Schwinger-Keldysh Action Principles for the Damped SHO & 4D Majorana Fermion

Motivation     How does one write down an action for the damped harmonic oscillator $\vec{x}$? Denoting each time derivative as an overdot,

(1):    $m \ddot{\vec{x}} + f \dot{\vec{x}} + \omega^2 \vec{x} = 0 ,$

where $m$ is the mass, $f$ is friction, and $\omega$ is the oscillation frequency of the particle in the limit of zero friction.

More generally, how does one write down an action, not necessarily for particle mechanics, that does not require specifying boundary values?

It turns out that these questions are intimately related to the Schwinger-Keldysh formalism behind the computation of expectation values of quantum operators, as well as the treatment of out-of-equilibrium and/or open quantum systems. Here, I will merely focus on the (semi-)classical limit of two specific examples.

General Strategy     The strategy goes as follows. First double the number of degrees of freedom. For example, if $\vec{x}$ is the trajectory of the simple harmonic oscillator particle, we would now have $\vec{x}_1$ and $\vec{x}_2$. The full action then takes the form

(2):    $S_{\text{total}} \equiv S_0[\vec{x}_1,\dot{\vec{x}}_1] - S_0[\vec{x}_2,\dot{\vec{x}}_2] + S_{\text{IF}}[\vec{x}_1,\dot{\vec{x}}_1; \vec{x}_2,\dot{\vec{x}}_2 ] ,$

where the two $S_0$ are the same except one is evaluated on $\vec{x}_1$ and the other on $\vec{x}_2$; while the “influence action” $S_{\text{IF}}$ couples the $\vec{x}_1$ and $\vec{x}_2$ but has to obey the anti-symmetry property:

(2′):    $S_{\text{IF}}[ \vec{x}_1,\dot{\vec{x}}_1; \vec{x}_2,\dot{\vec{x}}_2 ] = -S_{\text{IF}}[ \vec{x}_2,\dot{\vec{x}}_2; \vec{x}_1,\dot{\vec{x}}_1 ] .$

The difference between the action $S_0$ evaluated on copy-1 and that on copy-2 in eq. (2) arises, within the quantum context, from the Schwinger-Keldysh path integral when describing the time evolution of the density matrix, which plays a key role in the computation of expectation values of quantum field operators. Additionally, the influence action in eq. (2′) can be argued to arise from “integrating out” degrees of freedom.

The full action involves integrating the degrees of freedom from some initial time $t_i$ to the final time $t_f$ and — if fields (as opposed to particles) are involved — over the appropriate spatial domain. However, instead of the usual boundary values, one now requires that the copy-“1” and copy-“2” of the degrees of freedom to be specified at the initial time $t_i$. At the final time $t_f$ we do not fix their trajectories but merely demand that the two copies coincide there:

(3):    $\vec{x}_1[t_f] = \vec{x}_2[t_f]; \qquad\qquad \dot{\vec{x}}_1[t_f] = \dot{\vec{x}}_2[t_f] .$

This necessarily means their variation must also coincide:

(3′):    $\delta\vec{x}_1[t_f] = \delta\vec{x}_2[t_f]; \qquad\qquad \delta\dot{\vec{x}}_1[t_f] = \delta\dot{\vec{x}}_2[t_f] .$

With these conditions in mind, we then demand that the total action $S_{\text{total}}$ be stationary under the variation of both copies of the degrees of freedom. Only after the ensuing equations-of-motion are obtained, do we set the two copies to be equal.

Damped SHO     Let us now proceed to show that the damped harmonic oscillator of eq. (1) follows from the action

(4):    $S_{\text{DSHO}} \equiv \int_{t_i}^{t_f} d t \left\{ \left(\frac{1}{2} m \dot{\vec{x}}_1^2 - \frac{1}{2} \omega^2 \vec{x}_1^2\right) - \left(\frac{1}{2} m \dot{\vec{x}}_2^2 - \frac{1}{2} \omega^2 \vec{x}_2^2\right) - \frac{f}{2} (\vec{x}_1 - \vec{x}_2)\cdot(\dot{\vec{x}}_1 + \dot{\vec{x}}_2) \right\} .$

Demanding the action $S_{\text{DSHO}}$ be stationary under variation with respect to both $\vec{x}_1$ and $\vec{x}_2$,

(5):    $0 = \delta S_{\text{DSHO}} \\ = \int_{t_i}^{t_f} d t \left\{ \delta\vec{x}_1 \cdot \left( -m \ddot{\vec{x}}_1^2 - \omega^2 \vec{x}_1 - f \dot{\vec{x}}_2 \right) - \delta\vec{x}_2 \cdot \left( -m \ddot{\vec{x}}_2^2 - \omega^2 \vec{x}_2 - f \dot{\vec{x}}_1 \right) \right\} + \text{BT};$

with the boundary terms

(5′):    $\text{BT} = \Big[ m (\delta\vec{x}_1\cdot\dot{\vec{x}}_1 - \delta\vec{x}_2\cdot \dot{\vec{x}}_2) - \frac{f}{2} (\vec{x}_1 - \vec{x}_2)\cdot (\delta\vec{x}_1+\delta\vec{x}_2) \Big]_{t_i}^{t_f} .$

Remember, from eq. (3), that we fix the initial conditions $\delta \vec{x}_1[t_i] = 0 = \delta \vec{x}_2[t_i]$; this sets to zero all the terms in the lower limit. Whereas, for the upper limit, we are to set $\vec{x}_1[t_f] = \vec{x}_2[t_f]$; $\delta \vec{x}_1[t_f] = \delta \vec{x}_2[t_f]$; $\dot{\vec{x}}_1[t_f] = \dot{\vec{x}}_2[t_f]$ as well as $\delta \dot{\vec{x}}_1[t_f] = \delta \dot{\vec{x}}_2[t_f]$; and only then does it vanish.

With the boundary terms vanishing, the principle of stationary action then yields the two independent equations

(5”):     $m \ddot{\vec{x}}_1 + f \dot{\vec{x}}_2 + \omega^2 \vec{x}_1 = 0$

and

(5”’):     $m \ddot{\vec{x}}_2 + f \dot{\vec{x}}_1 + \omega^2 \vec{x}_2 = 0.$

Setting $\vec{x}_1 = \vec{x}_2$ in equations (5”) and (5”’) then returns the DSHO equation of eq. (1).

4D Majorana Fermion     For the second example, let us turn to the Majorana fermion, which unlike its Dirac cousin, only requires either the chiral left or chiral right SL$[2,\mathbb{C}]$ spinor — but not both. One such version is provided by the equation

(6):     $i \bar{\sigma}^\mu \partial_\mu \psi = m \epsilon \cdot \psi^* ,$

where $\psi$ is a 2-component spinor, $\bar{\sigma}^0$ is the $2 \times 2$ identity matrix; $\bar{\sigma}^i \equiv -\sigma^i$ with $\{\sigma^i\}$ being the Hermitian Pauli matrices; $m$ is the fermion’s mass, and $\epsilon$ is the 2D Levi-Civita tensor. At the semi-classical level, and at first sight, you might think that the right hand side of eq. (6) could arise from a Lagrangian density of the form

(6′):     $\mathcal{L}_{\text{Majorana mass}} = -\frac{m}{2} \left( \psi^\dagger \epsilon \psi^* + \psi^{\text{T}} \epsilon^\dagger \psi \right)$

But upon closer examination you’d discover this Lagrangian density is identically zero*, as the Levi-Civita tensor is anti-symmetric and therefore

(6”):     $\psi^\dagger \epsilon \psi^* = 0 = \psi^{\text{T}} \epsilon^\dagger \psi .$

But as it turns out, the doubled-field formalism allows one to write down a Lagrangian density. It is given by

(7):     $S_{\text{Majorana}} = \int_{t_i}^{t_f} dt \left( \psi_1^\dagger i \bar{\sigma}^\mu \partial_\mu \psi_1 - \psi_2^\dagger i \bar{\sigma}^\mu \partial_\mu \psi_2 + \mathcal{L}_{\text{IF Majorana Mass}} \right) ,$

where the Majorana mass term is now part of the `influence Lagrangian’  that couples the two copies:

(7′):     $\mathcal{L}_{\text{IF Majorana Mass}} = - \frac{m}{2} \left( \psi_1^\dagger \epsilon \psi_2^* + \psi_1^{\text{T}} \epsilon^\dagger \psi_2 \right) .$

Notice the terms in eq. (7′) are similar to those in eq. (6′) but they do not vanish despite the anti-symmetric nature of $\epsilon$, because we now have two distinct copies of the spinor field.

A similar variational calculation to the one performed for the DSHO would yield eq. (6) from the action in eq. (7). The primary difference from the DHO is that, fermionic systems are first order ones, and therefore only the two copies of the fields — but not their derivatives — need to match at $t_f$, to ensure the boundary terms (analogous to the ones in eq. (5′)) vanishes.

I don’t yet know of any potential physical applications of such a perspective. What sort of open quantum systems would yield eq. (7′)?

* Upon quantization — as the referee of my paper below correctly emphasized — these Majorana fermion fields would still obey anti-commutation relations and, hence, Fermi-Dirac statistics. In fact, this is usually how the Majorana mass Lagrangian in eq. (6′) is justified: unlike the case of the Dirac mass terms, one has to introduce Grassmannian variables from the outset, so that eq. (6”) is no longer true .

References

• J. Schwinger, “Brownian Motion of a Quantum Oscillator,” J. Math. Phys. 2, 407 (1961).
• L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 (1964), [English translation, Sov. Phys. JEPT 20, 1018 (1965)].
• R. D. Jordan, “Effective field equations for expectation values,” Phys. Rev. D33, 444 (1986).
• C. R. Galley, D. Tsang and L. C. Stein, “The principle of stationary nonconservative action for classical mechanics and field theories,” arXiv:1412.3082 [math-ph].
• C. R. Galley, “Classical Mechanics of Nonconservative Systems,” Phys. Rev. Lett. 110, no. 17, 174301 (2013) doi:10.1103/PhysRevLett.110.174301 [arXiv:1210.2745 [gr-qc]].
• J. Polonyi, “Environment Induced Time Arrow,” arXiv:1206.5781 [hep-th].
• Y.-Z. Chu, “A Semi-Classical Schwinger-Keldysh Re-interpretation Of The 4D Majorana Fermion Mass Term,” J. High Energ. Phys., (2018) 2018: 13; arXiv:1708.00338 [hep-th].

Gender Differences for the Lazy/Busy

When I was a teenager (or younger?) I had already read that there are brain differences between men and women. Unfortunately, having spent nearly 2 decades in the US, the information I received regarding gender differences was often muddled; and only later on I began to realize, this was probably due to ideology on the Left. I wish to report that this unscientific behavior can be found throughout Physics and Astrophysics, driven by Leftist politics and radical feminism. That the noble demand for equal rights for men and women does not imply nor require that the genders have to be the same in every aspect — this is clearly not properly appreciated by many in Academia. Unfortunately, much of the hypersensitivity to gender issues is driven by the unfounded desire to see equal representation of women and men in physics, instead of allowing them free rein to choose their careers and judging people purely by merit, as the Scientific Method requires.

Recently, particle theorist Alessandro Strumia gave a talk at CERN’s 1st Workshop on High Energy Theory and Gender. The main thrust of his talk was to challenge the mainstream narrative that High Energy Theoretical Physics has much fewer women than men because of rampant discrimination. He points out there are more women than men in say Education while the reverse is true in STEM fields; and, the more egalitarian a society appears to be, the greater the difference in gender differences when it comes to career choices; moreover, this is consistent with men preferring “things” and women “people”. A glance at his slides would tell you he did some serious analysis/number crunching using bibliometric data collected from the High Energy search engine INSPIRE. I’m not able to independently verify the chronology of events, but there was outrage on social media regarding his talk; even press coverage; and his talk slides/videos were officially censored and the physicist himself was suspended from CERN itself — see CERN’s press release. (Soon after, Strumia’s funding agency, the European Research Council, as well as his home institute University of Pisa, both initiated an investigation against him.) At the end of his slides, Strumia said

PS: many told me “don’t speak, it’s dangerous”. As a student, I wrote that weak-scale SUSY is not right, and I survived. Hope to see you again.

The closest I could find to a justification of such a drastic action of suspension — I’ve been fired once and nearly fired another time over the course of my own 14-year post-Bachelor’s degree academic career, both times without good reasons, so I know full well how that feels like! — is Strumia’s slide 15, where he compared his citation counts with the female Comissar (as I understood it, who was also the organizer of the conference) and another female physicist whom CERN had recently hired, whereas Strumia himself was not offered the same job. This was apparently construed as “attacks on individuals,” which in turn breached CERN’s Code of Conduct. (If there are any misunderstandings on my part, I’d like to hear it; it’s difficult to decipher precisely what happened using information gleaned from the news media and the outrage-driven social media.) However, Strumia’s slide 15 clearly shows, with links to the INSPIRE database so the reader may readily verify the facts for herself, that he did in fact have an order of magnitude more citations than both women: his 30K versus the women’s 2-3K.

Now, I’m just as sensitive as the next human being, and I do consider such a manner of communication to be a rather blunt one. But the scientific ethos requires that, whenever presented with actual evidence, we should address it head on, and not let our personal offense get the better of us as scientists. Namely, “Why wasn’t Strumia hired when he had ten times more citations than the women?” appears to be a legitimate scientific question here. (On the other hand, I was told there were other hires Strumia omitted, and if so I wish he had put everyone on the list for comparison.) Furthermore, observe that was not even the only point on slide 15: he went on to show, of the CERN fellows present, the males had more citations, research papers and years of experience. To suspend him due to the top half of one slide out of 26; to quickly censor his videos such that concerned members of the scientific community (such as I) and of the public cannot independently ascertain what he expressed verbally; and to coat the press release with platitudes regarding “diversity” — altogether does not bode well for the scientific integrity of the particle physics laboratory on our Planet, when it comes to gender issues. The only physicists I am aware of who have actually tried to re-analyze Strumia (and Torre)’s work is Sabine Hossenfelder and her graduate student Tobias Mistele (though using arXiv data, not INSPIRE ones); I believe that is the only true way to respond constructively to the dialog. The rest, I’m afraid, has merely contributed to the Social Justice Warrior far Left Wing I-am-fuming-mad-and-I-need-no-justification culture that infests much of Western Academia these days. If you have been following the news for the past decade or so, Strumia is only but one of many academics/scientists who have been mobbed due to their non-politically-correct views.

I do not think it is unreasonable to postulate, all subsequent High Energy Theory and Gender workshops at CERN — recall that was the first! — will be saturated with talks that will dutifully tow the “women are oppressed/discriminated against” line. This is what such a harsh treatment of Strumia would produce. As scientists, we really need to do some self-examination and ask: is this the scientific outcome we wish to see if truth and intellectual integrity are to prevail? I’m sure it is possible to find sexist individuals, but if calling into question the mainstream narrative of systemic discrimination against women in STEM disciplines is considered taboo, then we have lost our way as scientists.

To be able to think critically through any important issue — particularly complicated and sensitive ones such as gender differences — it is paramount that one is able to hear from and debate against a broad range of views. This way, their relative strengths and weaknesses may be weighed and rational responsible thinkers could propose ideas based on the best available information at hand. This is why freedom of speech is fundamental to any serious democracy. Specifically, it is precisely to allow for contrarian views — popular ones don’t fear backlash from public and/or government persecution! — that is why liberal Western democracies, of which the US is a prime example, provides legal protection for the freedom of expression. (The US Constitution has enshrined this right within its First Amendment.) However, this freedom of expression should not be mere government law. Every one of us is responsible for upholding the right atmosphere within the organizations/societies we belong to, if we wish for there to be an uninhibited exchange of ideas, in order to approach the truth as closely as possible. I want to put on the record, this was why I was motivated to sign the following petition I found online:

CERN: Return Prof. Strumia to office!

Petition to Fabiola Gianotti, Director General CERN, Geneva

Professor Alessandro Strumia, CERN, spoke on Friday 28th September 2018 at a workshop in Geneva on gender and high energy physics. In his presentation he provided evidence for employment policies in physics that were discriminatory toward men and data supporting his opinion that women were given advantages in the academic world purely on the basis of their gender.

As a result, Professor Strumia was suspended with immediate effect by CERN on the grounds that his remarks were antithetical to its code of conduct and to its values.

There can be no free research and freedom of expression if any person must live in fear of existential threat simply for expressing his or her opinion.

Quite independently of the truth of Professor Sturmia‘s statements, none of them can be construed as defamatory, insulting or discriminatory. The opinion he expresses has been expressed many times in multiple research papers and by many other men and women of professional standing.

We cannot and will not tolerate opinions being censored simply because they are in contradiction with mainstream opinion. To do so would be to encourage a totalitarian trend our democracy should not allow.

As an example of political bias in Academia, the particlesforjustice letter — which even contains a thinly veiled threat to destroy Strumia — was posted on the Facebook group Astronomers, whose members are primarily professional astronomers / astrophysicists / physicists. This passed moderation despite the explicit rule that political and non-scientific postings are prohibited. While commenting against the letter, I was challenged to set up my own petition. Even though I did not do so, I did find the above petition and decided to post it in response — the commentary was later shut down by one of the moderators simply because the petition “did not originate in the scientific community and is not appropriate for this forum”. My private messages to the moderators have thus far not been replied to. Ironically, soon after that, someone posted a link to the selection committee for the Breakthrough and New Horizons Prize in Fundamental Physics — and, instead of celebrating the breadth and depth of the scientific expertise assembled — outrage ensued regarding the lack of women on the panel. Of course, no moderation whatsoever was imposed, despite the highly political and un-scientific nature of the discussion.

Update 19 November 2018: I found a very careful article written by a high energy physicist debunking many of the points raised by the particlesforjustice “Community” letter I linked to above. It speaks to the sad state of affairs in the Physics and Astrophysics communities that the author felt the need to remain anonymous.

Update 4 December 2018: There is now another article rebutting the particlesforjustice letter.

Yet another “gender bias” article (this time from Nature) was posted on the Facebook group Astronomers, and I tried to challenge the mainstream gender ideology narrative. This got me banned from the group permanently. An old classmate of mine from graduate school wrote to me to tell me she has unfriended me on Facebook because my comments were “problematic”.

Is Strumia a crackpot when it comes to the science of gender discrimination in STEM fields? Is there the slightest possibility that men could be discriminated against in STEM disciplines such as High Energy Theory? That his science is horrendous has been asserted repeatedly in the above letter and throughout social media. I cannot speak to the detailed analysis he had done; but I have been aware, since a few years ago and also cited by Strumia in his slides, that Williams and Ceci (faculty at the Department of Human Development, Cornell University) had found a preference for hiring women over men at the tenure-track level. In their youtube video, they also debunked the mainstream claim there is a ton of evidence to support discrimination against women — at least when it came to hiring them as professors.

“… We were really quite shocked, in poring over this literature — it took us many months to digest it all — how little evidence there was. And in fact, there was no experimental evidence. There were experiments, many of them, showing sex biases in hiring, but not of professors, not of tenure-track professors. … [After Wendy M. Williams spoke.] … But there actually was a lot of actuarial evidence that actually went opposite to the bias claims. By that I mean, there were a lot of very large scale studies that looked at who got hired. And these studies — again going back to the mid 1980’s — showed that, over and over again, women were hired at a higher rate than their fraction of the applicant pool. So women were less likely to apply for jobs in math intensive areas, but if they did apply they were more likely to be interviewed and more likely to be hired. … ” — Stephen J. Ceci

One thing I wish they had done was to include physics and/or astrophysics in their analysis. (Of all the “math intensive” groups they analyzed, only male economists were gender neutral.)

My own sense is that my fellow scientists really need to take a hard look at the planks in their own eyes and recognize they are — whether consciously or not — taking part in science denial themselves, despite often ridiculing the Right for climate-science denial. Evolutionary forces have shaped how the genders metamorphosed throughout humanity’s existence, due to the different roles they have played for the majority of that duration; and, hence, it would be shocking if men and women were truly identical. I urge the open minded amongst my fellow scientists: please, educate yourselves a tad. (That includes myself, of course — I am no expert in evolutionary biology.) In particular, women and men on average have different interests, life priorities; and therefore make distinct career choices. There really is no good reason to expect a 1:1 ratio in women to men in various careers, such as Physics versus Nursing, and to force it so would in fact necessarily involve discrimination.*

Fortunately, for the busy/lazy physicist / astrophysicist / astronomer out there, there are now plenty of readily accessible youtube videos discussing gender differences known to science, from the experts themselves. (If readers wish to contribute more links, please do post them in the comments section below.)

Let’s begin with the cognitive psychologist Steven Pinker, who wrote the book The Blank Slate: The Modern Denial of Human Nature. Here, Pinker tells us men tend to “chase status at the expense of family” whereas women tend to value family over career. Women gravitate towards “people-oriented” careers whereas men towards “things-oriented” ones — even at the PhD level, more women are pursuing degrees in Education than in Physics, say; even though the total number women pursuing higher education has been growing significantly over the past decades.** Men tend to be the risk-taking ones. Men are better at three dimensional mental rotations, spatial perception and visualization. “Women are better at mathematical calculation” and “men score better on mathematical word problems and tests of mathematical reasoning”. Pinker goes on to explain why there are good reasons to believe many of these sex differences are biological; i.e., they cannot be accounted for solely due to “socialization”. There are large differences in exposure to sex hormones starting prenatally; and small differences in size, density, cortical asymmetry, hypothalamic nuclei of men versus women’s brains. Gender differences in personality transcends “ages, years of data collection, education levels, and nations”. Many of these gender differences has not changed with time; are also seen in other animals; and in fact emerge in early childhood. He even spoke about cases where boys without penises (due to accident or otherwise) and who were brought up as girls, still ended up exhibiting male typical behavior. He also advertises other popular level books like his, that explains the scientific evidence for the biological factors behind gender differences. Steven Pinker can also be found speaking with Dave Rubin here and here about related issues.

Debra Soh, who has a PhD in neuroscience, has been discussing how far Left politics has made discussing the science of transgendered people very difficult, even within academia. Herehere, and here (among other similarly humor-tinged clips) she explains that exposure to testosterone before birth (i.e., “prenatal exposure”) have serious impacts on why the different genders develop different interests. Higher levels lead to “male-typical activities” such as mechanical stuff; whereas lower levels are associated with “socially-engaging” ones. Women are higher in agreeableness and neuroticism, and lower in stress tolerance. (“Neuroticism is simply a technical term for someone’s likelihood to experience negative moods,” according to Soh.) Rates of depression are higher in women. Testosterone is related to greater risk-taking by men. When it comes to brain structure, certain portions are larger in men than in women; there are more front-to-back connections in men’s brains but more left-to-right-hemisphere connections in women’s brains. She goes on to femsplain why James Damore (who was fired by Google for his now infamous memo regarding his reading of what the scientific literature says about gender differences) in fact got his facts/scientific literature right.

Gad Saad, who founded the field of evolutionary psychology applied to marketing and consumer behavior, runs a youtube channel to counter what he likes to call the “tsunami of lunacy crashing against the shores of reason” — i.e., politically correct culture that has become so illiberal and irrational — can be found speaking about gender differences, for instance, here, here, and here.

Heterodox academy, which was founded by NYU psychologist Jonathan Haidt and others, in an effort to counter the strong left wing illiberal culture of Western academia, contains a page on the abovementioned “Google memo”. They performed a literature review to examine how robust Damore’s claims were. Towards the end of this page,

In conclusion, based on the meta-analyses we reviewed and the research on the Greater Male Variability Hypothesis, Damore is correct that there are “population level differences in distributions” of traits that are likely to be relevant for understanding gender gaps at Google and other tech firms. The differences are much larger and more consistent for traits related to interest and enjoyment, rather than ability. This distinction between interest and ability is important because it may address  one of the main fears raised by Damore’s critics: that the memo itself will cause Google employees to assume that women are less qualified, or less “suited” for tech jobs, and will therefore lead to more bias against women in tech jobs. But the empirical evidence we have reviewed should have the opposite effect. Population differences in interest and population differences in variability of abilities may help explain why there are fewer women in the applicant pool, but the women who choose to enter the pool are just as capable as the larger number of men in the pool. This conclusion does not deny that various forms of bias, harassment, and discouragement exist and may contribute to outcome disparities, nor does it imply that the differences in interest are biologically fixed and cannot be changed in future generations.

If our three conclusions are correct then Damore was drawing attention to empirical findings that seem to have been previously unknown or ignored at Google, and which might be helpful to the company as it tries to improve its diversity policies and outcomes.

There was also a Quillette article written by 4 scientists —  Jussim, Schmitt, Miller, and Soh — on James Damore’s “Google Memo”.

Ellis et al.: there is an entire book — Sex Differences: Summarizing More than a Century of Scientific Research.

Let me close with the following two examples which I find illustrative of the current far-Left Wing culture within the West and its Academy.

Physics postdoc Jess Wade’s Twitter post and her New Scientist article have both compared Strumia’s talk to the memo Damore put out, as if that would decisively rule out any credibility in Strumia’s presentation. For instance, in her New Scientist article, she states:

Unlike my talk, backed by evidence, he [Strumia] cited a bunch of poorly thought out gender science from right-wing thinkers. These included James Damore, who was fired from Google last year for holding similar views.

Is using bibliometric data from INSPIRE slide after slide the same as citing “a bunch of poorly thought out gender science from right-wing thinkers”? More importantly, why did she grossly misrepresent Strumia’s talk as “un-backed by evidence”? In her Twitter thread, Jess Wade was challenged on whether she had read Damore’s memo, but as of this writing I could not find any substantive response from her end. I’m sorry to state, it is quite clear who is being ideologically driven — and it is not Strumia nor Damore, as far as I can tell. And, given the current climate, it is also highly unlikely for her to face any serious push back from other physicists.

Another example comes from no less than the former governor of Vermont, Howard Dean. At an event at Kenyon College, Dean misrepresented not only the awful conduct of Yale students towards faculty Nicholas and Erika Christakis; but also what James Damore said about women in STEM careers. I was glad to see both Steven Pinker and Heather Mac Donald pushing back with the relevant facts; but as far as I could tell, Dean was simply unwilling to concede his serious errors.

The lack of intellectual integrity and honesty exhibited by the Left when it comes to gender issues is precisely the evidence for its commitment to ideology. From the outrage mob that quickly followed the news of Strumia’s talk, it is clear the strongly illiberal tendencies of the far Left has infiltrated Physics / Astrophysics.

The illiberal, irrational, gender-science-denying and identity politics obsessed character of the Left Wing, which Western Academia firmly belongs, form the key impetus behind why I no longer wish to consider myself part of it.

* I believe the Scientific Method requires that, if we are interested in attracting the most competent and creative scientific minds, people should be judged based solely on merit — for e.g., their past accomplishments. I’m afraid I do find a lot of these academic “diversity” initiatives to be primarily identity politics driven, and not merit driven. If we are genuinely keen in “diversity” we should be investing in thought diversity.

** Overall, in the West, women now outnumber men when it comes to obtaining advanced degrees. See here, for example.

Statement of Author Contributions

What are the actual scientific values practiced by the (theoretical) physics community? In particular, how seriously do we think there should be proper accounting of the intellectual effort of each scientist involved — regardless of seniority — in a given project? Namely, who actually did the damned work? In addition to who came up with the ideas, who wrote the code/did the calculations that made those ideas reality? Who found the means to overcome the technical difficulties? Who carried out the check(s) of the results? Does one’s supervisor deserve to be on one’s paper just because the supervisor is paying the salary; i.e., can intellectual credit be bought? What are the incentive structures in place that reflect our values?

Physical Review D (PRD) — supposedly one of the most highly regarded journals in high energy theory, cosmology and gravitation — recently sent out a survey regarding its “performance”: see here if you’re interested in participating. Now, one of the primary purposes of journals is that of conducting peer review, but I have personally found the process itself to be quite arbitrary and frankly mostly a going-through-the-motion just to acquire a superficial “stamp of approval”. For papers from mid-1990’s onward, I go to the arXiv, not the journals, to read them. If the sub-standard level of refereeing in theoretical physics I have experienced is reflective of the current climate, and if it continues not to improve substantially, I do fear journals will become obsolete in the near future.

One feedback I gave PRD concerned that of intellectual credit. Paraphrasing myself, I wish to advocate:

That theoretical physics journals mandate a description of contributions from each and every individual author on the paper.

I believe both Science and Nature require it — so, I have never understood, why not the top theoretical physics journals? To play devil’s advocate, I am curious: what potential negative consequences could such a mandate have on research collaborations?

What happens to the simple harmonic oscillator path integral at half periods?

Previously we had stated but not elucidated the result for the transition amplitude of the simple harmonic oscillator (SHO) at half periods. The SHO Hamiltonian, with $X$ and $p$ denoting the position and momentum operators respectively, reads

(1):    $H_\text{SHO} \equiv \frac{1}{2} p^2 + \frac{1}{2} \omega^2 X^2 .$

In most quantum mechanics textbooks, we are told that the transition amplitude from the spatial location $x'$ to $x$ over the time period $t-t'$ is

(1′):    $\langle x \vert \exp[-i(t-t') H_{\text{SHO}}] \vert x' \rangle = \sqrt{ \frac{\omega}{2\pi i \sin[\omega(t-t')]} } \exp\left[ \frac{i\omega}{2\sin[\omega(t-t')]} \left((x^2+x'^2)\cos[\omega(t-t')]-2xx'\right) \right] .$

One can see that, when $t-t'$ equals some multiple of $\pi/\omega$, this formula does not make much sense.

The classical solution for $X$ is a linear combination of $\sin[\omega t]$ and $\cos[\omega t]$. The period $T$ is therefore $\omega T = 2\pi \Rightarrow T = 2\pi/\omega$. Over a half period $\pi/\omega$, i.e., upon replacing $t \to t \pm \pi/\omega$, we have $\sin[\omega t] \to - \sin[\omega t]$ and $\cos[\omega t] \to - \cos[\omega t]$. In other words, starting at time $t$ the motion performs a parity flip $X \to -X$ over a half-oscillation.

We will now discuss why the quantum SHO particle does exactly the same thing. If it begins from the location $x'$ at time $t$, then it has to be at $x \equiv (-)^n x'$ at time $t + n \pi/\omega$ for integer $n$. Namely, we have the path integral (aka transition amplitude)

(2):    $\langle x \vert \exp[-i H_{\text{SHO}} \Delta t_n ] \vert x' \rangle = \frac{1}{i^n} \delta[x - (-)^n x'] ;$

with the time interval

(2′):     $\Delta t_n \equiv \frac{n\pi}{\omega}$.

One might otherwise think there is an inherent ‘fuzziness’ to quantum motion, but at half-periods, all the quantum aspects of the dynamics seem to make their appearance only in the Maslov index $1/i^n$ multiplying the Dirac delta-function on the right hand side of eq. (2). As we will now see, eq. (2) is a direct consequence of the exact invariance of the SHO Hamiltonian in eq. (1) under parity. If $P$ is the parity operator that implements $P X P^{-1} = -X$, then the commutator $[P, H_{\text{SHO}}]$ vanishes.  Therefore the SHO energy eigenstates must be simultaneous eigenstates of the parity operator; specifically, the $\{ \vert E_\ell \rangle \}$ in

(2′):     $H_{\text{SHO}} \vert E_\ell \rangle = \left(\ell + \frac{1}{2}\right) \omega \vert E_\ell \rangle$

obey

(3):     $P \vert E_\ell \rangle = (-)^\ell \vert E_\ell \rangle, \qquad \ell = 0,1,2,\dots$.

Now, by inserting a complete set of energy eigenstates, we may begin from the left hand side of eq. (2), to consider motion over a single half period $\Delta t_1 = \pi/\omega$:

(4):    $\langle x \vert \exp[-i H_{\text{SHO}} (\pi/\omega)] \vert x' \rangle = \sum_{\ell=0}^\infty \langle x \vert E_\ell \rangle \langle E_\ell \vert x' \rangle \exp[-i (\ell+1/2) \pi]$ .

Observe that $e^{-i\pi/2} = 1/i$; whereas $e^{-i\ell \pi} = (-)^\ell$. Invoking the parity property of the energy eigenstates in eq. (3) to say $e^{-i\ell \pi} \langle x \vert E_\ell \rangle = \langle -x \vert E_\ell \rangle$, we may thus write

(4′):    $\langle x \vert \exp[-i H_{\text{SHO}} (\pi/\omega)] \vert x' \rangle = \frac{1}{i} \sum_{\ell=0}^\infty \langle -x \vert E_\ell \rangle \langle E_\ell \vert x' \rangle$ .

The completeness relation in the position representation is $\sum_\ell \langle u \vert E_\ell \rangle \langle E_\ell \vert v \rangle = \delta[u-v]$; comparing this to eq. (4′) we arrive at

(5):    $\langle x \vert \exp[-i H_{\text{SHO}} (\pi/\omega)] \vert x' \rangle = \frac{1}{i} \delta[(-) x - x'] = \frac{1}{i} \delta[x - (-) x']$ .

By recognizing the unitary nature of the time-evolution operator $\exp[-i H_{\text{SHO}} t ]$, and applying the result in eq. (5) $n$ times to evolve a quantum particle from $x'$ to some other location $x$ over $n$ half periods, one will recover the primary result in eq. (2).

Reference

• P. A. Horvathy, “The Maslov correction in the semiclassical Feynman integral,” Central Eur. J. Phys. 9, 1 (2011) doi:10.2478/s11534-010-0055-3 [quant-ph/0702236].

Synge World Function & Shapiro Delay

Geodesics     Consider all possible spacetime trajectories joining the points $x'$ and $x$. Suppose you found a trajectory $Z^\mu[0 \leq \lambda \leq 1]$ (obeying $Z[0]=x'$ and $Z[1]=x$) such that any perturbation away from it yields a slightly longer or shorter path — then such a path is said to extremize the distance between $x'$ and $x'$. In differential geometric/general relativity lingo, such a path is called a geodesic.

In a given spacetime metric $g_{\mu\nu}$ and given a geodesic trajectory $Z^\mu$ joining $x'$ to $x$, half the square of the geodesic distance between $x'$ and $x$ can be written as

(1):    $\sigma[x,x'] = \frac{1}{2} \int_0^1 g_{\mu\nu}[Z] \frac{d Z^\mu}{d\lambda} \frac{d Z^\nu}{d\lambda} d\lambda , \qquad \qquad Z[0]=x', \ Z[1]=x ;$

where $\sigma[x,x']$ is dubbed “Synge’s World Function” in the literature.

Conversely, if we view eq. (1) as a functional of the trajectory $Z$, and we demand it to be extremized, then it would yield the (affinely parametrized) geodesic equation

(1′):    $\frac{d^2 Z^\mu}{d \lambda^2} + \Gamma^\mu_{\phantom{\mu}\alpha\beta} \frac{d Z^\alpha}{d \lambda} \frac{d Z^\beta}{d \lambda} = 0 .$

To sum: Synge’s world function is the action principle for affinely parametrized geodesic motion; and when evaluated on a given geodesic $Z^\mu[0 \leq \lambda \leq 1]$ joining $x'$ to $x$, it hands us half the square of the geodesic distance between this pair of spacetime points.

Perturbation Theory     In a weakly curved spacetime of the form

(2):    $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu},$

where the components of $h_{\mu\nu}$ are assumed to be much less than unity; the Synge’s world function may be used to find an $\mathcal{O}[h]$ accurate integral solution for geodesic distances merely from the geodesic solutions in flat spacetime — namely, a straight line $\bar{Z}^\mu$ joining $x'$ to $x$ — precisely because $\sigma[x,x']$ is the geodesic action principle. To see this, we first express the geodesic solution $Z^\mu$ of the geometry in eq. (2) as a perturbation away from a straight line:

(2′):    $Z^\mu = \bar{Z}^\mu + \delta Z^\mu,$

where the straight line itself is

(2.S):    $\bar{Z} = x' + \lambda (x-x') .$

Up to first order in $\delta Z$, Synge’s world function is

(2”):    $\sigma[x,x'] = \int_0^1 \left( \frac{1}{2} \left( \eta_{\mu\nu} + h_{\mu\nu}[\bar{Z}] \right) (x-x')^\mu (x-x')^\nu - \delta Z^\kappa \left( \eta_{\kappa\mu} + h_{\kappa\mu}[\bar{Z}] \right) \frac{D^2 \bar{Z}^\mu}{d \lambda^2} + \mathcal{O}\left[(\delta Z)^2\right] \right) d\lambda .$

Here, the boundary conditions $Z[0]=x', \ Z[1]=x \Leftrightarrow \delta Z[0]=0=\delta Z[1]$ were used to set to zero the boundary terms; we have used eq. (2.S) to infer $\dot{Z} = x-x'$; and, finally, the ‘geodesic operator’ $D^2/d \lambda^2$ reads

(2”’):    $\frac{D^2 Y^\mu}{d\lambda^2} \equiv \frac{d^2 Y^\mu}{d \lambda^2} + \Gamma^\mu_{\phantom{\mu}\alpha\beta}[g=\eta+h] \frac{d Y^\alpha}{d \lambda} \frac{d Y^\beta}{d \lambda} .$

Both $\delta Z$ and $D^2 \bar{Z}/d \lambda^2$ in eq. (2”) must scale as order $h$ or higher since they vanish in the limit as $h_{\mu\nu} \to 0$. (The arguments for $D^2 \bar{Z}/d \lambda^2 \sim \mathcal{O}[h]$ and $\delta Z \sim \mathcal{O}[h]$ can be made explicit by direct computation for the former; and, for the latter, by first converting the geodesic equation of eq. (1′) into an integral equation, followed by employing the Born-series-approximation iteration technique.) Therefore the $\delta Z (\dots) D^2 \bar{Z}/d \lambda^2$ group of terms on the right in eq. (2”) must scale as $\mathcal{O}[h^2]$ or higher and may thus be dropped if all we are seeking is a first order accurate expression.

To summarize: at first order in the metric perturbation, half the square of the geodesic distance between the pair of spacetime points $x'$ and $x$ in a weakly curved spacetime (cf. eq. (2)) is given by the integral

(2.Synge):    $\sigma[x,x'] = \frac{1}{2} (x-x')^\mu (x-x')^\nu \int_0^1 \left( \eta_{\mu\nu} + h_{\mu\nu}[\bar{Z}] \right) d\lambda + \mathcal{O}[h^2],$

with the straight line already given in eq. (2.S).

Linearized Einstein’s Equations     If we choose the de Donder gauge

(3):    $\partial^\sigma \bar{h}_{\sigma\mu} = 0 ,$

where we are moving indices with the flat Cartesian metric and

(3′):    $\bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{\eta_{\mu\nu}}{2} \eta^{\sigma\rho} h_{\sigma\rho} ;$

Einstein’s field equations $G_{\alpha\beta} = 8\pi G_\text{N} T_{\alpha\beta}$ linearized about flat spacetime yields

(3”):    $\partial^2 \bar{h}_{\mu\nu} = -16\pi G_\text{N} \bar{T}_{\mu\nu} ,$

with $\partial^2$ denoting the flat spacetime wave operator and $\bar{T}_{\mu\nu}$ is the portion of the matter stress-energy tensor $T_{\mu\nu}$ that does not contain any metric perturbations. Now, in the non-relativistic limit, stress-energy-momentum is dominated by the energy density; if $v$ is some characteristic speed of the internal dynamics of the source (in its rest frame), we usually have $\bar{T}_{0i}/\bar{T}_{00} \sim \mathcal{O}[v]$ and $\bar{T}_{ij}/\bar{T}_{00} \sim \mathcal{O}[v^2]$. In such a scenario, we may parametrize the metric perturbation as a unit matrix proportional to the Newtonian potential

(3.N):    $h_{\mu\nu} = 2\Phi \delta_{\mu\nu}$

such that eq. (3”) is now dominated by the Poisson equation

(3”’):    $\vec{\nabla}^2 \Phi = \frac{16\pi G_\text{N}}{d} \bar{T}_{00}.$

($d$ is the spacetime dimension.) What’s crucial for the Shapiro delay discussion below is that this Newtonian potential is strictly negative — provided the energy density is positive ($\bar{T}_{00} \geq 0$) — since the Euclidean Green’s function $1/\vec{\nabla}^2$ is strictly negative:

(3.NS):    $\Phi[t,\vec{x}] = - \frac{16\pi G_\text{N}}{d} \int_{\mathbb{R}^{d-1}} d^{d-1}\vec{x}' \frac{\Gamma[\frac{d-3}{2}]}{4\pi^{\frac{d-1}{2}} |\vec{x}-\vec{x}'|^{d-3}} \overline{T}_{00}[t,\vec{x}'] .$

Shapiro Delay     Consider two observers at spatial locations $\vec{x}$ and $\vec{x}'$ sending signals to each other via null rays (e.g., high frequency electromagnetic waves). Suppose the null rays pass through a region of spacetime near an isolated non-relativistic matter source, we may compute the time-of-flight between emission at $(t',\vec{x}')$ to reception at $(t,\vec{x})$ using the Synge’s world function. Since null rays are involved, that means the Synge’s world function in eq. (2.Synge) is zero:

(4):    $0 = (x-x')^\mu (x-x')^\nu \int_0^1 \left( \eta_{\mu\nu} + 2\Phi[\bar{Z}] \delta_{\mu\nu} \right) d\lambda , \qquad \qquad x\equiv(t,\vec{x}), \ x'\equiv(t',\vec{x}') \\ = T^2 - R^2 + 2 (T^2 + R^2) \int_0^1 \Phi[\bar{Z}] d\lambda .$

(Remember, $\bar{Z}^\mu = x'+\lambda(x-x')$.) We have used the shorthand $T\equiv t-t'$ for the time elapsed; and $R\equiv |\vec{x}-\vec{x}'|$ for the Euclidean coordinate distance between the observers. We see that, if there were no matter source, so that $\Phi = 0$, the Minkowski light cone condition $T^2 = R^2$ would be recovered. That $T^2 = R^2(1 + \mathcal{O}[\Phi])$ in turn means the $T^2$ multiplying the $\Phi$-integral may be replaced with $R^2$, since the error incurred would be of second order. This leads us to deduce from eq. (4)

(4′):    $T^2 = R^2 \left( 1 - 4 \int_0^1 \Phi[\bar{Z}] d\lambda + \mathcal{O}[\Phi^2] \right).$

Taking the positive square root on both sides, we find that the time elapsed is

(4.Shapiro):    $t-t' = |\vec{x}-\vec{x}'| \left( 1 - 2 \int_0^1 \Phi[\bar{Z}] d\lambda + \mathcal{O}[\Phi^2] \right) .$

We have arrived at the main result: Shapiro time delay. If the matter source were absent, the spacetime would be completely flat and $T=R$. But now that $\Phi$ is non-trivial, we see it increases the time-of-flight because the $-2 \Phi$ in eq. (4.Shapiro) is positive; which in turn is due to the positive character of the energy density $\bar{T}_{00}$ in eq. (3.NS). In fact, if energy density were strictly negative, $\bar{T}_{00} < 0$, notice this would decrease the time-of-flight and the effective speed of light would be faster than that in flat spacetime!

References

• I.I. Shapiro, “Fourth Test of General Relativity,” Phys. Rev. Lett. 13, 789 (1964). doi:10.1103/PhysRevLett.13.789
• I.I. Shapiro, G.H. Pettengill, M.E. Ash, M.L. Stone, W.B. Smith, R.P. Ingalls and R.A. Brockelman, “Fourth Test of General Relativity: Preliminary Results,”
Phys. Rev. Lett. 20, 1265 (1968). doi:10.1103/PhysRevLett.20.1265
• I.I. Shapiro, M.E. Ash, R.P. Ingalls, W.B. Smith, D.B. Campbell, R.B. Dyce, R.F. Jurgens and G.H. Pettengill, “Fourth test of general relativity – new radar result,” Phys. Rev. Lett. 26, 1132 (1971). doi:10.1103/PhysRevLett.26.1132
• M.J. Pfenning and E. Poisson, “Scalar, electromagnetic, and gravitational selfforces in weakly curved space-times,” Phys. Rev. D 65, 084001 (2002) doi:10.1103/PhysRevD.65.084001 [gr-qc/0012057].
• Y. Z. Chu and G.D. Starkman, “Retarded Green’s Functions In Perturbed Spacetimes For Cosmology and Gravitational Physics,” Phys. Rev. D 84, 124020 (2011) doi:10.1103/PhysRevD.84.124020 [arXiv:1108.1825 [astro-ph.CO]].

Simple Harmonic Oscillator Path Integral

If you have taken quantum mechanics up to graduate school, you’d certainly have learned about the following transition amplitude — i.e., the path integral — for the simple harmonic oscillator (SHO):

(1):    $\langle x \vert \exp[-i(t-t') H_{\text{SHO}}] \vert x' \rangle = \sqrt{ \frac{\omega}{2\pi i \sin[\omega(t-t')]} } \exp\left[ \frac{i\omega}{2\sin[\omega(t-t')]} \left((x^2+x'^2)\cos[\omega(t-t')]-2xx'\right) \right] ,$

where we assume $t-t' \geq 0$; and the Hamilton of the SHO is

(1′):    $H_{\text{SHO}} = \frac{1}{2} p^2 + \frac{\omega^2}{2} X^2 .$

I will focus on the 1-dimensional case for simplicity. The key point in this post is that the discussion found in textbooks is usually not complete. For one, what does the square root of $1/i$ mean in eq. (1)? And, what happens when $\omega(t-t')$ is an integer multiple of $\pi$ — the sine goes to zero, and does that imply the path integral becomes ill defined?

The full path integral — the quantum mechanical transition amplitude for the SHO particle to propagate from $x'$ to $x$ is actually as follows. When the time elapsed lie within the ‘first half-period’, namely $0 < t-t' < \pi/\omega$, we have

(2):    $\langle x \vert \exp[-i(t-t') H_{\text{SHO}}] \vert x' \rangle = e^{-i\frac{\pi}{4}} \sqrt{ \frac{\omega}{2\pi |\sin[\omega(t-t')]|} } \exp\left[ \frac{i\omega}{2\sin[\omega(t-t')]} \left((x^2+x'^2)\cos[\omega(t-t')]-2xx'\right) \right] ,$

where here and below the square roots of real quantities are the positive ones. When the time elapsed lie between the $n$th and the $(n+1)$th half-periods, $n \pi/\omega < t-t' < (n+1) \pi/\omega$, we have

(2′):    $\langle x \vert \exp[-i(t-t') H_{\text{SHO}}] \vert x' \rangle = \frac{e^{-i\frac{\pi}{4}}}{i^n} \sqrt{ \frac{\omega}{2\pi |\sin[\omega(t-t')]|} } \exp\left[ \frac{i\omega}{2\sin[\omega(t-t')]} \left((x^2+x'^2)\cos[\omega(t-t')]-2xx'\right) \right] .$

In more detail, if we define $K_0[t-t';x,x'] \equiv e^{-i\frac{\pi}{4}} \sqrt{ \frac{\omega}{2\pi |\sin[\omega(t-t')]|} } \exp\left[ \frac{i\omega}{2\sin[\omega(t-t')]} \left((x^2+x'^2)\cos[\omega(t-t')]-2xx'\right) \right]$,

(2′.1):    $\langle x \vert \exp[-i(t-t') H_{\text{SHO}}] \vert x' \rangle = K_0[t-t';x,x'], \qquad\qquad 0 < t-t' < \frac{\pi}{\omega},$

(2′.2):     $\langle x \vert \exp[-i(t-t') H_{\text{SHO}}] \vert x' \rangle = \frac{1}{i} K_0[t-t';x,x'], \qquad\qquad \frac{\pi}{\omega} < t-t' < \frac{2\pi}{\omega},$

(2′.3):     $\langle x \vert \exp[-i(t-t') H_{\text{SHO}}] \vert x' \rangle = \frac{1}{i^2} K_0[t-t';x,x'], \qquad\qquad \frac{2\pi}{\omega} < t-t' < \frac{3\pi}{\omega},$

(2′.4):     $\langle x \vert \exp[-i(t-t') H_{\text{SHO}}] \vert x' \rangle = \frac{1}{i^3} K_0[t-t';x,x'], \qquad\qquad \frac{3\pi}{\omega} < t-t' < \frac{4\pi}{\omega} \dots$

These $1/i$ jumps in the phase factor when transitioning across half-periods are known as Maslov indices, and come about because one additional eigenvalue of the operator $(d/dt)^2 + \omega^2$ flips sign whenever the time elapsed $t-t'$ crosses over from say the $n$th to $(n+1)$th half-period.$\,^\star$

What happens when the time elapsed is some multiple of $\pi/\omega$? When $t-t' = n\pi/\omega$, for integer $n$, the path integral collapses to a Dirac delta function:

(2”):    $\langle x \vert \exp[-i(t-t') H_{\text{SHO}}] \vert x' \rangle = \frac{1}{i^n} \delta\left[ x - (-)^n x' \right] .$

As I hope to show in an upcoming post, the $(-)^n$ occurring within the Dirac delta function in eq. (2”) is intimately related to the fact that the energy eigenstates of the SHO system are also eigenstates of the parity operator.

$\,^\star$   I learned about this subtle aspect of the path integral from a comment left on Peter Woit’s excellent blog. The Maslov index, which falls within the more general rubric of Morse theory, is a generic feature of path integrals. It also shows up in the description of geometric optics and caustics.

References

• P. A. Horvathy, “The Maslov correction in the semiclassical Feynman integral,” Central Eur. J. Phys. 9, 1 (2011) doi:10.2478/s11534-010-0055-3 [quant-ph/0702236].
• L. S. Schulman, “Techniques and Applications of Path Integration,” Dover Publications (December 27, 2005)
• H. Kleinert, “Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets.” World Scientific Publishing Company; 5 edition (May 18, 2009)