The Lagrangian – possibly the most fundamental mathematical principle in physics

This is something I wrote on Quora, which was ‘upvoted’ by a physics tutor (Mike Milner) and someone with an MSc (Dimitrios Kalemis), which gives it some credence.

I’ve written about all of this before in previous posts, but probably not as succinctly, which hopefully makes it easier to follow.


How does an electron know beforehand that it's a single slit or double slit so it decides whether to create an interference pattern or not?

Obviously it doesn’t. It’s like asking how does a ball thrown in the air know what path to follow? These 2 questions have more in common than you might think.

There is a fundamental principle in physics called the principle of least action, and Richard Feynman used it to describe the trajectory of a ball in a gravitational field and also as the basis for his path integral method of quantum mechanics (QM).

The principle of least action is that the difference between the potential energy and the kinetic energy of a particle will always be a minimum and, mathematically, this is called a Lagrangian. In his book, Six Not-So-Easy Pieces, Feynman demonstrates how this applies to a body in a gravitational field when it follows the path dictated by a geodesic, which, in Einstein’s theory of relativity, is the path of maximum relativistic time. It turns out that this is the shortest path and also the path of least action, as determined by the Lagrangian.

Feynman gives the following analogy. Imagine a lifesaver needing to run along a beach and then swim out to rescue a bather in distress in the surf. The lifesaver could run along the beach (at a diagonal) until he (or she) is perpendicular to the swimmer in the waves and swim out. Or the lifesaver could run straight into the surf and swim diagonally to the swimmer. But the optimum path is something in between these 2 and that’s the path of least action or least time. It’s also the path of light when it refracts through glass or any other medium.

It was Paul Dirac who originally wrote a Lagrangian for QM and Feynman used his result to derive Schrodinger’s equation. Feynman’s approach to the 2 slit problem or any other QM problem was to combine all the possible paths the electron (or a photon) could take. By ‘combine’ this means adding all the phases of the wave function, most of which cancel each other out. Then, using Born’s rule, he derived the probabilities of where the electron would hit the screen on the other side of the slit(s).

In his book, QED, he provides a graphic demonstration using this method to derive the path of a photon hitting a mirror. He says ‘the light goes where the time is least.’

In response to your specific question, the electron’s path is only determined retrospectively after it hits the screen on the other side of the slit(s). Freeman Dyson (who collaborated with Feynman) argues that QM cannot describe the past but only the future. So prior to the electron hitting the screen, QM describes the probabilities of where it will go, which is mathematically dependent on it being able to go everywhere at once. If there are 2 slits then this means it can go through both and if there is only one slit then it can only go through one. So the observation made retrospectively confirms this.

Addendum: Sabine Hossenfelder gives a much more erudite exposition in this video. And I agree with her - it's the closest we have to a 'theory of everything'.

Christianity and Buddhism

Last month’s issue of Philosophy Now (Issue 132, June/July 2019) had as its theme ‘West meets East’, so it was full of articles about Eastern philosophies and comparative philosophy. It led me to revisit this essay I wrote when I was a student some 20 years ago, studying philosophy, and specifically, when I took a unit called Religious Studies.

I should point out that I was brought up Christian, which I rejected in my mid to late teens, and in my 30s I took an interest in Buddhism and neo-Confucianism. So I had some background knowledge before I took the course. One can already see my existentialist leanings. 

Whilst I have previously written a post on Jesus, I haven’t written a post on Siddhārtha Gautama specifically, though I’ve made references to Buddhism in various posts.


‘Most religions envisage the spiritual path as a journey away from the false claims of the illusory self towards an understanding of the Real Self.’ Critically discuss this in relation to at least two of the three traditions studied in this course.

I will address this topic with respect to two of the religious systems under study: Christianity and Buddhism. The terms ‘illusory self’ and ‘Real’ or ‘True Self’ are open to wide interpretation within both systems, but if we perceive life as a journey, then what we are discussing is nothing less than the purpose of that journey as interpreted by both these religions.

This essay is not about the relative merits of Buddhism and Christianity, nevertheless it compares philosophical doctrines and points of view in relation to man’s mortal existence and his destiny. It also compares two views of a metaphysical universe which of course directly impact on how man perceives himself.

Buddhism and Christianity are both religions that evolved from earlier religions: Hinduism and Judaism respectively. Both arise from a distinct personality who remains central to the beliefs of their respective systems. Accordingly, I think there are two parts to these religions, and I intend to discuss both parts. Firstly, there is the part concerned with the personae: their lives as exemplars; and secondly their teachings and the philosophies that evolved therefrom.

Any great man, any personality who had an immense impact on a large body of people, eventually becomes mythologised, and it is the myth that continues and lives in people’s consciousness until it completely displaces the original persona. This is no different with Jesus and Siddhārtha Gautama, but as I will explain later there are more mythic qualities associated with the Christ than with the Buddha.

Historically, myth and religion have been synergistic. A myth, often but not always, includes factual elements, but it is not my intention to distil truth from fiction. For the purpose of this discussion, I’m taking another tact, where the mythic elements are not the focus.

When people refer to ‘The Buddha’, it is generally acknowledged that they are referring to Gautama Buddha or Siddhartha, even though he is not the first or only Buddha. Siddhartha was a prince born of the Ksatriya caste, a warrior and ruling caste, who became an ascetic when he juxtaposed his privileged style of living with the suffering of ordinary people. His impulse was not as simplistic as that however, because he was also aware that sickness, old age and death were burdens on human life that neither privilege nor wealth could avert.

As a result, he spent his entire life searching for the means, psychologically rather than physically, in releasing man’s spirit from this burden. At the age of 35 he achieved a state of enlightenment or awakening: an event which defines Buddhism in its essence. ‘The portrait of the Buddha...  is thus one of a man of both great wisdom and great compassion moved by the spectacle of human suffering and determined to free men from its fetters by a rational system of thought and a way of life.’ (ref. Encyclopaedia Brittanica)

Jesus’ story on the other hand, is told within the context of an enormous history: the history of the Jewish people. But it is more than that because it has mythic consequences relating to Divine judgement and the end of mortal history. But I would prefer, for the purposes of this discussion, to look at Jesus in a human context because I believe that is where his greatest message lies.

Jurgen Moltmann in Man gives a very good account of Jesus that reminds us of Jesus’ basic humanity and how he related to the lowest strata of society rather than those privileged by birth. Jesus provided a role, which to this day, very few people follow. I am not referring to the role of martyr, but to the role of facing the worst in human suffering and human weakness and human oppression, and revealing to such people his common humanity with them. There is a resonance here with Simone Weil’s Essay: On Human Personality; which reminds us that the intelligent person recoils from affliction in the same way ‘flesh recoils from death’.

I believe this is the greatest lesson Christ ever taught: that he was superior to all people, yet he gave his Grace to those least fortunate, regardless of creed, background or social position.

Buddha was not mythologised in the way that Christ was, neither was he a martyr, but in the final analysis these differences are of less significance than the hope they provide to all people through their example, their teachings and their lives. Both Christ and Buddha are not heroes in the traditional sense. They were antiheroes and pacifists, who were both renowned for their incomparable compassion to their fellow man. In this way, by their very lives, they both point to an identity and a destiny that ordinary people can emulate. This of course, is not how either of these religions are defined, but the lives of these men hold as much significance, perhaps even more, than their teachings.

On the other hand, to approach the destiny of the Self from a purely philosophical viewpoint, in either Christianity or Buddhism, one needs to go to the core of their respective beliefs. In Buddhism this is the concept of karma, and in Christianity it is a relationship with God through Christ. This also highlights the fundamental, and some would say irreconcilable differences between their philosophical and religious viewpoints.

Karma is generally understood as a causal connection between man’s actions and his destiny or fate. This causal relationship has metaphysical consequences, because it traverses lives. In other words, action in this life can affect destiny in the next life, which infers that some aspect of the Self is reborn. In Buddhist philosophy this leads to a contradiction because the Buddha explicitly preached a philosophy of no-self: that is no attachment, but also no soul.

Karma is a concept common to Hinduism, and is used as an explanation and rationalisation of the caste system, but Buddha considered the caste tradition inequitable.

More significantly, there is another way of perceiving karma that is best explained by John Hick in Death and Eternal Life, where he discusses the concept of a world karma. Hick explains with this concept that there is no need to consider an individual karma or rebirth, and so overcomes the contradiction. With or without the contradiction, the idea of a universal karma has a certain appeal and finds resonance in other concepts like Jung’s ‘collective unconscious’.

Masao Abe also makes reference to a similar, if not the same concept, when he cites Shin’ichi Hisamatsu’s notion of FAS: “Awakening to the Formless Self”. What Hick and Abe are both inferring is that there is a collective karma of the whole of mankind: past, present and future. What Abe describes as the ‘depth, breadth, and length of human existence’ According to Abe, Hisamatsu identifies this awakening as the same experience as satori (enlightenment in Zen Buddhist terminology).

To many Buddhists, satori, enlightenment or nirvana, is the whole purpose of man’s existence as an individual, and this is what is meant by finding the ‘True Self.’ Personally, I believe there are other perspectives to this question, without denying the significance of satori, and I will return to them later.

But another significant attribute of karma in Buddhist philosophy is that it deals with good and evil in human life without acknowledging a Deity or a Devil. I think this is fundamental in understanding the differences in Buddhist and Christian beliefs and also how they approach the question of the Self and its destiny.

To elaborate we need to examine the other obvious distinction between Christianity and Buddhism, which is that Christianity fundamentally requires a relationship with God. To a large extent, this philosophical nexus also determines the role of Christ.

It is Christ that makes Christianity unique in a way that Buddha doesn’t. As Fritz Buri says: ‘But in distinction to the Buddha, Jesus is not only teacher, but also an actor in the history of existence.’ It is Christ’s resurrection that places him mythically above man, though not immortal. It places him perfectly between God and man. In the Christian perspective, Christ is our connection with God, with Heaven and with a consciousness beyond death. This is the Christian response to both karma and nirvana.

Much of contemporary Christian belief revolves around the idea of being born again; of ‘finding Christ’. Many believers maintain that without this rebirth, which includes the acceptance of Christ as their saviour, there is no possibility of achieving the kingdom of Heaven. Yet according to Matthew this is not enough. In Matthew 7:21, Jesus says it is not enough to use his name: ‘It is not anyone who says to me: “Lord, Lord”, who will enter the Kingdom of heaven, but the person who does the will of my Father in Heaven’.

But in Christian doctrine, it is the metaphorical rebirth that signals a change in spiritual identity. To the orthodox Christian, this is the only path, the only destiny for the Self to consider.

Once again, I believe there are other perspectives to be considered, and it is another passage in Matthew which provides a clue. In Matthew 12:33-37, Jesus maintains that what comes from a man’s mouth (in words) comes from his heart - in this way one can tell evil from good. Specifically: Matthew 12:34; ‘You brood of vipers, how can your speech be good when you are evil? For words flow out of what fills the heart.’

‘What fills the heart’ is perhaps what the True Self is all about, and has resonances with Buddhism as well as other Eastern philosophies, but more importantly, is a key factor in Augustine’s neo-Platonic influenced philosophy: ‘...to reach the good, which is the real, one must “return into” oneself; for it is the spirit at the heart of man’s inmost self that links him to the ultimate reality.’ (ref. Encyclopaedia Brittanica)

In Christianity, the essential element of life’s journey is man’s relationship with God. This relationship is obviously deeply individualistic and despite the rituals and liturgies of the traditional churches, can really only be achieved within an individual’s consciousness. Again, in reference to Augustine: ‘Grace awakens the dormant power of the mind to see God’s image in itself, to see itself, that is, as God’s image.’ In other words, God is found only by looking inside ourselves, not by a leap of imagination into the unknown, conjuring images of a supreme being or a pantheistic spirit. That is not to say that Augustine didn’t recognise God as creator of the Universe, but man’s conscious accessibility to God is an inner journey, not an external relationship.

This, I believe, provides the best insight into the Christian perspective of understanding the Self and its destiny. The state of Grace that the Christian strives for, is to my interpretation, the same state as satori or nirvana, that is the Buddhist’s highest goal.

In Buddhist philosophy, as perceived from a Western perspective, the biggest conceptual hurdle is the belief in karma but not the soul. To overcome this paradox, Buddhist philosophers invoke the concept of no-self, but it tends to create more confusion than resolution.

If one simply dwells on the self or no-self paradox in Buddhism, then I believe one misses the point. The point of the journey of life is to acquire meaning and perhaps also an identity. In Christianity the notion of identity is very clear: it is achieved in a metaphorical rebirth (finding one’s identity in Christ). In Buddhism the purpose of the journey is to achieve satori or nirvana. But if the emphasis is changed from the destination to the journey itself, then it gives a different perspective. It is then concerned with the way we live our lives. It is the notion of karma that gives substance to Buddhist belief, not a concern with self or no-self. Buddha’s teachings on the no-self, I believe, reflect his concern with man’s preoccupation with the self and its unhealthy consequences. Whilst karma can be seen as a stick and carrot approach to religious teaching, this is a misplaced emphasis. If karma is seen instead as man’s connection to the rest of humanity, including past and future humanity, then one begins to grasp the point.

‘Interconnection between the individual and the whole universe is stressed in the Buddhist doctrine of karma.’ (ref. Encylopaedia Brittanica) From this conceptual viewpoint, the notion of individual karma and rebirth can be taken as a secondary consideration, and is neither denied nor affirmed.

But perhaps more relevantly, individual karma and therefore the Self, should not be considered as being independent of our universal or collective karma. That, at least, is my interpretation.

There is still another perspective of the Self, which is man’s purpose given by God. An idea that finds resonance in both Christian and Eastern beliefs. Tu Wei-Ming, a Confucian scholar, expresses it best: ‘...we are guardians of the good earth, the trustees of the mandate of Heaven....embedded in our human nature is the secret code for Heaven’s self-realisation.’

Humankind is above all else, the caretaker of the planet Earth. If one believes in a God, Christian or otherwise, as a creator who explicitly places man in charge of his creation, then the responsibility is huge indeed. Buddhist doctrine, on the other hand, ignores any explicit reference to this responsibility; nevertheless man’s karmic relationship, either individualistic or holistic, points him in the same direction - Earth’s fate has a causal dependency on man’s fate. From this point of view, one cannot ignore that the individual’s journey has a connection to humankind’s collective journey, with or without a heaven, with or without rebirth. From this perspective, the difference between the illusory self and the True Self is perhaps not one of identification but of awareness. An awareness not of Divine inheritance but of responsibility to our inheritance.

In the final analysis, I believe that religion or religious viewpoint is not so much a belief as an attitude. An attitude towards the Universe, towards one’s life and life in general, but above all, an attitude that reflects the Self at its deepest core rather than at a superficial level.

The spiritual journey is a euphemism for the search inside oneself to discover the true nature of the Self so that it may ‘light the world’  (Budda’s last words, purportedly). This is why the artist who has the most impact on us, is the one who digs deepest into his or her psyche. Augustine was right when he said the search for God was an inner journey. It is the inner journey which finds the True Self not the journey in the material world. Both Buddhists and Christians agree that the desire to create a position or an identity for ourselves in the world of business, commerce or social environment is the illusory self. The True Self, through which we engage our relationships to others and to the world at large, is, in the final analysis, the means by which we gain satisfaction from living.





References:

Abe M., Transformation in Buddhism, Buddhist-Christian Studies, Vol.7, 1987, pp.15-20.

Augustine, Encyclopaedia Britannica, Edition 15, 1989, Vol.14, pp.286-390.

Balthasar H.U. von, Engagement with God, trans. J. Halliburton, SPCK, London, 1975, Part 2, ch.4, pp.67-80.

Bultmann R. Existence and Faith, trans. Schubert M. Ogden, Hodder and Stoughton, London, 1961, pp. 248-66.

Buri F., Ingram P. & Streng F. (Eds), Buddhist-Christian Dialogue - Mutual Renewal and Transformation, University of Hawaii Press, Honolulu, 1986.

Burnaby J., ‘Introduction’, in J.Baillie, J.T.McNeill et al. (eds), The Library of Christian Classics, Vol.III, SCM Press, London, 1955, pp.23-31,31-6.

Ching J., Paradigms of the Self in Buddhism and Christianity, Buddhist-Christian Studies, Vol. 4, 1984, pp.31-50.

Cochrane C., Christianity and Classical Culture, Oxford University Press, London, 1944, pp.399-411.

Collins S., Selfless Persons: Imagery and Thought in Therevada Buddhism, Cambridge University Press, 1982, Part IV, pp.218-24.

Coward H., Psychology and Karma, Philosophy East and West, Vol.33, No 1, 1984, pp.54-58.

Fenner P. ‘Cognitive theories of the emotions in Buddhism and Western psychology’, Psychologia, vol.30, 1987, pp.217-27.

Fenner P., The self and its destiny in Buddhism, Religious Systems B,
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Hick J. Death and Eternal Life, Collins, London, 1976, ch.18, pp.347-60.

Hopkins J. & Rinbochay L., Death, Intermediate State and Rebirth in Tibetan Buddhism, Rider, London, 1979, Introduction, pp.13-21.

Howard W., Christianity according to St. John, Duckworth, London, 1943, ch. IV, pp.81-105.

Lichter D. & Epstein L., ‘Irony in Tibetan notions of the good life’, in C.F. Keyes & E.V. Daniel (eds), Karma. An Anthropological Enquiry, University of California Press, Berkeley, Calif., 1983, Part 2, sn 9, pp.233-38.

McLellan D., Utopian Pessimist: The Life and thought of Simone Weil, 
Poseidon Press,1990.

Moltmann J. Man, SPCK, London, 1974, pp. 16-21, 105-17.

The New Jerusalem Bible - New Testament, Darton, Longman & Todd, Reader’s Edition, 1991.

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God is not a mathematician; God is in the mathematics

I’ve just read a recently published book, The Universe Speaks in Numbers; How Modern Maths Reveals Nature’s Deepest Secrets by Graham Farmelo. The title says it all really, but the content takes you places you’ve probably never been, intellectually. This book, possibly more than any other book I’ve read, gives you access to the minds of the superstars in theoretical physics, both past and present, but mostly present. It’s like listening to a group of virtuoso musicians and becoming aware of how inadequate you are in the presence of such talent.

It’s written in an historical context, from Newton through Maxwell to Einstein, but rather than lingering in the so-called golden age of physics, it focuses on the last 50 years.

Farmelo covers what he calls ‘the long divorce’, a term coined by Freeman Dyson, covering much of the latter 20th Century, when pure mathematicians and theoretical physicists, not only avoided each other, but were not very accommodating of each other’s contributions.

That all changed very late in the 20th Century, and since then there has been a synergy that has benefited both disciplines. The problem is that theoretical physics has leapfrogged ahead of experimental physics, and, consequently, physicists at the leading edge of high energy physics have become more dependent on mathematical solutions to provide veracity to their ideas. ‘High energy’ means highly confined dimensions, so we are talking about the smallest constituents of matter we currently know or can imagine.

I’ve made the point in previous posts that the ultimate arbiter of truth is evidence, but Farmelo’s book has made me re-evaluate that position. Instead of testing theories with experiments that can’t be performed, theoretical physicists simply adhere to 2 founding principles: quantum mechanics and special relativity. Because those 2 specific theories have been experimentally tested in extremis, they simply ensure that any new theory, no matter at what level, or using whatever mathematical tools they have, must meet both criteria.

The book necessarily covers string theory, from its inception to its current status. But what I found most fascinating was all the discoveries that seemed to have occurred on the side so-to-speak, where mathematics appears to continually underlie the fundamentals of nature in unexpected ways.

I admit to being a sceptic of string theory, due to its 6 additional dimensions that can’t be observed and its plethora of 10⁵⁰⁰ universes. But I must also admit that the people exploring this mathematical world leave me stranded when it comes to intellectual wizardry.

Farmelo repeatedly refers to 2 talks given by Albert Einstein and Paul Dirac, respectively, where they effectively gave a call-to-arms, arguing that mathematics is the key to new theories in physics, with experimental physics providing confirmation rather than having the leading role. Whether by design or accident, this is how physics has evolved in the last 5 decades. As Farmelo expounds, not everyone has been happy with this development, yet there have been successes.

To give one example, mathematical devices called twistors (developed by Roger Penrose in the 1960s) have led to providing accurate predictions in the amplitudes of scattering gluons, which are the mediating particles for quarks in atomic nuclei. This short description belies the convoluted story, involving many theorists in the UK and the US, and the many unexpected discoveries made along the way; including a connection with a mathematical object discovered by Hermann Grassmann in 1844, called eponymously a Grassmannian. It led to another mathematical object called an 'amplituhedron'. One of the co-discoverers, Nima Arkani-Hamed (an American born Iranian, at the Princeton Institute for Advanced Study) said:

This is a concrete example of a way in which the physics we normally associate with space-time and quantum mechanics arises from something more basic.

The ‘something more basic’ is only known mathematically, as opposed to physically. I found this a most compelling tale and a history lesson in how mathematics appears to be intrinsically linked to the minutia of atomic physics.

In the same context, Arkani-Hamed says that ‘the mathematics of whole numbers in scattering-amplitude theory chimes… with the ancient Greeks' dream: to connect all nature with whole numbers.’

There is an assumption by non-physicists that the role of mathematics in understanding nature is a consequence of the fact that we need to measure everything. A common criticism is that people who emphasise the role of mathematics in their theories have ‘mistaken the map for the terrain’.

Einstein was probably the first to use mathematics alone to sculpture a theory independently of observation and experimentation, when he developed his masterpiece, the general theory of relativity. It was his mathematical prediction that gravity would bend light that clinched his theory when few people believed that relativity reflected reality.

In reference to the abovementioned metaphor about the ‘map and terrain’, there is an axiomatic inference that the map is derived from ‘surveying’ the terrain. However, it’s becoming increasingly apparent that the map is discovered before the terrain is even 'explored', which turns the metaphor on its head. It’s not a metaphor I would choose to use, but if you insist, you might have to consider the possibility that the map pre-exists the terrain.

In reference to the title, I’ll retell a joke by mathematical physicist, Robbert Dijkgraaf, from the Princeton Institute for Advanced Study:

What is the difference between a physicist and a mathematician?
A physicist studies the laws that God chose for nature to obey.
A mathematician studies the laws that God has to obey.