Mentorship Irresponsibilities of Physics Graduate Schools

Why go to graduate school in theoretical physics? If you are thinking about doing so, and particularly if you reside in North America or Europe, my recommendation would be: don’t do it! Find a career where your quantitative/analytical-thinking/programming skills are in demand, so that — relative to your chances within academia — you’ll likely find much higher standards of professionalism and more plentiful opportunities to grow.

In the US, your chances of getting admitted into a given physics graduate school would be curtailed if you indicate interest in pursuing theoretical as opposed to experimental physics. (There are usually many more people wishing to do theory than experiment.) This is why theory-inclined applicants misrepresent themselves, stating they wish to do experiment instead. (I am, in fact, quite curious how many current theoretical physics faculty lied on their graduate school statements.) However, once you do get in, you’ll be told that you are free to “explore your interests”. This is euphemism for the physics departments’ unwillingness to take any responsibility to help match you to the mentor of your indicated interests — despite judging you, often more harshly, based on it. International applicants should take particular note of this: you are essentially entering a wild lottery, going across the world thinking you are going to pursue your passion in theoretical physics when your fate may turn out very differently. In the current climate of over supply of graduate students and under supply of both jobs and genuinely good research problems (especially in high energy theory), there is little incentive for potential advisors to seek out students to mentor/nuture, let alone take them based on their merit. Instead, the personalities of both student and faculty play an outsized role; and the process of advisor-seeking often feels much more like a dating game.

♦     While I was a postdoc at U of MN Duluth (UMD), which does not award PhD degrees, one of the top MS students of the faculty who hired me had written 5 papers by time he was applying to PhD programs. Knowing the competitive nature of the admission process, however, I encouraged him to visit his top choices to deliver talks on his research. Given his track record, caliber, and experience, I told him he should not shy away from directly and aggressively approaching potential advisors to start working with them as soon as possible. The reader may not be surprised to hear — for, otherwise I would not be writing about this! — my advice was not taken; and while this student did get into a decent program he hadn’t found an advisor nearly a year into his tenure as a PhD student. I did hear from his former UMD classmate, this student did eventually find a mentor, but not within his original field of interest.

♦    Another ex-student of my research supervisor at UMD, whom I had co-mentored rather closely, spent 2+ years working with him — writing a robust numerical Python code to solve a class of Euclidean Quantum Field Theories — only to end up falling out with his advisor. He did end up writing his MS Project on the same topic, but with a completely different faculty. As I understood it, his supervisor falsely claimed all the 2+ years’ of work by the student was “trivial”. The only plausible explanation I have been able to come up with is that, due to his well-known conflict with the only other research-active theorist in the Department and out of his own pride, my then-supervisor did not want his students to only complete an MS degree with a Project. Rather he required them to produce something publishable (the student did not produce enough for a paper) in order to graduate from his theory group.

♦    What happens when an advisor produces an ill conceived, misguided project for his PhD student? Answer: his student struggles mightily with it, displaying great competence and scrupulousness — with little substantive aid from the advisor himself — only to end up wasting 2+ years and facing poor job prospects. Zero consequences for the advisor, as far as I can tell, other than padding his CV and grant applications with the paper the student did still manage to produce despite the trying circumstances.

♦    If you work with a senior and/or famous faculty, you may notice they publish very frequently and easily. This is at least partly because of the social-political dynamics of theoretical physics — it often takes just several conversations or so (especially if it involves junior people) for senior/famous folks to get their names on papers. Moreover, if you are senior/famous, it is easy to find junior scientists to explain things to you… Whereas, as already alluded to, as a graduate student you may have to struggle on your own with the technical implementation of your advisor’s ideas — and the burden of failure often falls entirely upon your shoulders.

♦    I took a year off after obtaining my MS degree to properly search for a graduate school to pursue my interests in theoretical cosmology, gravitation and field theory. During my visits to various schools I ran into quite a number of unhappy graduate students. One of them brought me inside a conference room, closed the door, and proceeded to warn me: if I ever came here, I should make sure not to work with this student’s advisor! The advisor is the sort that would barely write a single paper with his graduate student; and then leave them to find their own projects to work on. In theoretical physics, finding your own project as a graduate student is usually not only very difficult, it is a very poor strategy if one wishes to mature as a scientist. Unfortunately, this student’s experience is not an uncommon phenomenon.

♦    I obtained my MS degree in Physics from Yale. One of my classmates there was an international scholar who had graduated valedictorian from a small but elite school in the US. Like my UMD research supervisor’s top student mentioned above, however, he struggled to find an advisor that matched his research interests. He had to quit Yale, took a break from school, and eventually returned to his undergraduate alma mater to complete his PhD degree. The last time I heard from him, he was a physics and math private tutor in Texas. I suppose it is not too unreasonable to assert: a person of his abilities does not need to travel across the Atlantic (he’s European) to waste his time getting the most advanced degree possible in theoretical physics just to become a private tutor in a land so distant from home. (As a side note: I do personally know of at least 1-2 other PhD classmates of mine who have gone on to become high school teachers — a respectable profession, of course; but not one requiring a doctorate in theoretical physics. This side remark is not one about poor mentorship per se, but about the dearth of job opportunities.)

♦    When I was at Yale I came to appreciate that high energy theory was in the decline due to the severe lack of guidance from experiment: as of this writing, there are still no discoveries in particle physics that can illuminate the way forward for theorists working to extend the Standard Model to account for Dark Matter and other mysteries. I decided to turn my attention towards cosmology. I was temporarily relieved that the high energy theory group had just hired a new cosmologist; and I quickly made sure to approach him to indicate my interest in working with him. The way things worked at Yale, I was supposed to find an internship of sorts with a potential advisor during my first summer there. But the cosmologist told me he would be away and thus cannot take me. It turned out he was around, and even took on an undergraduate student. I repeated my interest in working with him during the following Fall. He asked me what classes I was taking and I happily told him I was auditing a Numerical Methods course offered by the Astronomy Department. This cosmologist was a heavily numerical one, and I had taken the initiative to pick up some numerical/programming skills. He told me he would take me “over all other students at Yale” but also did remark that numerical analysis was a very “portable skill,” a puzzling comment that I only understood later on. As far as Life itself was concerned, I guess I was still in kindergarten then, not recognizing that when a faculty is repeatedly vague with you, that most likely indicates a “No”. After more than 1+ years of pinning my hopes upon that cosmologist, I finally learned he — as brand new faculty at Yale — was taking his one and only student from a completely different institution. Only during Spring of my second year at Yale did the cosmologist reveal he was not inclined to take Yale students because he was worried they have a tendency to become quants right after graduation. When discussing this with my white American classmate later on, he then informed me he had overheard conversations between faculty that they believed Chinese graduate students have a tendency to go into finance directly after obtaining their PhD.

I had thought that this student the cosmologist had taken from another institution must be extraordinary — well trained in cosmology and quantum field theory perhaps. During the student’s first year at Yale, however, he had to take a cosmology reading course with his advisor and quantum field theory with mine. And, he did not do spectacularly in the latter. (I know, because I was his TA; I am glad to report, on the other hand, the top student in that QFT course is now a faculty at an elite US institution.)

At Yale, if you do not find an advisor by the end of the second year of graduate school, you’re fucked your support will cease completely. After graduating from Kindergarten into Elementary School of Life, I proceeded to pursue the cosmologist’s next-door neighbor out of panic — despite hearing he had already taken a student. Now, a dating advice: you should be very wary of partners who are ready to find a new fling at the slightest temptation. (Did I mention advisor seeking is like dating?) My experiences with this advisor employer, whom I worked with for a whole year before taking a year off from school, can easily fill a blog post or two. Suffice to say, theoretical high energy physicists aren’t the nicest lot. A gravitation theorist once told me he chose not to do particle theory because of its hard-nosed culture; I find the latter to be an understatement.

These examples merely illustrate a single aspect of the deeply flawed reward structure that exists within the academia I am familiar with: the extraordinarily low standards of mentorship within the theoretical physics community. Unfortunately, nowadays, Western academia — including physics/astrophysics — enjoys emphasizing the need for “diversity,” but given how left-leaning illiberal/biased it has grown, however, I fear this just translates into an obsession with identity politics that is highly hypocritical and destructive.


Green’s Functions in Translation Symmetric and Parity Invariant Spaces

Setup     In D spatial dimensions, suppose we wish to solve the following Green’s function equation

(1)    \left(\mathcal{W} - \vec{\nabla}^2_{\vec{x},D}\right) G_D = \left(\mathcal{W} - \vec{\nabla}^2_{\vec{x}',D}\right) G_D = M \cdot \delta^{(D)}[\vec{x}-\vec{x}'] .

The \mathcal{W} is some linear differential operator and \vec{\nabla}^2_{\vec{x},D} and \vec{\nabla}^2_{\vec{x}',D} are Laplacians in flat/Euclidean space with respect to some observer at \vec{x} and point source at \vec{x}'. The \delta^{(D)}[\vec{x}-\vec{x}'] is the D dimensional Dirac delta function in flat space. Note that we are not assuming anything about the geometry of spacetime itself at this point; but we shall assume that both \mathcal{W} and M do not depend on D nor on the spatial coordinates \vec{x} and \vec{x}', so that the system is invariant under spatial translations

(2)    \vec{x} \to \vec{x} + \vec{a} \qquad \text{and} \qquad \vec{x}' \to \vec{x}' + \vec{a}

for constant \vec{a}; as well as parity flips

(3)    \vec{x} \to -\vec{x} \qquad \text{and} \qquad \vec{x}' \to -\vec{x}' .

We then expect that the Green’s function G_D be a function of space solely through the Euclidean distance

(4)    R \equiv |\vec{x}-\vec{x}'| .

Result     The central assertion of this post is that

The Green’s function in odd spatial dimensions may be obtained from its 1-dimensional counterpart

(4′)    G_{\text{odd }D \geq 1}[R] = \left(-\frac{1}{2\pi R} \frac{\partial}{\partial R}\right)^{\frac{D-1}{2}} G_1[R] .

The Green’s function in even spatial dimensions may be obtained from its 2-dimensional counterpart

(4”)    G_{\text{even }D \geq 2}[R] = \left(-\frac{1}{2\pi R} \frac{\partial}{\partial R}\right)^{\frac{D-2}{2}} G_2[R] .

Proof     In D+2 spatial dimensions

(5)    \left(\mathcal{W} - \vec{\nabla}^2_{\vec{x},D+2}\right) G_{D+2} = \left(\mathcal{W} - \vec{\nabla}^2_{\vec{x}',D+2}\right) G_{D+2} = M \cdot \delta^{(D+2)}[\vec{x}-\vec{x}'] .

Let us consider applying the D dimension version of the differential operator in eq. (1) onto the following integral involving the G_{D+2}:

(5′)     \widehat{G}_{D} \equiv \int_{\mathbb{R}^2} d x'^{D+1} d x'^{D+2} G_{D+2} .

By translation symmetry, note that we may also integrate with respect to (x^{D+1},x^{D+2}).

(5”)     \widehat{G}_{D} = \int_{\mathbb{R}^2} d x^{D+1} d x^{D+2} G_{D+2} .

We have

(6)     \left( \mathcal{W} - \vec{\nabla}_{\vec{x},D} \right) \widehat{G}_D \\ = \left( \mathcal{W} - \vec{\nabla}_{\vec{x},D} - \partial_{x^{D+1}}^2 - \partial_{x^{D+2}}^2\right) \int_{\mathbb{R}^2} d x'^{D+1} d x'^{D+2} G_{D+2} ;

where the -\partial_{x^{D+1}}^2-\partial_{x^{D+2}}^2 was inserted without any impact because, if we perform the integral of (x'^{D+1},x'^{D+2}) the result is independent of (x^{D+1},x^{D+2}). However, if we now interchange the order of integration and differentiation, followed by employing eq. (5),

(7)     \left( \mathcal{W} - \vec{\nabla}_{\vec{x},D} \right) \widehat{G}_D \\ = M \int_{\mathbb{R}^2} d x'^{D+1} d x'^{D+2} \delta^{(D)}[\vec{x}-\vec{x}'] \delta[x^{D+1}-x'^{D+1}] \delta[x^{D+2}-x'^{D+2}] \\ = M \cdot \delta^{(D)}[\vec{x}-\vec{x}'].

Similar arguments would demonstrate that

(8)     \left( \mathcal{W} - \vec{\nabla}_{\vec{x}',D} \right) \widehat{G}_D = M \cdot \delta^{(D)}[\vec{x}-\vec{x}'] .

In other words, eq. (5′) is in fact G_D itself, i.e., we may drop the \widehat{\cdot}. Strictly speaking, if we were dealing with spacetime Green’s functions of wave equations, we would also have to argue that integrating retarded (advanced) Green’s functions G_{D+2} would yield retarded (advanced) Green’s functions G_D. That this is true can be seen by viewing the integral over the 2 ‘extra dimensions’ as an instantaneous 2D sheet source — and the observer confined on the D dimensional space receives a retarded (advanced) signal from the 2D sheet of uniform ‘charge density’.

By translation symmetry, we may write equations (5′) and (5”) as

(8)    G_D[R] = \int_{\mathbb{R}^2} d (x-x')^{D+1} d (x-x')^{D+2} G_{D+2}[R] ,

Note that this integral relation on its own does not require parity invariance, but the assumption that G is a function of space through R does. Now, on the right hand side of eq. (8),

(9)    R \equiv R_{D+2} = \sqrt{R_{D} + \rho^2}


(9′)    R_D \equiv \sqrt{\sum_{i=1}^{D} ((x-x')^i)^2}, \qquad \qquad \rho \equiv \sqrt{((x-x')^{D+1})^2+((x-x')^{D+2})^2} .

Switching the Cartesian integration variables of eq. (8) to 2D cylindrical coordinates (\rho,\phi),

(8)    G_D[R] = 2\pi \int_{0}^\infty d \rho \cdot \rho  G_{D+2}\left[ \sqrt{R^2+\rho^2} \right] = 2\pi \int_{R}^\infty d R' \cdot R' G_{D+2}\left[ R' \right] .

At this point, differentiating both sides with respect to R yields the differential recursion relation

(9)    G_{D+2}[R] = -\frac{1}{2\pi R} \frac{\partial}{\partial R} G_D[R] .

By applying the ‘raising operator’ \mathcal{D}_R \equiv - (2\pi R)^{-1} \partial_R n times on G_D[R], we have

(10)    G_{D+2n}[R] = \left(-\frac{1}{2\pi R} \frac{\partial}{\partial R}\right)^n G_D[R] .

We arrive at equations (4′) and (4”) by setting, respectively, D=1 and D=2.

Application to Wave Equations     The fundamental solution of wave equations lie in their Green’s functions, which in turn can be thought of the field engendered by a spacetime-point source. The actual solution will then involve the superposition of all spacetime point sources weighted by the physical source in the setup in question. In spatial translation symmetric and parity invariant spacetimes, we would expect the Green’s functions of the wave operator in d spacetime dimensions, namely G_d \equiv 1/\Box, to depend on space only through the function R. Thus, they should all obey, from equations (4′) and (4”),

(11)    G_{\text{even }d}[t-t',R] = \left(-\frac{1}{2\pi R} \frac{\partial}{\partial R} \right)^{\frac{d-2}{2}} G_2[t-t',R]


(11′)    G_{\text{odd }d}[t-t',R] = \left(-\frac{1}{2\pi R} \frac{\partial}{\partial R} \right)^{\frac{d-3}{2}} G_3[t-t',R].

These statements include the Minkowski and spatially flat Friedmann–Lemaître–Robertson–Walker cosmological spacetimes. The flat spacetime case, in particular, takes the form

(12)    G^+_{\text{even }d \geq 4}[t-t',R] = \left(-\frac{1}{2\pi} \frac{\partial}{\partial R} \right)^{\frac{d-2}{2}} \left( \frac{\delta[t-t'-R]}{4\pi R} \right)


(12′)    G^+_{\text{odd }d \geq 3}[t-t',R] = \left(-\frac{1}{2\pi R} \frac{\partial}{\partial R} \right)^{\frac{d-3}{2}} \left( \frac{\Theta[t-t'-R]}{2\pi\sqrt{(t-t')^2-R^2}} \right).

These are the retarded Green’s functions, where cause (i.e., the source) at (t',\vec{x}') precedes the effect (i.e., the signal) at (t,\vec{x}); they satisfy

(12”)     \left(\partial_0^2 - \partial_i \partial_i \right) G = \delta[t-t'] \delta^{(d-1)}[\vec{x}-\vec{x}'] .

I have chosen to begin the even dimensional results at d = 4 to demonstrate the well known fact that waves associated with massless particles in even dimensional spacetimes higher than 2 propagate strictly on the light cone — as encoded by the Dirac delta function \delta[t-t'-R] of eq. (12); whereas, according to eq. (12′), with \Theta[z]=1 when z>0 and \Theta[z]=0 when z<0, massless signals do in fact travel inside the null cone — tails develop — in all odd dimensional flat spacetimes.

Speculations     Can the ‘raising operator’ -(2\pi R)^{-1} \partial_R be viewed as a genuine operator acting on a Hilbert space involving some form of union of Minkowski or spatially flat cosmological spaces of all even (or odd) dimensions? The reason for this speculation is that the presence of tails in eq. (12′) is likely due to the incomptability of strictly null propagation with spherical symmetry; in other words, in odd-dimensional Minkowski, there is no solution for an infinitesimally thin spherical shell propagating at the speed of light outward from some instantaneous point source at (t',\vec{x}'). Can this be elucidated more precisely — i.e., using equations — through such an operator approach?


  • Y.-Z. Chu, “More On Cosmological Gravitational Waves And Their Memories,” Class. Quant. Grav. 34, no. 19, 194001 (2017) doi:10.1088/1361-6382/aa8392
    [arXiv:1611.00018 [gr-qc]].
  • Y.-Z. Chu, “Transverse traceless gravitational waves in a spatially flat FLRW universe: Causal structure from dimensional reduction,” Phys. Rev. D 92, no. 12, 124038 (2015) doi:10.1103/PhysRevD.92.124038 [arXiv:1504.06337 [gr-qc]].
  • H. Soodak and M. S. Tiersten, Am. J. Phys. 61 (5), May 1993

Integration Via Differential Equations: Two Bessel(-Trigonometric) Integrals

It is rather easy to find integrals that cannot be expressed in “closed form”, which in this context means functions whose properties we know a lot about. I, for one, am rather grateful for the resources — and the people who compiled them! — such as the Table of Integrals, Series, and Products, DLMF, AMS55 and Wolfram Math World that we may consult for calculus results and properties of “special functions”.

In this post I will discuss how to evaluate the following two integrals by solving the relevant differential equations they satisfy.

(I): \frac{E_1[m]}{I_\nu[m]} \equiv \int_{-Z-\sqrt{Z^2-1}}^{-Z+\sqrt{Z^2-1}} \frac{d \rho'}{\sqrt{\rho'}} \frac{I_\nu[m \rho'] \cos\left[ m \sqrt{2 \bar{\sigma}} \right]}{I_\nu[m] \sqrt{2 \bar{\sigma}}} = \pi P_{\nu-\frac{1}{2}}\left[-Z\right]

(II): \frac{E_2[m]}{I_\nu[m]} \equiv \int_{-Z-\sqrt{Z^2-1}}^{-Z+\sqrt{Z^2-1}} \frac{d \rho'}{\rho'} \frac{I_{\nu}[m \rho']}{I_\nu[m]} J_0\left[m\sqrt{2\bar{\sigma}}\right] = \frac{1}{\nu} \left\{ \left( -Z + \sqrt{Z^2-1} \right)^\nu - \left( -Z - \sqrt{Z^2-1} \right)^\nu \right\}

Here, J_\nu is the Bessel function of the first kind; I_\nu is the modified Bessel function of the first kind; and

(III): \bar{\sigma} \equiv -\frac{1}{2} \left( \rho'^2 + 1 + 2 \rho' Z \right) \\ = -\frac{1}{2} \left\{ \rho' - \left( -Z - \sqrt{Z^2-1} \right) \right\} \left\{ \rho' - \left( -Z + \sqrt{Z^2-1} \right) \right\}.

Motivation       Since there are infinitely many intractable integrals anyway, you may wonder why you should pay attention to this result. There is in fact a physical reason for doing so. I hope to start writing about it more, but in curved spacetimes, waves associated with massless particles in Nature — including light itself — do not in fact travel strictly on the null cone. Results (I) and (II) describe the inside-the-light cone (aka “tail”) portion of massive scalar waves in de Sitter spacetime, by viewing the latter as a hyperboloid situated in 1 higher dimensional Minkowski spacetime. The massless wave tails can in turn be obtained by setting m=0.

Derivation of I and II       By a direct calculation, you may readily verify that

(D1): \mathcal{D}_m E[m] \equiv m^2 E''[m] + m E'[m] - (m^2+\nu^2) E[m] = 0

where here E[m] is either E_1 or E_2. Note that this is the ordinary differential equation (ODE) satisfied by I_\nu[m] itself; i.e., \mathcal{D}_m I_\nu[m] = 0.

In more detail, you should find that applying \mathcal{D}_m upon the left hand sides of equations (I) and (II) yields

(D2): \mathcal{D}_m E_1[m] = -2m \int_{-Z-\sqrt{Z^2-1}}^{-Z+\sqrt{Z^2-1}} d\rho' \frac{\partial}{\partial \rho'} \left( \sqrt{\rho'} \sin\left[ m \sqrt{2\bar{\sigma}} \right] I_\nu[m\rho'] \right)


(D3): \mathcal{D}_m E_2[m] = -2m \int_{-Z-\sqrt{Z^2-1}}^{-Z+\sqrt{Z^2-1}} d\rho' \frac{\partial}{\partial \rho'} \left( \sqrt{2\bar{\sigma}} I_\nu[m\rho'] J_1\left[ m \sqrt{2\bar{\sigma}} \right] \right) .

That is, acting \mathcal{D}_m on E_{1,2} converts the integrands into total derivatives, which then tells us the result is simply these integrands evaluated at the end points \rho_\pm \equiv -Z \pm \sqrt{Z^2-1}. But from eq. (III) we see these \rho_\pm are precisely the zeroes of \bar{\sigma} and hence of \sin[m \sqrt{2\bar{\sigma}}] and \sqrt{2\bar{\sigma}} J_1[m \sqrt{2 \bar{\sigma}}]; which in turn means the integrals are zero. In other words, \mathcal{D}_m E[m] = 0. But as already alluded to, this is precisely the ODE satisfied by I_\nu[m] itself. However, since there are two linearly independent solutions, we still need to show that our integrals satisfy E[m] \propto I_\nu. For non-integer \nu, note that I_{\pm\nu} are linearly independent. Moreover, we may check from equations (I) and (II) that E_{1,2}[m] are in fact power series in m that begin with an overall m^\nu pre-factor arising from the I_\nu[m \rho']. Since I_{\pm\nu}[m] is m^{\pm\nu} times a positive power series in m^2, this tells us there cannot be a I_{-\nu}[m] term in our E_{1,2}. What remains is to figure out the \chi_{1,2} in

(D4): E_{1,2}[m] = \chi_{1,2} I_\nu[m] .

To do so, notice \chi_{1,2} cannot depend on m. We may therefore extract their values through the limits

(D5): \chi_{1,2} = \lim_{m \to 0} \frac{E_{1,2}[m]}{I_\nu[m]} .

From equations (I) and (II), and utilizing the Taylor series results

I_\nu[z] = \frac{(z/2)^\nu}{\Gamma[\nu+1]} \left( 1 + \mathcal{O}[z^2] \right)


J_0[z] = 1 + \mathcal{O}[z^2]

we have

(D6): \lim_{m \to 0} \frac{E_1[m]}{I_\nu[m]} = \int_{-Z-\sqrt{Z^2-1}}^{-Z+\sqrt{Z^2-1}} d \rho' \frac{\rho'^{\nu-\frac{1}{2}}}{\sqrt{-\rho'^2 - 1 - 2\rho' Z}}


(D7): \lim_{m \to 0} \frac{E_2[m]}{I_\nu[m]} = \int_{-Z-\sqrt{Z^2-1}}^{-Z+\sqrt{Z^2-1}} d \rho' \rho'^{\nu-1} = \frac{1}{\nu} \left\{ \left( -Z + \sqrt{Z^2-1} \right)^\nu - \left( -Z - \sqrt{Z^2-1} \right)^\nu \right\} .

From equations (D4), (D5) and (D7), we may see that equations (II) has been proven for non-integer \nu. What remains, therefore, is to tackle eq. (D6). If we put

(D8): \rho' \equiv - Z + \cos[u] \sqrt{Z^2-1}, \qquad\qquad u \in [0,\pi]

so that

\frac{d\rho'}{\sqrt{2 \bar{\sigma}}} = \frac{d\rho'}{\sqrt{-\rho'^2 -1 - 2\rho' Z}} = \frac{d(\cos u) \sqrt{Z^2-1}}{\sin[u] \sqrt{Z^2-1}};

eq. (D6) now reads as

(D9): \lim_{m \to 0} \frac{E_1[m]}{I_\nu[m]} = \int_{u=0}^{u=\pi} d u \left( -Z + \sqrt{Z^2-1} \cos u\right)^{\nu-\frac{1}{2}}.

Since the cosine is an even function we may extend the integration limit to u = -\pi, namely

(D9′): \lim_{m \to 0} \frac{E_1[m]}{I_\nu[m]} = \frac{1}{2}\int_{u=-\pi}^{u=\pi} d u \left( -Z + \sqrt{Z^2-1} \cos u\right)^{\nu-\frac{1}{2}} .

At this point, referring to the integral representation of the Legendre function P_{\nu-1/2}[z] — see here, for instance — tells us we have arrived at the right-hand-side of eq. (I), at least for non-integer \nu. Our results are very likely true for integer \nu as well, since I believe it is safe to assume they are continuous functions of \nu.


  • Y.-Z. Chu, “A line source in Minkowski for the de Sitter spacetime scalar Green’s function: massive case,” Class. Quant. Grav. 32, no. 13, 135008 (2015); doi:10.1088/0264-9381/32/13/135008; [arXiv:1310.2939 [gr-qc]].

First Research Grant!

I’m entering my 5th month living/working in Taiwan. My colleagues have been very kind to me thus far; and this semester I am enjoying teaching Differential Geometry and Physics in Curved Spacetimes. (The official course title is “General Relativity”.)

I am also very excited to report: my very first research proposal as faculty has just been approved! I wish to express my gratitude towards all Taiwan tax payers for supporting my research! 😀

Can Self-Interactions Screen the 5th Force?

We live in a Universe where the rules of quantum mechanics reign. Even though we usually associate the quantum world with the microscopic one, it is not true that quantum mechanics bears no consequences on physical phenomenon on astrophysical or cosmological scales. Such an example is that of astrophysical neutrinos. Humanity can now directly verify that the Sun runs off nuclear reactions driven by the weak force, because there are several detectors around the world sensitive to neutrinos generated deep within the core of our parent star as the result of its dominant pp cycle. However, due to quantum mechanics — and despite the neutrinos propagating over the macroscopic Earth-Sun distance — there is a non-zero probability that the original electron-type neutrinos engendered within the Sun are found instead as muon-type or tau-type neutrinos here on Earth.

Motivated by the cosmological constant problem and the discovery that the universe is experiencing accelerated expansion, many cosmologists spend their time studying alternate theories of gravitation that oftentimes involve adding to Einstein’s General Relativity scalar fields that couple — as universally as scalar fields can — to ordinary matter. One possible goal of these theories is to do away with the need to introduce a cosmological constant \Lambda term in Einstein’s equations, because many Quantum Field Theorists find its presence un-`natural’​. In particular, the measured value of the (square root of the) cosmological constant \sqrt{\Lambda}, assuming it accounts for all of Dark Energy, is significantly smaller than the energy scales set by the fundamental microscopic interactions among elementary particles. However, because General Relativity is well tested from millimeter to Solar System length scales, there is a need to screen (i.e., make as weak as possible) such a hypothetical `5th force’ in Nature mediated by one of these scalar fields, so that they are not already ruled out by available data.

It turns out there are several such candidate theories on the market today. To screen the force it mediates, near the source of gravity, such a scalar field \phi typically acquires one of the following characteristics: (I) grow large relative to some energy scale E, namely, \phi/E > 1; or (II) its gradients grow large relative to E, namely \nabla\phi/E^2 > 1 and/or \nabla^2 \phi/E^3 > 1. (See for instance Section 3.2 of arXiv: 1407.0059 and 2.2.4 of arXiv: 1601.06133.) Because these field theories are designed to drive the observed acceleration of the universe, a cosmological phenomenon, the length scale \ell occurring within them — i.e., the \ell \sim 1/E — are often macroscopic ones. This makes contact with the quantum nature of the Universe we live in, because the scalar self-interactions responsible for such a supposed screening effect, involving \phi/E, \nabla\phi/E^2 and \nabla^2\phi/E^3, are often irrelevant in the Quantum Field Theory (QFT) sense. As I will now argue qualitatively (and non-rigorously), this means quantum effects might be so important over macroscopic scales that the very mechanism invented to screen the 5th force cannot be relied on to work as intended.

Vainshtein & Galileons     For concreteness, let’s examine the Galileons, a class of scalar field theories that Vainshtein screen the 5th force it mediates by having the second derivatives be large, specifically \nabla^2 \phi/E^3 > 1. About a 4-dimensional (4D) Minkowskian background, their dynamics is described by the following Lagrangian density:

(1):   \mathcal{L}_{\text{G}} = \frac{1}{2} (\partial \phi)^2 \left( 1 + a_3 \frac{\partial^2 \phi}{E^3} + a_4 \left( \frac{\partial^2 \phi}{E^3} \right)^2 + a_5 \left( \frac{\partial^2 \phi}{E^3} \right)^3 \right) .

In 4D flat spacetime, an `irrelevant’ term in a Lorentz invariant QFT has coefficients that scale as (1/\text{Mass})^{\text{positive power}}. This indicates the a_{3,4,5} interaction terms in eq. (1) are irrelevant; note that I have displayed them in schematic form because my arguments below do not for the most part depend on their details. Their detailed form — see equations (34) through (38) of arXiv: 0811.2197does ensure that the associated equation-of-motion (EoM) of the Galileon is second-order in time derivatives, so it does not suffer from Ostrogradsky instability (see arXiv:1506.02210 for a review): non-degenerate higher order Lagrangians would yield Hamiltonians that are linear in the highest-order momentum, and are therefore unbounded from below. (In plain language: the energy of systems suffering from the instability can — pathologically! — approach negative infinity unhindered, usually by imparting infinite positive energy to other systems it interacts with.)

The primary issue with `irrelevant’ terms in a field theory’s dynamics is that, once they become necessary to describe the physics at hand, you’d need to include an increasing number of them the more accurately you want to model your physical system. In fact, the rules of perturbative QFT tell us one may have no choice. Even if you forgot to include certain irrelevant terms in your Lagrangian density, the results of perturbative QFT calculations — because of the quantum nature of our world and the self-interactions of the screened scalar field — would become infinite. And as a result, irrelevant terms involving higher powers of 1/Mass would have to be added as counter-terms to cancel these infinities. For the Galileon, these considerations yield at the very least the following infinite series, which I again present in schematic form to emphasize that this is a series involving the power counting parameter \partial^2 \phi/E^3:

(2):   \mathcal{L}'_{\text{G}} = \frac{1}{2} (\partial \phi)^2 \left( 1 + \sum_{s=1}^\infty a_{s+2} \left( \frac{\partial^2 \phi}{E^3} \right)^s \right) ,

where all the \{ a_s \vert s = 3,4,5,\dots\} are dimension-less pure numbers. There are other terms such as

(2′):   E^4 \left(\frac{\Box \phi}{E^3}\right)^s,   E^2 \Box \left( \frac{\Box \phi}{E^3} \right)^s   and   \Box^2 \ln [\Box/E^2] \left(\frac{\Box \phi}{E^3} \right)^s .

(See arXiv: 1310.0187 for a pure Galileon quantum correction calculation.) Now, the expert reader may complain I have ignored in eq. (2) the “Galilean” symmetry enjoyed by the Galileon — the invariance, up to a total divergence, of the Lagrangian density in eq. (1) under the replacement \phi \to \phi + b_\mu x^\mu + c, where b_\mu and c are constants. However, this symmetry is no longer respected by the matter-Galileon coupling (see for e.g., the \phi T/M_{\text{pl}} in eq. (3) below) nor in curved spacetime (see arXiv: 0901.1314). As such, I do not believe one needs to regard it as fundamental.

While there are non-renormalization theorems guaranteeing that a_{3,4,5} do not themselves receive any quantum corrections, I believe the same theorems do not prohibit the existence of non-zero \{ a_{s > 5} \}. It is also my understanding that these non-renormalization theorems were proven without considering interactions with generic forms of matter; but Galileons, being a 5th force that “modifies” the equivalence-principle respecting gravitational interaction, really should couple to all matter. Since \partial^2 \phi/E^3 > 1, this indicates we likely have in eq. (2) a non-convergent or very slowly converging series precisely where we need the screening to be effective. In other words, our Galileon theory cannot provide a sensible example of Vainshtein screening because it cannot be trusted precisely where we need its self-interactions to substantially weaken the Galileon force relative to that of gravity itself. A physical parallel of this Galileon situation, is to employ the Euler-Heisenberg effective dynamics to model quantum electrodynamics at energy scales significantly higher than the electron mass. Or, use the Fermi theory to describe electroweak interactions at energies far beyond the W and Z boson masses. A simple mathematical analogy is to attempt to evaluate f[x] = 1/(1-x) at say x=3/2 using its geometric series representation \sum_{\ell=0}^\infty x^\ell.

One may try to salvage this Galileon theory by imagining that the \{ a_{s > 5} \} could turn out to be tiny enough to render the \partial^2\phi/E^3 > 1 harmless; this is of course a distinct possibility but remember, though, that under renormalization group flow, the infinite number of coupling constants in eq. (2) do not generically remain zero unless some additional physical principle is uncovered/introduced.

Length scales     For Galileons to be relevant to cosmic acceleration, cosmologists usually set the Galileon energy scale E based on Newton’s gravitational constant G_{\text{N}}, as well as the current Hubble parameter H_0 (a measure of the Universe’s current rate of expansion):

E^{-1} \sim \left( \frac{\sqrt{G_{\text{N}}}}{H_0^2} \right)^{1/3} \sim 10^3 \text{km}.

You might think this indicates the theory ceases to be viable below length scales of a thousand kilometers or so, already a rather discouraging state of affairs on its own. The situation turns out to be far more dire, however. To be more specific, let us follow the literature and consider the lowest order coupling to ordinary matter:

(3):   \mathcal{L}_{\text{int}} = \frac{\phi}{M_{\text{pl}}} T^{\sigma}_{\phantom{\sigma}\sigma}, \qquad\qquad M_{\text{pl}} \sim 1/\sqrt{G_{\text{N}}}.

where T^{\sigma}_{\phantom{\sigma}\sigma} is the trace of the energy-momentum-stress-shear tensor of some astrophysical system. (As an aside, observe the Galileon does not couple to the photon at this order since the latter’s stress tensor is traceless; i.e., it violates the equivalence principle.) Modeling the Sun as a point mass M_{\odot}; placing it at the origin of our coordinate system; and ignoring the planets for now — a possibly very poor assumption given how important the nonlinearities will turn out to be — we find that

(4):   T^{\sigma}_{\phantom{\sigma}\sigma} = M_\odot \delta^{(3)}[\vec{x}].

Using 3D spherical coordinates (r,\theta,\phi), the ODE that arises from solving \mathcal{L}_\text{G} + \mathcal{L}_\text{int} in equations (1), (3), and (4) then reads:

(5):   \left( \frac{\phi'[r]}{r} \right) \left( 1 + \frac{a_3}{E^3} \left( \frac{\phi'[r]}{r} \right) + \frac{a_4}{E^6} \left( \frac{\phi'[r]}{r} \right)^2 \right) = \frac{M_\odot}{4\pi M_{\text{pl}} r^3} .

The detailed structure of the Galileon interaction (see discussion on p11 of arXiv:0811.2197) is responsible for the a_5 term dropping out for a time-independent \phi profile. (Also, to arrive at eq. (5), it is useful to recognize that the `left-hand-side’ of the ​equations-of-motion for Galileons take the `total-divergence’ form \nabla_r J^r, for an appropriately defined J^\mu. This allows Gauss’ theorem to be applied, reducing the EoMs to first order ones. On the other hand, this feature is unlikely to continue to hold once the a_{s > 5} terms in eq. (2) are included.) The a_{3,4} in eq. (5) tell us the higher powers of \phi'/r are directly inherited from the nonlinearities of the Galileon theory. Heuristically speaking, if we associate every derivative with one power of 1/r, then \phi'/r \sim \partial^2 \phi. We may use this to estimate eq. (5) to go as

(5′):   \left( \frac{\partial^2 \phi}{E^3} \right) \left( 1 + a_3 \left( \frac{\partial^2 \phi}{E^3} \right) + a_4 \left( \frac{\partial^2 \phi}{E^3} \right)^2 \right) \sim \left(\frac{r_v}{r}\right)^3 ,

where r_v is the Vainshtein radius of the Sun:

(6):   r_v \equiv \frac{1}{E} \left(\frac{M_\odot}{4\pi M_{\text{pl}}}\right)^{1/3} \sim \mathcal{O}\left( 10^2 - 10^3 \right) \text{light years}.

As one gets close to the Sun — i.e., as 1/r grows — we expect the highest order term to dominate:

a_4 \left( \frac{\partial^2 \phi}{E^3} \right)^3 \sim \left(\frac{r_v}{r}\right)^3 \qquad \Rightarrow \qquad  \frac{\partial^2 \phi}{E^3} \sim \frac{r_v}{r} \gg 1 .

Referring to eq. (6), we see that our power counting parameter is not only greater than one, it is much greater than unity for the inner Solar System. The Earth is roughly r \sim \mathcal{O}(10) light minutes from the Sun, compared to the hundreds of light years Vainshtein radius. Hence,

\frac{\partial^2 \phi}{E^3} \sim \frac{r_v}{r} \sim \mathcal{O}(10^7-10^8).

This teaches us that the series in eq. (2) likely does not converge, or does so very slowly, once r \lesssim r_v \sim \mathcal{O}(10^2-10^3) light years — instead of the original expectation of r \lesssim E^{-1} \sim 10^3 km. In the same vein, this result that \partial^2 \phi/E^3 \sim \mathcal{O}(10^7-10^8) indicates the quantum corrections from pure Galileon self-interactions, i.e., the terms in eq. (2′), are so large that the entire theory very likely ceases to be valid below r \lesssim r_v. That is, even if one disregards the Galileon symmetry-breaking infinite series of eq. (2), the following infinite series is probably very ill defined, or at least not useful at all for making physical predictions, within our Solar System:

\sum_{s = 1}^{\infty} \left( q_{s,1} E^4 \left( \frac{\partial^2 \phi}{E^3} \right)^s + q_{s,2} E^2 \partial^2 \left( \frac{\partial^2 \phi}{E^3} \right)^s + q_{s,3} \partial^4 \ln \left[\frac{\partial^2}{E^2}\right] \left(\frac{\partial^2 \phi}{E^3} \right)^s \right) ,

with \{ q_{s,1/2/3} \in \mathbb{R} \}.

Now, to be fair, this power counting parameter does get a tad smaller the higher the powers of \partial^2 \phi/E^3 we include. That is, let us consider modifying eq. (5′) as follows:

\left( \frac{\partial^2 \phi}{E^3} \right) \left( 1 + a_3 \left( \frac{\partial^2 \phi}{E^3} \right) + a_4 \left( \frac{\partial^2 \phi}{E^3} \right)^2 + \dots + a_{N+2} \left( \frac{\partial^2 \phi}{E^3} \right)^N \right) \sim \left(\frac{r_v}{r}\right)^3 .

Near the Sun, where the highest power term dominates, we have

(7):   \frac{\partial^2 \phi}{E^3} \sim \left( \frac{r_v}{r} \right)^{\frac{3}{N+1}} ,

which means \partial^2 \phi/E^3 \to 1 as N \to \infty. I’m not sure, though, if this inspires confidence in the utility of the \infty-series theory of eq. (2) — for, even if it converges, one still has an infinite number of coefficients to match against Nature, unless some underlying organizing principle is uncovered.

Vainshtein Screening     On the other hand, let us verify that the Vainshtein screening mechanism does work, provided we are willing to suspend our doubts about the validity of theory itself. Taking equations (6) and (7) into account, the Galileon force per unit mass due to the Sun is

\frac{\phi'}{M_{\text{pl}}} \sim \frac{\partial^2 \phi}{M_{\text{pl}}} r \sim \frac{E^3 r}{M_{\text{pl}}} \left(\frac{r_v}{r}\right)^{\frac{3}{N+1}} \sim \frac{M_\odot}{M_{\text{pl}} r^2} \left(\frac{r}{r_v}\right)^{\frac{3N}{N+1}}.

Since the Newtonian gravity force per unit mass due to the Sun is M_\odot/(M_{\text{pl}}^2 r^2), we readily verify that the Galileon force is indeed suppressed near the Sun:

\frac{\text{Galileon `5th Force'}}{\text{Newtonian Gravity}} \sim \left( \frac{r}{r_v} \right)^{\frac{3N}{N+1}}.

Superluminality     Finally, I should mention: small \phi waves propagating on top of its spherically symmetric solution sourced by a central mass can travel faster than the speed of light. This means the chronology of cause versus effect is ambiguous in these theories, even if one believes no closed timelike curves can be set up.

Skepticism     I have not examined too closely the other screening mechanisms on the market, involving \phi/E > 1 and \nabla \phi/E^2 > 1. However, I cannot help but express concerns of the same spirit as the Galileon case: why wouldn’t one face a non-convergent or very slowly converging series in either \phi/E or \nabla \phi/E^2?

I end with a question that sums up my skepticism:

Is it really possible to implement a mechanism for weakening a hypothetical `5th force’ in Nature near its material source, that involves only a small number of large self-interactions and that is simultaneously relevant for cosmic acceleration?

Update 5 November 2017:     To improve the precision of the arguments, I have added the phrase “very slowly converging” to my discussions on the “non-convergent” Galileon theories. Compare, for instance, \exp[x] to the first N terms of its Taylor series \sum_{\ell = 0}^{N-1} x^\ell/\ell!. For large x \gg 1, one would have to sum up many terms, i.e., N would have to be very large, in order to accurately capture the behavior of e^x. I’m not sure, however, if it is possible to prove rigorously that the Galileon field theory (or the Euler-Heisenberg, or Fermi 4-fermion theory) necessarily has to break down beyond their relevant energy scales — even if physical evidence strongly suggests they do — but it is reasonable to suspect, at the very least, the resulting series is very slowly converging and therefore useless for making physical predictions.


  • M. A. Luty, M. Porrati and R. Rattazzi, “Strong interactions and stability in the DGP model,” JHEP 0309, 029 (2003); doi:10.1088/1126-6708/2003/09/029; [hep-th/0303116].
  • A. Nicolis, R. Rattazzi and E. Trincherini, “The Galileon as a local modification of gravity,” Phys. Rev. D 79, 064036 (2009); doi:10.1103/PhysRevD.79.064036; [arXiv:0811.2197 [hep-th]].
  • C. Deffayet, G. Esposito-Farese and A. Vikman, “Covariant Galileon,” Phys. Rev. D 79, 084003 (2009) doi:10.1103/PhysRevD.79.084003 [arXiv:0901.1314 [hep-th]].
  • R. P. Woodard, “Ostrogradsky’s theorem on Hamiltonian instability,” Scholarpedia 10, no. 8, 32243 (2015) doi:10.4249/scholarpedia.32243 [arXiv:1506.02210 [hep-th]].
  • N. Brouzakis, A. Codello, N. Tetradis and O. Zanusso, “Quantum corrections in Galileon theories,” Phys. Rev. D 89, no. 12, 125017 (2014) doi:10.1103/PhysRevD.89.125017 [arXiv:1310.0187 [hep-th]].
  • K. Hinterbichler, M. Trodden and D. Wesley, “Multi-field galileons and higher co-dimension branes,” Phys. Rev. D 82, 124018 (2010); doi:10.1103/PhysRevD.82.124018; [arXiv:1008.1305 [hep-th]].
  • A. Joyce, L. Lombriser and F. Schmidt, “Dark Energy Versus Modified Gravity,” Ann. Rev. Nucl. Part. Sci. 66, 95 (2016); doi:10.1146/annurev-nucl-102115-044553; [arXiv:1601.06133 [astro-ph.CO]].
  • A. Joyce, B. Jain, J. Khoury and M. Trodden, “Beyond the Cosmological Standard Model,” Phys. Rept. 568, 1 (2015); doi:10.1016/j.physrep.2014.12.002; [arXiv:1407.0059 [astro-ph.CO]].

Academic Fraud at the University of Minnesota Duluth

Humanity may one day find alternate mathematical (or, even non-mathematical) formulations of the physics uncovered over the past few centuries. But until then: Complex Numbers, the abstract formulation of Linear Algebra, Calculus on the Complex Plane, and an introduction to Partial Differential Equations, together form the mathematical core of any solid undergraduate education in physics. To tell students such material is graduate-level mathematical physics is — at the very least — misleading. At an institute like the U of MN Duluth (UMD), where students would invariably use it as validation to disengage from such a course altogether, this sort of misinformation holds them back from acquiring the necessary mathematical skills for tackling physics. Such an act of deceit is therefore appropriately labeled as academic fraud. (Note added: It makes it even more fraudulent if it was driven by personal feud/political reasons, as I believe was the case here.)

It is particularly scandalous and egregious, in my opinion, that the perpetrators of such fraud came from a pair of theoretical physicists, i.e., scientists who should know very well what constitutes the mathematical foundations of Quantum Mechanics, Electrodynamics, etc.: my colleagues (and married couple) John R. Hiller and Sophia Chabysheva. During Fall 2015, they abused the trust that UMD physics students had placed in their seniority, and quite possibly stunted the academic development of roughly 11 of the 13 students in my Fall 2015 Analytical Methods in Physics course. They did so by repeatedly misleading the students that my course material was a graduate-level one.

(See here for a copy of my course webpage. The lecture notes may also be found here. I spent the majority of my time covering Chapters 1 through 4; and placed heavy emphasis on the abstract formulation of Linear Algebra.

Note for non-physicists: this is a math-for-physics course. Stating, for instance, that the abstract formulation of Linear Algebra is not needed for understanding Quantum Mechanics is like asserting one does not need to learn Biology to become a doctor. You might also wonder what the official description on the course catalog for Phys 3033, Analytical Methods in Physics, actually states:

A survey of analytical methods for the solution of fundamental equations of physics, such as those of Newton, Schrodinger, and Maxwell, and of the underlying mathematics, including complex variables, linear algebra, vector analysis, and ordinary and partial differential equations.

I should mention that John Hiller was in fact the one who set up this course for the first time roughly two years prior, i.e., Fall 2013. Hence this course description was very likely authored by him. This unfortunately only strengthens my hypothesis that the underlying reasons for the behavior by the husband and wife were driven by factors unrelated to academic content/value/merit.)

Towards the end of Spring 2015, John and Sophia had approached me to teach Analytical Methods instead of the Classical Mechanics course my then-Department Head Marc Seigar had initially assigned to me. Apart from trying to help out my colleagues, I agreed to swap classes with them because I am a theorist, and such a math-for-physics course lies very close to my heart. I ended up taking time off my research towards the end of the summer and wrote up a set of notes that became essentially a short textbook, so that the students did not have to purchase one. The textbook that was ordered by John Hiller, who was initially scheduled to teach the course, cost roughly 190 US dollars. I was particularly glad I saved the students quite a bit of money when I later found that one of them was already a father, and was simultaneously juggling school work, child care and a full-time job.

Once the Fall 2015 semester began, Sophia started speaking to me regularly, asking about my course and student attendance. I did not suspect foul play at the time, and transparently provided links to my course webpage, lecture notes, etc. Since she was teaching Classical Mechanics, and her course shared the same students as mine; I merely thought, perhaps, that she was trying to sync our courses somewhat. It was only much later that I was told John Hiller used this information his wife had obtained from me, to complain at the UMD Physics faculty meeting and to my College’s (then-)Dean Joshua Hamilton that I was teaching at the 8000 level (i.e., graduate-level) — all behind my back, and without ever having a discussion with me. During the same semester, on the other hand, I recall even inviting John and Sophia to the theory group’s journal club/seminar series to deliver a talk about their research.

On my end, despite bringing lots of energy to my classroom teaching, I could feel many of the students rapidly falling away. On the weakest students’ homework assignments, I even encouraged them — very politely! — to feel free to come seek my help, but to no avail. In fact, it was the top 2 students who came to me regularly. By mid-semester or so, only a few students remained engaged; and by the final weeks, only the top 2 were still plugged into my material. Even one of the latter 2 blurted out one day in class, that whatever I was teaching then was too hard — that certainly piqued my suspicion that something or someone must have sufficiently emboldened him to state his opinion so openly and forcefully. Given that this was my very first upper level course, and given how much time I had invested in it, my Fall 2015 teaching experience was extremely disheartening.

It was only towards the end of the semester, that two graduate students within the theoretical physics group (who were not taking my class) told me they had heard my course was too difficult. Upon my probing, they revealed they had heard it from Sophia. Everything then made sense to me immediately: since we shared the same students, she must have exploited her platform to tell them my course was inappropriately difficult! Not surprisingly, only 5 out of the 13 students turned up to fill in teaching evaluations at the end of the semester, and all 5 evaluations rated my course as unreasonably arduous. The only consolation was the following unsolicited feedback I received at the end of the semester, from a student double majoring in Math and Physics while he was turning in his final paper:

Thanks for a great semester. While the class was not easy by an [sic] means, it did push me. The linear algebra portion was particularly interesting. It was rigorous enough that three of us were allowed to register for advanced linear algebra (5000 level) as opposed to the 4000 level linear algebra class that’s required by the mathematics degree.

Now, during Fall 2015, I had submitted an internal UMD (EVCAA) grant proposal to support my polishing of the Analytical Methods lecture notes over summer 2016 (i.e., the following year). While I did not get the grant, my then-Department Head Marc Seigar (whom I recall was on the grant committee) did kindly, in response, put me down to teach Analytical Methods again during Fall 2016. I distinctly remembered Marc coming to my office to tell me so; and I even asked if Hiller would be comfortable with the arrangement since John was the one who started the course 2 years prior.

However, during Spring 2016, Marc Seigar brought me into a meeting with Vitaly Vanchurin — who hired me starting Fall 2014, so I was considered the latter’s research postdoc — and informed the both us that he had just held my annual performance evaluation meeting with our College’s Dean, Joshua Hamilton. Given the poor teaching evaluations and given Hiller had brought his complaints directly to the Dean himself, Hamilton was rather displeased with my teaching of the Fall 2015 Analytical Methods course and had applied pressure on Seigar to assign me to teach only discussion sections. It was then I informed both Marc and Vitaly that I strongly suspected foul play, that both Hiller and Chabysheva had back-meddled with my class, directly misleading the students about my material. I remembered Marc Seigar — looking down at the floor — mumbling to himself: why, given Hiller hardly showed up in the Physics Department and given he did not receive stellar teaching evaluations himself, John’s opinions should be valued at all. At the end of this meeting, I made sure to register with Marc Seigar my desire to teach Analytical Methods again in the Fall (of 2016). Later, in a one-on-one meeting with Marc, he re-confirmed that Hiller had indeed brought his complaints all the way up to Dean Hamilton. My publication rate was also criticized in my performance evaluation, according to Seigar; and during this 1-on-1 meeting with Marc, he revealed this came from John Hiller as well.

Prior to this turn of events, I had decided I would simply be upfront with my next batch of Analytical Methods students and let them know, without naming names, they would likely be misinformed by other faculty that mine was a 8000-level course. But I would go on to assure them that, while challenging, the class would prove very useful to their understanding Quantum Mechanics, Electrodynamics, etc. With this turn of events, however, I decided to write to my former (Fall 2015) students to obtain firm evidence of back-meddling. To this end, I created the following anonymous online survey:

Did Sophia Chabysheva (and possibly John Hiller) openly discuss her views on how and what was being taught in Phys 3033 [the Analytical Methods course], during Fall 2015 itself (i.e., while the class was in session)? If so, can you elaborate on what she (or they) said? And, how frequently did she (or they) do so?

Altogether, 3 students responded; the results can be found here. This was towards the end of Spring 2016, so an intervening semester had gone by. Let me highlight the third student’s testimony. Not only did Sophia Chabysheva mislead my students “a couple times a month” he went on to corroborate my fears, that students’ development were indeed held back:

Dr Chabysheva remarked that … he (Dr Chu) was pushing too hard of a topic. At the time, I often agreed with Dr Chabysheva, blaming the difficulty of material for my struggles rather than my own lack of ambition.

Is there any ounce of truth to the assertion that I was teaching at the graduate level? Physicists amongst my readers can readily discern it for yourselves; but here, the student in fact debunks it quite unequivocally:

After getting through the course, I am extremely grateful to Dr Chu for having put in the extra effort in order to teach us a quality Analytical Methods course. The topics discussed have come up numerous times in my other undergraduate courses, and I owe my increased GPA to Dr Chu.

Armed with such concrete evidence, in an e-mail dated 25 April 2016, I went on to write to my then-Department Head Marc Seigar to file a formal complaint against John R. Hiller and Sophia Chabysheva. I not only detailed how Sophia and John sabotaged my course and held back my students’ academic growth; I further pointed out John Hiller’s problematic attitude towards education likely extended beyond my Analytical Methods course. Specifically, a graduate student had told me how, when taking Hiller’s course, John would become unhappy when the former attempted to use methods not taught in class to solve problems. Additionally, Hiller made the same student jump through hoops — solve miscellaneous problems — just to prove the latter was worthy of access to some supplementary material provided to John by Emeritus Professor Thomas Jordan. [Edit 22 November 2017: I was just informed the problems were not “miscellaneous”.]

Unfortunately, Marc Seigar — for political and personal career-advancement reasons that require some explanation; see side story below — had very likely by then, already dismissed all of my concerns and reduced them to a mere “differences in opinion” between Hiller/Chabysheva and I regarding how to teach Analytical Methods. During a short meeting between Marc, Vitaly, and I soon afterwards, Marc Seigar, in fact, asked Vitaly precisely that; i.e., “Isn’t it just a difference of opinion?” It was only after posing his rhetorical question, that Seigar revealed he was not able to access the above online survey whose link I had provided in my e-mail. In other words, he likely had an opinion/verdict already, without bothering to assimilate the concrete evidence I had provided. On the other hand, in the e-mail response Marc Seigar sent on the same day of my formal complaint, he did not address the central issues regarding the academic fraud itself, but thought it was “important to clarify” that Dean Hamilton had “never directly told me to remove you as instructor for Phys 3033”. (What was the above-mentioned post-performance-evaluation Chu-Seigar-Vanchurin meeting about, then?) Seigar also went on to state that he had “no evidence” Hiller went to the Dean regarding my teaching.

Puzzled, I wrote back to ask:

… as far as timeline goes, is it not accurate to assert, after my submission of the EVCAA R&S grant (last semester) you kindly placed me to teach Phys 3033 again in the Fall; but after your meeting with the Dean a couple of weeks ago, I was removed from the schedule?

I never received a reply. Furthermore, afterwards, Seigar never held a single discussion with me addressing the specifics of either my course nor the details of my allegations against John and Sophia.

Marc Seigar did state in the above e-mail he would attempt to arrange a meeting with everyone involved. But that never happened either. What he did do instead was to hold a meeting with John, Sophia, and Vitaly; very likely to isolate me, politically speaking. (Vanchurin was on his way to tenure that year, and thus had significant incentives to go along with Seigar’s political moves and to withdraw his support for me.) The official reason why the meeting did not occur — revealed to me by Seigar only much later, was that Hiller had simply refused to meet me, because the latter thought it was “not worth his time”.

On the other hand, it was nearly a year after my course was sabotaged by Hiller and Chabysheva, and after 3 months of waiting for a substantive reply from my then-Department Head Seigar to my end-April formal complaint — did I decide to take matters into my own hands. I wrote a harsh e-mail, dated 27 July 2016, directly to both Hiller and Chabysheva, detailing their academic fraud. Chabysheva retorted back, essentially blaming the difficulty of my course for poor attendance in her class. As for Hiller — and here, the perils of tenure were on full display — he hit back with self-righteous nonsense, talking down to me:

If you are to move forward with an academic career, or any career, you need to learn to take advice from those with more experience, rather than following only your own ideas and then lashing out when your approach isn’t accepted.

In the same rebuttal, he also lied rather blatantly that he had advised me on how to teach the course but I had ignored it “both in content and methods”. (In reality, the only exchange Hiller and I had about the course content was John’s explanation of why he chose the ~$190 textbook I mentioned earlier.)

The action was swift: Marc Seigar — for the sake of professional transparency, I had copied the e-mails to him — went straight to Human Resource and wrote a formal cease-and-desist letter to shut down the exchange. When I tried to clarify with him what that was about, he conveniently pointed out, he saw no reason why I should be further involved in the physics curriculum since I had a 100% research appointment in the upcoming 2016-2017 academic year. (Joshua Hamilton had, in the meantime, removed me from teaching duties for my 3rd and final year as a postdoctoral associate at UMD because he did not want to deal with me; see side story below.) Marc then went on to state he had spent “many hours talking to all parties involved” — as already stated above, after my formal complaint, I do not recall having a single substantive discussion with him about my Analytical Methods course — and has concluded “… this is simply a difference in opinion and philosophy over how a particular course should be taught. It is nothing more than that.” He did, however, take the opportunity to chide me, saying I had a tendency to “send out e-mails at odd hours of the day/morning” — why this is of any relevance, I do not know — and as my “mentor” he tells me I need to learn to “accept the situation” and “move on”.

When Fall 2016 finally came around, I noticed Marc Seigar had been promoted to Associate Dean — i.e., working directly under Joshua Hamilton.

A Side Story

What happened to the Marc Seigar of Fall 2015, whom I believe recognized my teaching effort then, and had put me down to teach Phys 3033 again during Fall 2016?

During mid Spring 2016, the Dean of my College, Joshua Hamilton, tried to fire me by the end of December 2016.  The verbal excuse provided, as transmitted via my then-Department Head Marc Seigar, was that UMD’s College of Science and Engineering was under severe budget constraints. Initially, over e-mail, I tried to negotiate in good faith: for instance, offering to have my salary voluntarily cut by 25% (the amount the school was supposedly unable to fork out) as long as I could remain employed full time till the end of the academic year. This latter condition, that my job was to last till end May 2017, was actually the very first sentence of my offer letter.

I took the effort to explain, as a foreigner in the US, this would be a severe disruption because my stay here was tied to my employment status; and being mid-Spring the postdoc application season was largely over. I also tried to point out the value I believe I had brought to the academic community here. However, Hamilton appeared to be stonewalling, and unfortunately I had to let him know, if he was not interested in negotiating in good faith, I had no choice but to sue him and UMD. (As a side note, I have never met Joshua Hamilton in person in a professional setting. Vitaly had suggested that I meet him in person to negotiate but Marc Seigar discouraged me from doing so, hinting that the Dean himself was not a particularly pleasant person to deal with face-to-face.)

Thankfully, the leaders of the postdoctoral associate organization based in the U of MN Twin Cities campus referred me to the Office of Conflict Resolution (OCR), which I promptly contacted. Julie Showers of OCR kindly informed me that the U of MN needed to prove it was suffering a financial emergency in order to legitimately fire a postdoc before her/his term was up.

The upshot was that Joshua Hamilton had no choice but to reinstate my position. But in doing so, he deliberately removed me from my teaching duties. Later on, he got my then-Department Head Marc Seigar to tell me in person, this was because he (Hamilton) “did not want to deal with” me.

It had turned out the performance evaluation meeting Seigar had held with Hamilton, which led to the Chu-Seigar-Vanchurin meeting delineated above, was actually not part of protocol. Instead, my performance review had to be written by Seigar himself. I had thought it to be a mere formality at the time, but was slightly taken aback when I noticed Marc Seigar took the opportunity to write a scathing report on me — from criticizing my e-mails to the Dean to stating my teaching needed improvement. In a face-to-face meeting I politely pushed back, and only then did Seigar tone down his wording. Marc Seigar’s demeanor and body language over the course of Spring 2016 left me really quite bewildered; and things only made much more sense when I saw him promoted to Associate Dean during Fall 2016. During the time while I was still fighting to keep my job, Marc Seigar once gleefully remarked to Vitaly Vanchurin (in my presence), that if I were fired soon Seigar would not have to deal with my complaints of academic fraud.

I cannot help but wonder, did Joshua Hamilton really have such severe budget constraints in the first place? Half of my first 2 years’ of wages came, in fact, from my teaching duties — and hence, at least as I understood it, directly from the UMD Physics Department itself (as opposed to the College). Now that I did not have to teach, Hamilton must be spending more of the College’s money just so that he did not have to deal with me? If so, has the Dean himself put the College in more debt — and likely wasted tax payers’ money, given U of MN is a public school — solely for his personal vendetta? And, why is this not financial fraud?

Honor does not pay

UMD was my 3rd postdoctoral position, after my 2nd one at the University of Pennsylvania. I received my postdoctoral offer from Vitaly Vanchurin sometime in January 2014 and was given very little time to decide. A week after I accepted his offer, I got an offer from CEA Saclay, a far more prestigious institute than UMD. As a scientist, however, I value integrity very highly and turned it down right away, explaining I had already accepted another offer.

Given the appallingly low academic and professional standards held by faculty and leadership at UMD, I am afraid I can only state: honor does not pay — I regretted that decision!

During the January 2014 Skype meeting/interview with Vanchurin, he told me he was trying to build a theory group. (He also hired two others, and implied he will try to keep all of us around, but my other two colleagues left after only a year due to unfavorable circumstances here at UMD.) He told me he would treat his hires as “equals” but in reality Vitaly treated me as his graduate student. Whereas — while we did not get the money — I even wrote 2 grants with him in which I contributed significant ideas. On the other hand, Vanchurin’s rather abrasive style had led him to step on both Joshua Hamilton and John Hiller’s toes very badly. I do not think it is unreasonable to guess, I was probably political cannon fodder for their conflicts, and this is why I experienced so much trouble at UMD.

As of summer 2017, all 3 of Vanchurin’s M.S. students had left his group. One of them — whom I co-mentored — spent the past 2 years or so working with him, only to be told his [the student’s] work was essentially trivial. In truth, the student successfully wrote a robust numerical code to solve a large class of 1-dimensional Euclidean Quantum Field Theories, certainly enough work to graduate with a Master’s degree from UMD.

An Open Letter to (former) Dean Joshua Hamilton and Associate Dean Marc Seigar


In accordance to local Minnesotan culture, I had shown you a lot of deference while you were serving as Department Head. But now that I am no longer under your purview, I’m afraid I have to be rather straightforward here.

  • As an astrophysicist by training, I am sure you know enough of Quantum Mechanics to know how fundamental the abstract formulation of Linear Algebra is. Why did you repeatedly talk past me, phrasing it as either a “conflict resolution” issue or a “physics curriculum” issue — but not addressing the detailed complaints I lodged, when the evidence is clear: John and Sophia repeatedly lied to my students I was teaching at the 8000-level. How could you say in good faith you spent “hours speaking to everyone involved” when you never spoke to me after I lodged my complaints? Did you even bother to read the evidence? For that matter, did you even bother to read my complaint e-mail?
  • Did you feel the strong need to align yourself with your then-upcoming boss Joshua Hamilton — in order to secure your promotion to Associate Dean — given how vindictive he appears to be? This is not a frivolous question. Your answer would indicate where your self-interests were, and whether as Dept Head you were still acting in the interests of upholding the highest standards of learning and science education. To pose the questions bluntly: did you lie about Hiller back-stabbing me in front of Hamilton — you clearly indicated that to me in private — and did you also lie about Hamilton asserting implicit or explicit pressure on you to remove me from teaching Analytical Methods again during Fall 2016?

Josh and Marc:

I am fully aware that Vitaly Vanchurin is on rather poor terms with both you, Joshua Hamilton, and John R. Hiller. I am equally sensitive to the distinct possibility that the multiple difficulties the three hires by Vanchurin faced — even, for instance, having to fight for the right to apply for NSF grants, despite promised the opportunities to do so before we arrived — is due to his severe lack of diplomatic skills.

But both of you are leaders at your respective academic institutions. Should you not be the adults in the room?

Allow me to also make a few somewhat more personal remarks here.

Academia in the United States enjoys paying lip service towards the notion of “diversity”. I have Moebius Syndrome, a one-in-100,000-births-rare disability, which affects my appearance; clarity of speech; and ability to communicate with facial expressions. Despite that, I have not allowed it to hold me back from showing tremendous enthusiasm towards teaching/mentoring both undergraduate and graduate students. Here at UMD, I regularly receive positive feedback for my discussion sections — for instance, during the end of my final semester of teaching here (i.e., Spring 2016), one of my students made sure to thank me, in person, for “giving a damn”. (While I very much appreciated the gratitude, his comments reinforced my worry regarding the academic culture here at UMD, something that both of you are not helping at all to improve.) More importantly — as an experienced theoretical physicist — by writing a short textbook for the students, I brought my personal perspective on the mathematical tools required for understanding physics. I saved them the ~190 dollars they each would have otherwise paid for the course. By raising the academic standards, 3 of them double majoring in Math and Physics were placed into the more advanced version of the Math Department’s Linear Algebra course. Additionally, the student quoted above clearly stated he felt my course helped raised his GPA. On top of all these, within the theoretical physics group led by Vanchurin, I have consistently taken time away from my own research to co-mentor a number of his graduate students, one of whom has recently been admitted into a PhD program in Canada. Research-wise, I have written a number of single-author papers on gravitational wave physics while at UMD; and also helped find exact analytic solutions to a class of theories constructed by Vitaly Vanchurin and his former MS Thesis student Daniel Schubring.

Despite the value I have brought to UMD’s academic community, I have been threatened with premature and illegimate termination; received scathing performance reviews in retailation for pushing back; had my teaching duties revoked; was told by my then-Department Head that the Dean of my College does not want to deal with me; had my detailed evidence-based assertions that colleagues had misled my students summarily dismissed by a Department Head who was more intent on talking past me — calling himself my “mentor” just so he can gas light me on several occasions — and more interested in telling me I needed to learn to “move on”. Whereas the very persons who committed serious academic fraud, i.e., John R. Hiller and Sophia Chabysheva, and who helped sabotage my Analytical Methods course, were instead assigned to teach it during Fall 2016. By Fall 2016, we had at least 3 other theoretical physicists in the Department, not including me, who could have taught the class — allowing the perpetrators of academic fraud to teach the very same course is a fine example of how perverted the reward structure is within academia.

As the leaders of UMD’s College of Science and Engineering — and, especially for Marc Seigar, who has repeatedly called yourself my “mentor” — does your professional conduct serve as an excellent example of how to treat with respect disabled members of your academic community who exert extra effort to challenge students to grow intellectually? You had the power and authority to set the proper tone for the College, to ensure that the highest standards of professionalism and academic inquiry were upheld. All it would have taken is a firm but civil conversation with John Hiller and Sophia Chabysheva, to let them know they had, by providing students false information, violated basic standards of academic integrity; and by disrespectfully sabotaging my course without ever once attempting to communicate with me while I was running it, they had also set an extremely poor example of professional conduct. Instead, it appears you have chosen to play petty politics and catered to your personal career interests.

I don’t know about you both; but in my view, science education itself is a far higher calling that ought to transcend the petty but damaging politics that I am witnessing here. What is the point of holding your privileged positions at UMD and RIC, if you do not value with the highest priority the integrity of serious information transmitted to students by faculty? Both of you are scientists: a toxicologist/biochemist and an astrophysicist. Would you like to point your graduate students to my blog post, so that they can learn from you as role models of academic, professional and scientific honesty?

Yi-Zen Chu, October 2017

Update     I followed up with a letter to the leaders at U of MN Twin Cities & Duluth, as well as Rhode Island College.

Infinitesimal Volume & Particle Number in Fluids

Coordinates as scalar fields     ‘Equi-potential’ surfaces of a scalar field lay out parameterizations of the space(time) it inhabits. In 3D flat space, for instance, the radial coordinate r is a scalar field whose equi-potential surfaces are spherical shells of constant radius from the origin.

Differential forms as infinitesimal volume     Consider a collection of scalar fields \{ \Phi^1, \dots, \Phi^N \}, where N is less than or equal to the dimension of space(time). Modern integration theory uses differential forms to describe infinitesimal volume. For instance, if \{x^1,\dots,x^D\} are Cartesian coordinates in Euclidean space, instead of writing \int f[\vec{x}] d^D\vec{x} as is common in the physics literature, one would instead write

\int f[\vec{x}] dx^1 \wedge dx^2 \wedge \dots \wedge dx^{D-1} \wedge dx^D .

The translation between physicists’ math-speak and differential forms notation is that under an integration sign

d^N \vec{\Phi} \equiv d\Phi^1 \wedge d\Phi^2 \wedge \dots \wedge d\Phi^{N-1} \wedge d\Phi^N.

This needs to be supplemented with the constraint that the differential form on the right hand side is completely antisymmetric in its indices. (The physics notation, in fact, hides this need to choose an orientation of one’s coordinate system.) With the Levi-Civita symbol \epsilon_{\mu_1 \dots \mu_N}, this can be expressed through the relation

d\Phi^{i_1} \wedge d\Phi^{i_2} \wedge \dots \wedge d\Phi^{i_{N-1}} \wedge d\Phi^{i_N} = \epsilon_{i_1 \dots i_N} d\Phi^1 \wedge d\Phi^2 \wedge \dots \wedge d\Phi^{N-1} \wedge d\Phi^N .

The N=1 case, d\Phi^1, is an infinitesimal line segment in the ambient (d \equiv (D+1))-dimensional spacetime. The N=2 case, d\Phi^1 \wedge d\Phi^2, is an infinitesimal area of the parallelogram spanned by 2 infinitesimal line segments, in the following manner. If we choose locally flat and orthonormal coordinates \{ y^\mu \}, so that the I-th line segment can be expanded as \partial_{\mu} \Phi^{\text{I}} dy^\mu, we may view the \nuth term \partial_{\nu} \Phi^{\text{I}} dy^\nu (i.e., no sum over \nu) as the \nuth component of a locally Cartesian vector:

d\Phi^{\text{I}} \leftrightarrow \left( \partial_0 \Phi^{\text{I}} dy^0, \dots, \partial_{d-1} \Phi^{\text{I}} dy^{d-1} \right) .

For the N=2 case, since we are dealing with a 2D vector space — at least when the d\Phi^{1,2} are spacelike — we should be free to choose coordinates such that there are only 2 non-trivial Cartesian components: namely, for I \in \{1,2\}, \Phi^{\text{I}} \to ( 0, \partial_1 \Phi^{\text{I}} dy^1, \partial_2 \Phi^{\text{I}} dy^2, 0, \dots, 0 ). Their antisymmetric wedge product then yields

d\Phi^1 \wedge d\Phi^2 = \left( \partial_1 \Phi^1 \partial_2 \Phi^2 - \partial_2 \Phi^1 \partial_1 \Phi^2 \right) dy^1 \wedge dy^2 = \det\left[ \begin{array}{cc} \partial_1 \Phi^1 & \partial_2 \Phi^1 \\ \partial_1 \Phi^2 & \partial_2 \Phi^2 \end{array}\right] d^2\vec{y}.

At this point, we can indeed recognize the determinant as the area of the parallelogram spanned by d\Phi^1 and d\Phi^2. By extending this argument to arbitrary N, we may see rather explicitly for spacelike \{ d\Phi^1, \dots, d\Phi^N \} that their wedge product yields the volume of the N-parallelepiped they define.

Particle Number and Fluids          Imagine the following continuum approximation of a fluid made of microscopic point particles (in space). Select a small enough packet of the fluid such that, in its instantaneous rest frame, one may choose to lay out within it a grid — i.e., a coordinate system \{ \Phi^{\text{I}} \} — such that there are equal number of particles per infinitesimal “D-parallelepiped”. In other words, for spacelike \{ d\Phi^{\text{I}} \} and up to possibly an overall constant normalization, the d\Phi^1 \wedge \dots \wedge d\Phi^D is the co-moving number of particles in the corresponding infinitesimal volume. For d\Phi^1 \wedge \dots \wedge d\Phi^{D-1}, one may view it as the co-moving flux/number of strings piercing the corresponding infinitesimal area, if one instead has a fluid made of microscopic 1D strings (in space). Generically, for integer 1 \leq N \leq D-1, d\Phi^1 \wedge \dots \wedge d\Phi^{D-N} describes the number of N-dimensional objects penetrating the associated (D-N)-dimensional infinitesimal cross section. In the same vein, the (\mu,\nu) component of the Faraday tensor, namely F_{\mu\nu}, can be viewed as the number of electromagnetic field lines — in other words, flux — through the infinitesimal area d x^\mu \wedge d x^\nu.

Notice, for the particle fluid case, that one may readily write down a vector, dual to the d\Phi^1 \wedge \dots \wedge d\Phi^D, such that it is automatically perpendicular to the spacelike co-moving fluid element:

(1):   n^\mu \equiv \widetilde{\epsilon}^{\mu \alpha_1 \dots \alpha_D} \partial_{\alpha_1} \Phi^1 \dots \partial_{\alpha_D} \Phi^D .

Our scalar field description of co-moving particle number can be further used to formulate the dynamics of simple fluids. Let us write down an action that depends on this co-moving particle number. To form a scalar quantity from it, i.e., the wedge product d\Phi^1 \wedge \dots \wedge d\Phi^D, we may use the “square” of its dual n^\mu in eq. (1). That is,

(2):   S_{\text{particle fluid}} \equiv \int d^d x\sqrt{|g|} \mathcal{L}\left[ n^2/2 \right], \qquad \qquad n^2 \equiv n^\mu n_\mu .

The resulting D equations-of-motion are

(2′):   \partial_\sigma\left( \mathcal{L}'[n^2/2] n_\mu \epsilon^{\mu \alpha_1 \dots \alpha_{\text{I}-1} \sigma \alpha_{\text{I}+1} \dots \alpha_D} \partial_{\alpha_1} \Phi^1 \dots \partial_{\alpha_{\text{I}-1}} \Phi^{\text{I}-1} \cdot \partial_{\alpha_{\text{I}+1}} \Phi^{\text{I}+1} \dots \partial_{\alpha_D} \Phi^{D} \right) = 0 .

(The \epsilon is the Levi-Civita symbol, and not its pseudo-tensor counterpart.)

Perfect Fluid Stress-Energy Tensor          The result may not surprise you, but you may verify via a direct calculation that the energy-momentum-shear-stress tensor of eq. (2) yields that of a perfect fluid:

(3):   \,^{(\text{f})}T^{\mu\nu} = - g^{\mu\nu} \left( \mathcal{L}[n^2/2] - \mathcal{L}'[n^2/2] n^2 \right) - \mathcal{L}'[n^2/2] n^\mu n^\nu .

This allows us to identify the Lagrangian density itself to be negative the energy density \rho_{\text{f}}:

\rho_{\text{f}} = - \mathcal{L} ;

and the pressure density of the fluid p_{\text{f}} as

p_{\text{f}} = \mathcal{L}[n^2/2] - n^2 \mathcal{L}'[n^2/2] .

Note:          One issue with the dynamics of eq. (2) is, it does not always constrain the \{ d\Phi^{\text{I}} \} to be spacelike.

Example: Equilibrium in Flat Spacetime          That d\Phi^1 \wedge \dots \wedge \Phi^D describes particle number can be seen transparently when the fluid in question is in equilibrium in flat spacetime. Specifically, one solution to eq. (2), in an inertial Lorentz frame with Cartesian coordinates \{ x^\mu \}, is given by

(4):   \Phi^{\text{I}} = x^{\text{I}}, \qquad \qquad \text{I} \in \{ 1,2,\dots,D \} .

One may verify that this is a solution quite readily: from the results \partial_\alpha \Phi^{\text{I}} = \delta^{\text{I}}_\alpha and  n^\mu = \delta^\mu_0 (cf. eq. (1)), we note that everything inside the parenthesis of \partial_\sigma ( \dots ) is a constant.

Equation (4) describes a fluid at rest with respect to the Cartesian coordinate grid and has constant particle number density throughout its volume:

d\Phi^1 \wedge \dots \wedge d\Phi^D = d^D\vec{x} .

The associated stress tensor (cf. eq. (3)) is

\,^{(\text{f})}T^{\mu\nu} = - \eta^{\mu\nu} \left( \mathcal{L}[1/2] - \mathcal{L}'[1/2] \right) - \mathcal{L}'[1/2] \delta^\mu_0 \delta^\nu_0 .

All off-diagonal components are zero; while the energy and isotropic pressure densities are respectively

\rho_{\text{f}} = \,^{(\text{f})}T^{00} = -\mathcal{L}[1/2]


p_{\text{f}} = \,^{(\text{f})}T^{ii} = \mathcal{L}[1/2] - \mathcal{L}'[1/2]

with no sum over i .


  • W. Kopczynski, “A fluid of multidimensional objects,” Phys. Rev. D 36, 3582 (1987)
  • N. Andersson and G. L. Comer, “Relativistic fluid dynamics: Physics for many different scales,” Living Rev. Rel. 10, 1 (2007) doi:10.12942/lrr-2007-1 [gr-qc/0605010]
  • S. Dubovsky, L. Hui, A. Nicolis and D. T. Son, “Effective field theory for hydrodynamics: thermodynamics, and the derivative expansion,” Phys. Rev. D 85, 085029 (2012) doi:10.1103/PhysRevD.85.085029 [arXiv:1107.0731 [hep-th]]
  • D. Schubring and V. Vanchurin, “Field theory for string fluids,” Phys. Rev. D 92, no. 4, 045042 (2015) doi:10.1103/PhysRevD.92.045042 [arXiv:1410.5843 [hep-th]]