have to be dealt with using perturbation methods.  In retrospect, this shouldn’t have surprised me, since we generally can’t solve non-harmonic oscillator problems in closed form, and have to resort to numerical methods for most interesting problems.

There were a number of biting statements that triggered laughs while reading this book.  I wish that I’d made notes of more of of those while I read it, but here are two to whet your appetite:

  • If you’d been sucking away on a giant jawbreaker for a century, wouldn’t you hope to finally get close to the gum?
  • It’s easy enough for us to discard philosophy as useless — because it is useless.

On the picture above.

I like reading in the big living room chair behind my desk that our dog Tessa has claimed as her own, so as soon as I get up for coffee (or anything else), she will usually come and plop herself in the chair so that it’s no longer available to me.  If she was lying on the floor, and my wife sits on “her” chair, she will almost always occupy it once Sofia gets up.  Ironically, the picture above was taken just after I had gotten to the section where she was interviewing Chad Orzel, of “How to Teach Quantum Mechanics to your Dog” fame.

My youngest reader

August 23, 2019 Incoherent ramblings ,

My nephew Jake is a prodigy, and is already tackling QM!

Graduate Quantum Mechanics notes now available on paper from amazon

June 11, 2019 phy1520 ,

My notes for “Graduate Quantum Mechanics” (PHY1520H) taught by Prof. Arun Paramekanti, fall 2015. (435 pages), are now available on paper (black and white) through kindle-direct-publishing for $12 USD.

This book is dedicated to my siblings.

Kindle-direct-publishing is a print on demand service, and allows me to make the notes available for pretty close to cost (in this case, about $6 printing cost, $5 to amazon, and about $1 to me as a token royalty).  The notes are still available for free in PDF form, and the latex sources are also available should somebody feel motivated enough to submit a merge request with corrections or enhancements.

This grad quantum course was especially fun.  When I took this class, I had enjoyed the chance to revisit the subject.  Of my three round match against QM, I came out much less bloody this time than the first two rounds.

These notes are no longer redacted and include whatever portions of the problem I completed, errors and all.  In the event that any of the problem sets are recycled for future iterations of the course, students who are taking the course (all mature grad students pursuing science for the love of it, not for grades) are expected to act responsibly, and produce their own solutions, within the constraints provided by the professor.


phy1520.V0.1.9-3 (June 10, 2019)

  • First version posted to kindle-direct successfully.
  • Lots of 6×9 formatting fixes made.
  • Add commas and periods to equations.
  • Remove blank lines that cause additional undesired indenting (implied latex \par’s).

PHY2403H Quantum Field Theory. Lecture 4: Scalar action, least action principle, Euler-Lagrange equations for a field, canonical quantization. Taught by Prof. Erich Poppitz

September 23, 2018 phy2403 , , , , , , , , , , , , , , , , , , , , ,

[Click here for a PDF of this post with nicer formatting]

DISCLAIMER: Very rough notes from class. May have some additional side notes, but otherwise probably barely edited.

These are notes for the UofT course PHY2403H, Quantum Field Theory I, taught by Prof. Erich Poppitz fall 2018.

Principles (cont.)

  • Lorentz (Poincar\’e : Lorentz and spacetime translations)
  • locality
  • dimensional analysis
  • gauge invariance

These are the requirements for an action. We postulated an action that had the form
\int d^d x \partial_\mu \phi \partial^\mu \phi,
called the “Kinetic term”, which mimics \( \int dt \dot{q}^2 \) that we’d see in quantum or classical mechanics. In principle there exists an infinite number of local Poincar\’e invariant terms that we can write. Examples:

  • \( \partial_\mu \phi \partial^\mu \phi \)
  • \( \partial_\mu \phi \partial_\nu \partial^\nu \partial^\mu \phi \)
  • \( \lr{\partial_\mu \phi \partial^\mu \phi}^2 \)
  • \( f(\phi) \partial_\mu \phi \partial^\mu \phi \)
  • \( f(\phi, \partial_\mu \phi \partial^\mu \phi) \)
  • \( V(\phi) \)

It turns out that nature (i.e. three spatial dimensions and one time dimension) is described by a finite number of terms. We will now utilize dimensional analysis to determine some of the allowed forms of the action for scalar field theories in \( d = 2, 3, 4, 5 \) dimensions. Even though the real world is only \( d = 4 \), some of the \( d < 4 \) theories are relevant in condensed matter studies, and \( d = 5 \) is just for fun (but also applies to string theories.)

With \( [x] \sim \inv{M} \) in natural units, we must define \([\phi]\) such that the kinetic term is dimensionless in d spacetime dimensions

[d^d x] &\sim \inv{M^d} \\
[\partial_\mu] &\sim M

so it must be that
[\phi] = M^{(d-2)/2}

It will be easier to characterize the dimensionality of any given term by the power of the mass units, that is

[\text{mass}] &= 1 \\
[d^d x] &= -d \\
[\partial_\mu] &= 1 \\
[\phi] &= (d-2)/2 \\
[S] &= 0.
Since the action is
S = \int d^d x \lr{ \LL(\phi, \partial_\mu \phi) },
and because action had dimensions of \( \Hbar \), so in natural units, it must be dimensionless, the Lagrangian density dimensions must be \( [d] \). We will abuse language in QFT and call the Lagrangian density the Lagrangian.

\( d = 2 \)

Because \( [\partial_\mu \phi \partial^\mu \phi ] = 2 \), the scalar field must be dimension zero, or in symbols
[\phi] = 0.
This means that introducing any function \( f(\phi) = 1 + a \phi + b\phi^2 + c \phi^3 + \cdots \) is also dimensionless, and
[f(\phi) \partial_\mu \phi \partial^\mu \phi ] = 2,
for any \( f(\phi) \). Another implication of this is that the a potential term in the Lagrangian \( [V(\phi)] = 0 \) needs a coupling constant of dimension 2. Letting \( \mu \) have mass dimensions, our Lagrangian must have the form
f(\phi) \partial_\mu \phi \partial^\mu \phi + \mu^2 V(\phi).
An infinite number of coupling constants of positive mass dimensions for \( V(\phi) \) are also allowed. If we have higher order derivative terms, then we need to compensate for the negative mass dimensions. Example (still for \( d = 2 \)).
\LL =
f(\phi) \partial_\mu \phi \partial^\mu \phi + \mu^2 V(\phi) + \inv{{\mu’}^2}\partial_\mu \phi \partial_\nu \partial^\nu \partial^\mu \phi + \lr{ \partial_\mu \phi \partial^\mu \phi }^2 \inv{\tilde{\mu}^2}.
The last two terms, called \underline{couplings} (i.e. any non-kinetic term), are examples of terms with negative mass dimension. There is an infinite number of those in any theory in any dimension.


  • Couplings that are dimensionless are called (classically) marginal.
  • Couplings that have positive mass dimension are called (classically) relevant.
  • Couplings that have negative mass dimension are called (classically) irrelevant.

In QFT we are generally interested in the couplings that are measurable at long distances for some given energy. Classically irrelevant theories are generally not interesting in \( d > 2 \), so we are very lucky that we don’t live in three dimensional space. This means that we can get away with a finite number of classically marginal and relevant couplings in 3 or 4 dimensions. This was mentioned in the Wilczek’s article referenced in the class forum [1]\footnote{There’s currently more in that article that I don’t understand than I do, so it is hard to find it terribly illuminating.}

Long distance physics in any dimension is described by the marginal and relevant couplings. The irrelevant couplings die off at low energy. In two dimensions, a priori, an infinite number of marginal and relevant couplings are possible. 2D is a bad place to live!

\( d = 3 \)

Now we have
[\phi] = \inv{2}
so that
[\partial_\mu \phi \partial^\mu \phi] = 3.

A 3D Lagrangian could have local terms such as
\LL = \partial_\mu \phi \partial^\mu \phi + m^2 \phi^2 + \mu^{3/2} \phi^3 + \mu’ \phi^4
+ \lr{\mu”}{1/2} \phi^5
+ \lambda \phi^6.
where \( m, \mu, \mu” \) all have mass dimensions, and \( \lambda \) is dimensionless. i.e. \( m, \mu, \mu” \) are relevant, and \( \lambda \) marginal. We stop at the sixth power, since any power after that will be irrelevant.

\( d = 4 \)

Now we have
[\phi] = 1
so that
[\partial_\mu \phi \partial^\mu \phi] = 4.

In this number of dimensions \( \phi^k \partial_\mu \phi \partial^\mu \) is an irrelevant coupling.

A 4D Lagrangian could have local terms such as
\LL = \partial_\mu \phi \partial^\mu \phi + m^2 \phi^2 + \mu \phi^3 + \lambda \phi^4.
where \( m, \mu \) have mass dimensions, and \( \lambda \) is dimensionless. i.e. \( m, \mu \) are relevant, and \( \lambda \) is marginal.

\( d = 5 \)

Now we have
[\phi] = \frac{3}{2},
so that
[\partial_\mu \phi \partial^\mu \phi] = 5.

A 5D Lagrangian could have local terms such as
\LL = \partial_\mu \phi \partial^\mu \phi + m^2 \phi^2 + \sqrt{\mu} \phi^3 + \inv{\mu’} \phi^4.
where \( m, \mu, \mu’ \) all have mass dimensions. In 5D there are no marginal couplings. Dimension 4 is the last dimension where marginal couplings exist. In condensed matter physics 4D is called the “upper critical dimension”.

From the point of view of particle physics, all the terms in the Lagrangian must be the ones that are relevant at long distances.

Least action principle (classical field theory).

Now we want to study 4D scalar theories. We have some action
S[\phi] = \int d^4 x \LL(\phi, \partial_\mu \phi).

Let’s keep an example such as the following in mind
\LL = \underbrace{\inv{2} \partial_\mu \phi \partial^\mu \phi}_{\text{Kinetic term}} – \underbrace{m^2 \phi – \lambda \phi^4}_{\text{all relevant and marginal couplings}}.
The even powers can be justified by assuming there is some symmetry that kills the odd powered terms.

fig. 1. Cylindrical spacetime boundary.

We will be integrating over a space time region such as that depicted in fig. 1, where a cylindrical spatial cross section is depicted that we allow to tend towards infinity. We demand that the field is fixed on the infinite spatial boundaries. The easiest way to demand that the field dies off on the spatial boundaries, that is
\lim_{\Abs{\Bx} \rightarrow \infty} \phi(\Bx) \rightarrow 0.
The functional \( \phi(\Bx, t) \) that obeys the boundary condition as stated extremizes \( S[\phi] \).

Extremizing the action means that we seek \( \phi(\Bx, t) \)
\delta S[\phi] = 0 = S[\phi + \delta \phi] – S[\phi].

How do we compute the variation?
\delta S
&= \int d^d x \lr{ \LL(\phi + \delta \phi, \partial_\mu \phi + \partial_\mu \delta \phi) – \LL(\phi, \partial_\mu \phi) } \\
&= \int d^d x \lr{ \PD{\phi}{\LL} \delta \phi + \PD{(\partial_mu \phi)}{\LL} (\partial_\mu \delta \phi) } \\
&= \int d^d x \lr{ \PD{\phi}{\LL} \delta \phi
+ \partial_\mu \lr{ \PD{(\partial_mu \phi)}{\LL} \delta \phi}
– \lr{ \partial_\mu \PD{(\partial_mu \phi)}{\LL} } \delta \phi
} \\
\int d^d x
\delta \phi
\lr{ \PD{\phi}{\LL}
– \partial_\mu \PD{(\partial_mu \phi)}{\LL} }
+ \int d^3 \sigma_\mu \lr{ \PD{(\partial_\mu \phi)}{\LL} \delta \phi }

If we are explicit about the boundary term, we write it as
\int dt d^3 \Bx \partial_t \lr{ \PD{(\partial_t \phi)}{\LL} \delta \phi }
– \spacegrad \cdot \lr{ \PD{(\spacegrad \phi)}{\LL} \delta \phi }
\int d^3 \Bx \evalrange{ \PD{(\partial_t \phi)}{\LL} \delta \phi }{t = -T}{t = T}
– \int dt d^2 \BS \cdot \lr{ \PD{(\spacegrad \phi)}{\LL} \delta \phi }.
but \( \delta \phi = 0 \) at \( t = \pm T \) and also at the spatial boundaries of the integration region.

This leaves
\delta S[\phi] = \int d^d x \delta \phi
\lr{ \PD{\phi}{\LL} – \partial_\mu \PD{(\partial_mu \phi)}{\LL} } = 0 \forall \delta \phi.
That is

\PD{\phi}{\LL} – \partial_\mu \PD{(\partial_mu \phi)}{\LL} = 0.

This are the Euler-Lagrange equations for a single scalar field.

Returning to our sample scalar Lagrangian
\LL = \inv{2} \partial_\mu \phi \partial^\mu \phi – \inv{2} m^2 \phi^2 – \frac{\lambda}{4} \phi^4.
This example is related to the Ising model which has a \( \phi \rightarrow -\phi \) symmetry. Applying the Euler-Lagrange equations, we have
\PD{\phi}{\LL} = -m^2 \phi – \lambda \phi^3,
\PD{(\partial_\mu \phi)}{\LL}
\PD{(\partial_\mu \phi)}{} \lr{
\inv{2} \partial_\nu \phi \partial^\nu \phi } \\
\inv{2} \partial^\nu \phi
\PD{(\partial_\mu \phi)}{}
\partial_\nu \phi
\inv{2} \partial_\nu \phi
\PD{(\partial_\mu \phi)}{}
\partial_\alpha \phi g^{\nu\alpha} \\
\inv{2} \partial^\mu \phi
\inv{2} \partial_\nu \phi g^{\nu\mu} \\
\partial^\mu \phi
so we have
\PD{\phi}{\LL} -\partial_\mu
\PD{(\partial_\mu \phi)}{\LL} \\
-m^2 \phi – \lambda \phi^3 – \partial_\mu \partial^\mu \phi.

For \( \lambda = 0 \), the free field theory limit, this is just
\partial_\mu \partial^\mu \phi + m^2 \phi = 0.
Written out from the observer frame, this is
(\partial_t)^2 \phi – \spacegrad^2 \phi + m^2 \phi = 0.

With a non-zero mass term
\lr{ \partial_t^2 – \spacegrad^2 + m^2 } \phi = 0,
is called the Klein-Gordan equation.

If we also had \( m = 0 \) we’d have
\lr{ \partial_t^2 – \spacegrad^2 } \phi = 0,
which is the wave equation (for a massless free field). This is also called the D’Alembert equation, which is familiar from electromagnetism where we have
\lr{ \partial_t^2 – \spacegrad^2 } \BE &= 0 \\
\lr{ \partial_t^2 – \spacegrad^2 } \BB &= 0,
in a source free region.

Canonical quantization.

\LL = \inv{2} \dot{q} – \frac{\omega^2}{2} q^2
This has solution \(\ddot{q} = – \omega^2 q\).

p = \PD{\dot{q}}{\LL} = \dot{q}
H(p,q) = \evalbar{p \dot{q} – \LL}{\dot{q}(p, q)}
= p p – \inv{2} p^2 + \frac{\omega^2}{2} q^2 = \frac{p^2}{2} + \frac{\omega^2}{2} q^2

In QM we quantize by mapping Poisson brackets to commutators.
\antisymmetric{\hatp}{\hat{q}} = -i
One way to represent is to say that states are \( \Psi(\hat{q}) \), a wave function, \( \hat{q} \) acts by \( q \)
\hat{q} \Psi = q \Psi(q)
\hatp = -i \PD{q}{},
\antisymmetric{ -i \PD{q}{} } { q} = -i

Let’s introduce an explicit space time split. We’ll write
L = \int d^3 x \lr{
\inv{2} (\partial_0 \phi(\Bx, t))^2 – \inv{2} \lr{ \spacegrad \phi(\Bx, t) }^2 – \frac{m^2}{2} \phi
so that the action is
S = \int dt L.
The dynamical variables are \( \phi(\Bx) \). We define
\pi(\Bx, t) = \frac{\delta L}{\delta (\partial_0 \phi(\Bx, t))}
\partial_0 \phi(\Bx, t) \\
\dot{\phi}(\Bx, t),
called the canonical momentum, or the momentum conjugate to \( \phi(\Bx, t) \). Why \( \delta \)? Has to do with an implicit Dirac function to eliminate the integral?

&= \int d^3 x \evalbar{\lr{ \pi(\bar{\Bx}, t) \dot{\phi}(\bar{\Bx}, t) – L }}{\dot{\phi}(\bar{\Bx}, t) = \pi(x, t) } \\
&= \int d^3 x \lr{ (\pi(\Bx, t))^2 – \inv{2} (\pi(\Bx, t))^2 + \inv{2} (\spacegrad \phi)^2 + \frac{m}{2} \phi^2 },
= \int d^3 x \lr{ \inv{2} (\pi(\Bx, t))^2 + \inv{2} (\spacegrad \phi(\Bx, t))^2 + \frac{m}{2} (\phi(\Bx, t))^2 }

In analogy to the momentum, position commutator in QM
\antisymmetric{\hat{p}_i}{\hat{q}_j} = -i \delta_{ij},
we “quantize” the scalar field theory by promoting \( \pi, \phi \) to operators and insisting that they also obey a commutator relationship
\antisymmetric{\pi(\Bx, t)}{\phi(\By, t)} = -i \delta^3(\Bx – \By).


[1] Frank Wilczek. Fundamental constants. arXiv preprint arXiv:0708.4361, 2007. URL https://arxiv.org/abs/0708.4361.

Another aggregation of notes for phy1520, Graduate Quantum Mechanics.

December 15, 2015 phy1520, Uncategorized

I’ve posted a fourth (pre-exam) update of my aggregate notes for PHY1520H Graduate Quantum Mechanics, taught by Prof. Arun Paramekanti. In addition to what was noted previously, this contains the remainder of my lecture notes, more problem set solutions (not posted separately), and additional worked practice problems.

Most of the content was posted individually in the following locations, but those original documents will not be maintained individually any further.