Self-fulfilling prophecy

From Wikipedia, the free encyclopedia

Self-fulfilling prophecy refers to the socio-psychological phenomenon of someone "predicting" or expecting something, and this “prediction” or expectation comes true simply because one believes it will, and their resulting behaviors align to fulfil those beliefs. This suggests peoples' beliefs influence their actions. The principle behind this phenomenon is people create consequences regarding people or events, based on their previous knowledge toward that specific subject. Additionally, self-fulfilling prophecy is applicable to negative and positive outcomes.

American sociologist William Isaac Thomas was first to discover this phenomenon. In 1928 he developed the Thomas theorem (also known as the Thomas dictum), stating,

“

If men define situations as real, they are real in their consequences.

”

In other words, the consequence will come to fruition based on how one interprets the situation. Using Thomas' idea, another American sociologist, Robert K. Merton, coined the term self-fulfilling prophecy, popularizing the idea “...a belief or expectation, correct or incorrect, could bring about a desired or expected outcome.”

History

Merton applied this concept to a fictional situation. In his book Social Theory and Social Structure, he uses the example of a “bank run” to show how self-fulfilling thoughts can make unwanted situations happen. He mentions how a number of people falsely believe the bank was going to file for bankruptcy. Because of this false fear, many people decide to go to the bank and ask for all of their cash at once. These actions cause the bank to indeed go bankrupt because banks rarely have the amount of cash able to satisfy a multiple number of customers asking for all of their existing cash at once. 

Merton concludes this example with the analysis, “The prophecy of collapse led to its own fulfillment.”

While Merton’s example focused on self-fulfilling prophecies within a business, his theory is also applicable to interpersonal communication since it’s found to have a “potential for triggering self-fulfilling prophecy effects.” This is due to the fact “that an individual decides whether or not to conform to the expectations of others.” This makes people rely or fall into self-fulfilling thoughts since they are trying to satisfy other’s perception of them. 

Self-fulfilling theory can be divided into two subsections, one would be the Pygmalion effect which is when “one person has expectations of another, changes her behavior in accordance with these expectations, and the object of the expectations then also changes her behavior as a result.”

Additionally, Philosopher Karl Popper called the self-fulfilling prophecy the Oedipus effect:

One of the ideas I had discussed in The Poverty of Historicism was the influence of a prediction upon the event predicted. I had called this the "Oedipus effect", because the oracle played a most important role in the sequence of events which led to the fulfilment of its prophecy. … For a time I thought that the existence of the Oedipus effect distinguished the social from the natural sciences. But in biology, too—even in molecular biology—expectations often play a role in bringing about what has been expected.

An early precursor of the concept appears in Edward Gibbon's Decline and Fall of the Roman Empire: "During many ages, the prediction, as it is usual, contributed to its own accomplishment" (chapter I, part II).

Applications

Examples abound in studies of cognitive dissonance theory and the related self-perception theory; people will often change their attitudes to come into line with what they profess publicly.

Teacher expectations influence student academic performance. In the United States, the concept was broadly and consistently applied in the field of public education reform, following the "War on Poverty". Theodore Brameld noted: "In simplest terms, education already projects and thereby reinforces whatever habits of personal and cultural life are considered to be acceptable and dominant." The effects of teacher attitudes, beliefs and values, affecting their expectations have been tested repeatedly. Students may study more if they had a positive experience with their teacher. Or female students may perform worse if they expected their male instructor is a sexist.

The phenomenon of the "inevitability of war" is a self-fulfilling prophecy that has received considerable study.

The idea is similar to that discussed by the philosopher William James as "The Will to Believe." But James viewed it positively, as the self-validation of a belief. Just as, in Merton's example, the belief that a bank is insolvent may help create the fact, so too, on the positive side, confidence in the bank's prospects may help brighten them. Similarly, Stock-exchange panic episodes, and speculative bubble episodes, can be triggered with the belief that the stock will go down (or up), thus starting the selling/buying mass move, etc.

A more Jamesian example: a swain, convinced that the fair maiden must love him, may prove more effective in his wooing than he would had his initial prophecy been defeatist.

There is extensive evidence of "Interpersonal Expectation Effects", where the seemingly private expectations of individuals can predict the outcome of the world around them. The mechanisms by which this occurs are also reasonably well understood: it is simply that our own expectations change our behaviour in ways we may not notice and correct. In the case of the "Interpersonal Expectation Effects", others pick up on non-verbal behaviour, which affects their attitudes. A famous example includes a study where teachers were told arbitrarily that random students were "going to blossom". Oddly, those random students actually ended the year with significantly greater improvements. Rosenthal and Jacobson 1968 Other specific examples discussed in psychology include:

• 'Clever Hans' effect
• Observer-expectancy effect
• Hawthorne effect
• Placebo effect
• Pygmalion effect
• Stereotype threat

Sports

In Canadian ice hockey, junior league players are selected based on skill, motor coordination, physical maturity, and other individual merit criteria. However, psychologist Robert Barnsley showed that in any elite group of hockey players, 40% are born between January and March, versus the approximately 25% as would be predicted by statistics. The explanation is that in Canada, the eligibility cutoff for age-class hockey is January 1, and the players who are born in the first months of the year are older by 0–11 months, which at the preadolescent age of selection (nine or ten) manifests into an important physical advantage. The selected players are exposed to higher levels of coaching, play more games, and have better teammates. These factors make them actually become the best players, fulfilling the prophecy, while the real selection criterion was age. The same relative age effect has been noticed in Belgian soccer after 1997, when the start of the selection year was changed from August 1 to January 1.

Stereotype

Self-fulfilling prophecies are one of the main contributions to racial prejudice and vice versa. According to the Dictionary of Race, Ethnicity & Culture “Self-fulfilling prophecy makes it possible to highlight the tragic vicious circle which victimizes people twice: first, because the victim is stigmatized (STIGMA) with an inherent negative quality; and secondly, because he or she is prevented from disproving this quality.” To prove this, the author uses the example that Merton used in his book about how white workers expected that black people would be against the principles of trade unionism because white workers considered black workers to be “undisciplined in traditions of trade unionism and the art of collective bargain-ing.” This predictions caused the event to happen (black workers would be against trade unionism), because this forecasts became fact when all white people started to believe this and did not let the black workers get a job at any white men business. Which made black workers unable to learn or approve the principles of trade unionism since they were not given the chance of working in a work environment where these principles where seen or experienced.

  

In the article “The Accumulation of Stereotype-based self-fulfilling Prophecies.” The authors mention how teachers can encourage stereotype-based courses and can interact with students in a manner that encourages self-fulfilling thoughts. The example that was given was the one of a female student who seemed to do bad in math and her math teacher and counselor “channel her in the direction of confirming sex stereotypes” By this the author means that the teachers never encouraged her to improve her abilities in math. Instead, the teacher and the counselor recommended classes that were dominated by females.

Literature, media, and the arts

In literature, self-fulfilling prophecies are often used as plot devices. They have been used in stories for millennia, but have gained a lot of popularity recently in the science fiction genre. They are typically used ironically, with the prophesied events coming to pass due to the actions of one trying to prevent the prophecy (a recent example would be the life of Anakin Skywalker, the fictional Jedi-turned-Sith Lord in George Lucas' Star Wars saga). They are also sometimes used as comic relief.

Classical

Many myths, legends and fairy tales make use of this motif as a central element of narratives that are designed to illustrate inexorable fate, fundamental to the Hellenic world-view. In a common motif, a child, whether newborn or not yet conceived, is prophesied to cause something that those in power do not want to happen. This may be the death of the powerful person; in more light-hearted versions, it is often the marriage of a poor or lower-class child to his own. The events come about, nevertheless, as a result of the actions taken to prevent them: frequently child abandonment sets the chain of events in motion.

Greek

Oedipus in the arms of Phorbas.

 

The best known example from Greek legend is that of Oedipus. Warned that his child would one day kill him, Laius abandoned his newborn son Oedipus to die, but Oedipus was found and raised by others, and thus in ignorance of his true origins. When he grew up, Oedipus was warned that he would kill his father and marry his mother. Believing his foster parents were his real parents, he left his home and travelled to Greece, eventually reaching the city where his biological parents lived. There, he got into a fight with a stranger, his real father, killed him and married his widow, Oedipus' real mother. 

Although the legend of Perseus opens with the prophecy that he will kill his grandfather Acrisius, and his abandonment with his mother Danaë, the prophecy is only self-fulfilling in some variants. In some, he accidentally spears his grandfather at a competition—an act that could have happened regardless of Acrisius' response to the prophecy. In other variants, his presence at the games is explained by his hearing of the prophecy, so that his attempt to evade it does cause the prophecy to be fulfilled. In still others, Acrisius is one of the wedding guests when Polydectes tried to force Danaë to marry him, and when Perseus turns them to stone with the Gorgon's head; as Polydectes fell in love with Danaë because Acrisius abandoned her at sea, and Perseus killed the Gorgon as a consequence of Polydectes' attempt to get rid of Danaë's son so that he could marry her, the prophecy fulfilled itself in these variants. 

Greek historiography provides a famous variant: when the Lydian king Croesus asked the Delphic Oracle if he should invade Persia, the response came that if he did, he would destroy a great kingdom. Assuming this meant he would succeed, he attacked—but the kingdom he destroyed was his own. In such an example, the prophecy prompts someone to action because he is led to expect a favorable result; but he achieves another, disastrous result which nonetheless fulfills the prophecy.

When it was predicted that Cronos would be overthrown by his son, and usurp his throne as King of the Gods, Cronus ate his children, each shortly after they were born. When Zeus was born, Cronos was thwarted by Rhea, who gave him a stone to eat instead, sending Zeus to be raised by Amalthea. Cronos' attempt to avoid the prophecy made Zeus his enemy, ultimately leading to its fulfilment.

Roman

Romulus and Remus feeding from a wolf.

 

The story of Romulus and Remus is another example. According to legend, a man overthrew his brother, the king. He then ordered that his two nephews, Romulus and Remus, be drowned, fearing that they would someday kill him like he did to his brother. The boys were placed in a basket and thrown in the Tiber River. A wolf found the babies and she raised them. Later, a shepherd found the twins and named them Romulus and Remus. As teenagers, they found out who they were. They killed their uncle, fulfilling the prophecy.

Arabic

A variation of the self-fulfilling prophecy is the self-fulfilling dream, which dates back to medieval Arabic literature. Several tales in the One Thousand and One Nights, also known as the Arabian Nights, use this device to foreshadow what is going to happen, as a special form of literary prolepsis. A notable example is "The Ruined Man Who Became Rich Again Through a Dream", in which a man is told in his dream to leave his native city of Baghdad and travel to Cairo, where he will discover the whereabouts of some hidden treasure. The man travels there and experiences misfortune after losing belief in the prophecy, ending up in jail, where he tells his dream to a police officer. The officer mocks the idea of foreboding dreams and tells the protagonist that he himself had a dream about a house with a courtyard and fountain in Baghdad where treasure is buried under the fountain. The man recognizes the place as his own house and, after he is released from jail, he returns home and digs up the treasure. In other words, the foreboding dream not only predicted the future, but the dream was the cause of its prediction coming true. A variant of this story later appears in English folklore as the "Pedlar of Swaffham".

Another variation of the self-fulfilling prophecy can be seen in "The Tale of Attaf", where Harun al-Rashid consults his library (the House of Wisdom), reads a random book, "falls to laughing and weeping and dismisses the faithful vizier" Ja'far ibn Yahya from sight. Ja'far, "disturbed and upset flees Baghdad and plunges into a series of adventures in Damascus, involving Attaf and the woman whom Attaf eventually marries." After returning to Baghdad, Ja'far reads the same book that caused Harun to laugh and weep, and discovers that it describes his own adventures with Attaf. In other words, it was Harun's reading of the book that provoked the adventures described in the book to take place. This is an early example of reverse causality. In the 12th century, this tale was translated into Latin by Petrus Alphonsi and included in his Disciplina Clericalis. In the 14th century, a version of this tale also appears in the Gesta Romanorum and Giovanni Boccaccio's The Decameron.

Hinduism

Krishna playing his flute with Radha.

 

Self-fulfilling prophecies appear in classical Sanskrit literature. In the story of Krishna in the Indian epic Mahabharata, the ruler of the Mathura kingdom, Kansa, afraid of a prophecy that predicted his death at the hands of his sister Devaki's son, had her cast into prison where he planned to kill all of her children at birth. After killing the first six children, and Devaki's apparent miscarriage of the seventh, Krishna (the eighth son) was born. As his life was in danger he was smuggled out to be raised by his foster parents Yashoda and Nanda in the village of Gokula. Years later, Kansa learned about the child's escape and kept sending various demons to put an end to him. The demons were defeated at the hands of Krishna and his brother Balarama. Krishna, as a young man returned to Mathura to overthrow his uncle, and Kansa was eventually killed by his nephew Krishna. It was due to Kansa's attempts to prevent the prophecy that led to it coming true, thus fulfilling the prophecy.

Russian

Oleg of Novgorod was a Varangian prince who ruled over the Rus people during the early tenth century. As old East Slavic chronicles say, it was prophesied by the pagan priests that Oleg's stallion would be the source of Oleg's death. To avoid this he sent the horse away. Many years later he asked where his horse was, and was told it had died. He asked to see the remains and was taken to the place where the bones lay. When he touched the horse's skull with his boot a snake slithered from the skull and bit him. Oleg died, thus fulfilling the prophecy. In the Primary Chronicle, Oleg is known as the Prophet, ironically referring to the circumstances of his death. The story was romanticized by Alexander Pushkin in his celebrated ballad "The Song of the Wise Oleg". In Scandinavian traditions, this legend lived on in the saga of Orvar-Odd.

European fairy tales

Many fairy tales, such as The Devil With the Three Golden Hairs, The Fish and the Ring, The Story of Three Wonderful Beggars, or The King Who Would Be Stronger Than Fate, revolve about a prophecy that a poor boy will marry a rich girl (or, less frequently, a poor girl a rich boy). This is story type 930 in the Aarne–Thompson classification scheme. The girl's father's efforts to prevent it are the reason why the boy ends up marrying her. 

Another fairy tale occurs with older children. In The Language of the Birds, a father forces his son to tell him what the birds say: that the father would be the son's servant. In The Ram, the father forces his daughter to tell him her dream: that her father would hold an ewer for her to wash her hands in. In all such tales, the father takes the child's response as evidence of ill-will and drives the child off; this allows the child to change so that the father will not recognize his own offspring later and so offer to act as the child's servant. 

In some variants of Sleeping Beauty, such as Sun, Moon, and Talia, the sleep is not brought about by a curse, but a prophecy that she will be endangered by flax (or hemp) results in the royal order to remove all the flax or hemp from the castle, resulting in her ignorance of the danger and her curiosity.

Shakespeare

Shakespeare's Macbeth is another classic example of a self-fulfilling prophecy. The three witches give Macbeth a prophecy that Macbeth will eventually become king, but afterwards, the offspring of his best friend will rule instead of his own. Macbeth tries to make the first half true while trying to keep his bloodline on the throne instead of his friend's. Spurred by the prophecy, he kills the king and his friend, something he, arguably, never would have done before. In the end, the evil actions he committed to avoid his succession by another's bloodline get him killed in a revolution.

The later prophecy by the first apparition of the witches that Macbeth should "Beware Macduff" is also a self-fulfilling prophecy. If Macbeth had not been told this, then he might not have regarded Macduff as a threat. Therefore, he would not have killed Macduff's family, and Macduff would not have sought revenge and killed Macbeth.

Modern

Similar to Oedipus above, a more modern example would be Darth Vader in the Star Wars films, or Lord Voldemort in the Harry Potter franchise and the Big Three in Percy Jackson & the Olympians - each attempted to take steps to prevent action against them which had been predicted could cause their downfall, but instead created the conditions leading to it. Another, less well-known, modern example occurred with the character John Mitchell on BBC Three's Being Human. The Disney television series That's So Raven stars Raven-Symoné as the title character with the ability to see into the future with a strange situation. The extreme steps that the character takes to prevent the situation are almost always what lead to it. In George R. R. Martin’s book series a Song of Ice and Fire, Cersei Lannister kills a friend of hers after hearing a prophecy, from Maggy the Frog, that said friend will soon die.

New Thought

The law of attraction is a typical example of self-fulfilling prophecy. It is the name given to the belief that "like attracts like" and that by focusing on positive or negative thoughts, one can bring about positive or negative results. According to this law, all things are created first by imagination, which leads to thoughts, then to words and actions. The thoughts, words and actions held in mind affect someone's intentions which makes the expected result happen. Although there are some cases where positive or negative attitudes can produce corresponding results (principally the placebo and nocebo effects), there is no scientific basis to the law of attraction.

Causal loop

A self-fulfilling prophecy may be a form of causality loop, only when the prophecy can be said to be truly known to occur, since only then events in the future will be causing effects in the past. Otherwise, it would be a simple case of events in the past causing events in the future. Predestination does not necessarily involve a supernatural power, and could be the result of other "infallible foreknowledge" mechanisms. Problems arising from infallibility and influencing the future are explored in Newcomb's paradox. A notable fictional example of a self-fulfilling prophecy occurs in classical play Oedipus Rex, in which Oedipus becomes the king of Thebes, whilst in the process unwittingly fulfills a prophecy that he would kill his father and marry his mother. The prophecy itself serves as the impetus for his actions, and thus it is self-fulfilling. The movie 12 Monkeys heavily deals with themes of predestination and the Cassandra complex, where the protagonist who travels back in time explains that he cannot change the past.

Canonical quantum gravity

From Wikipedia, the free encyclopedia

 

In physics, canonical quantum gravity is an attempt to quantize the canonical formulation of general relativity (or canonical gravity). It is a Hamiltonian formulation of Einstein's general theory of relativity. The basic theory was outlined by Bryce DeWitt in a seminal 1967 paper, and based on earlier work by Peter G. Bergmann using the so-called canonical quantization techniques for constrained Hamiltonian systems invented by Paul Dirac. Dirac's approach allows the quantization of systems that include gauge symmetries using Hamiltonian techniques in a fixed gauge choice. Newer approaches based in part on the work of DeWitt and Dirac include the Hartle–Hawking state, Regge calculus, the Wheeler–DeWitt equation and loop quantum gravity.

Canonical quantization

In the Hamiltonian formulation of ordinary classical mechanics the Poisson bracket is an important concept. A "canonical coordinate system" consists of canonical position and momentum variables that satisfy canonical Poisson-bracket relations, 

${\displaystyle \{q_{i},p_{j}\}=\delta _{ij}}$

where the Poisson bracket is given by

${\displaystyle \{f,g\}=\sum _{i=1}^{N}\left({\frac {\partial f}{\partial q_{i}}}{\frac {\partial g}{\partial p_{i}}}-{\frac {\partial f}{\partial p_{i}}}{\frac {\partial g}{\partial q_{i}}}\right).}$

for arbitrary phase space functions ${\displaystyle f(q_{i},p_{j})}$ and ${\displaystyle g(q_{i},p_{j})}$. With the use of Poisson brackets, the Hamilton's equations can be rewritten as, 

${\displaystyle {\dot {q}}_{i}=\{q_{i},H\}}$,

${\displaystyle {\dot {p}}_{i}=\{p_{i},H\}}$.

These equations describe a ``flow" or orbit in phase space generated by the Hamiltonian ${\displaystyle H}$. Given any phase space function ${\displaystyle F(q,p)}$, we have 

${\displaystyle {d \over dt}F(q_{i},p_{i})=\{F,H\}.}$

In canonical quantization the phase space variables are promoted to quantum operators on a Hilbert space and the Poisson bracket between phase space variables is replaced by the canonical commutation relation: 

${\displaystyle [{\hat {q}},{\hat {p}}]=i\hbar .}$

In the so-called position representation this commutation relation is realized by the choice:  

${\displaystyle {\hat {q}}\psi (q)=q\psi (q)}$ and ${\displaystyle {\hat {p}}\psi (q)=-i\hbar {d \over dq}\psi (q)}$

The dynamics are described by Schrödinger equation: 

${\displaystyle i\hbar {\partial \over \partial t}\psi ={\hat {H}}\psi }$

where ${\displaystyle {\hat {H}}}$ is the operator formed from the Hamiltonian ${\displaystyle H(q,p)}$ with the replacement ${\displaystyle q\mapsto q}$ and 

${\displaystyle p\mapsto -i\hbar {d \over dq}}$.

Canonical quantization with constraints

Canonical classical general relativity is an example of a fully constrained theory. In constrained theories there are different kinds of phase space: the unrestricted (also called kinematic) phase space on which constraint functions are defined and the reduced phase space on which the constraints have already been solved. For canonical quantization in general terms, phase space is replaced by an appropriate Hilbert space and phase space variables are to be promoted to quantum operators. 

In Dirac's approach to quantization the unrestricted phase space is replaced by the so-called kinematic Hilbert space and the constraint functions replaced by constraint operators implemented on the kinematic Hilbert space; solutions are then searched for. These quantum constraint equations are the central equations of canonical quantum general relativity, at least in the Dirac approach which is the approach usually taken. 

In theories with constraints there is also the reduced phase space quantization where the constraints are solved at the classical level and the phase space variables of the reduced phase space are then promoted to quantum operators, however this approach was thought to be impossible in General relativity as it seemed to be equivalent to finding a general solution to the classical field equations. However, with the fairly recent development of a systematic approximation scheme for calculating observables of General relativity (for the first time) by Bianca Dittrich, based on ideas introduced by Carlo Rovelli, a viable scheme for a reduced phase space quantization of Gravity has been developed by Thomas Thiemann. However it is not fully equivalent to the Dirac quantization as the `clock-variables' must be taken to be classical in the reduced phase space quantization, as opposed to the case in the Dirac quantization. 

A common misunderstanding is that coordinate transformations are the gauge symmetries of general relativity, when actually the true gauge symmetries are diffeomorphisms as defined by a mathematician – which are much more radical. The first class constraints of general relativity are the spatial diffeomorphism constraint and the Hamiltonian constraint (also known as the Wheeler-De Witt equation) and imprint the spatial and temporal diffeomorphism invariance of the theory respectively. Imposing these constraints classically are basically admissibility conditions on the initial data, also they generate the `evolution' equations (really gauge transformations) via the Poisson bracket. Importantly the Poisson bracket algebra between the constraints fully determines the classical theory – this is something that must in some way be reproduced in the semi-classical limit of canonical quantum gravity for it to be a viable theory of quantum gravity. 

In Dirac's approach it turns out that the first class quantum constraints imposed on a wavefunction also generate gauge transformations. Thus the two step process in the classical theory of solving the constraints ${\displaystyle C_{I}=0}$ (equivalent to solving the admissibility conditions for the initial data) and looking for the gauge orbits (solving the `evolution' equations) is replaced by a one step process in the quantum theory, namely looking for solutions ${\displaystyle \Psi }$ of the quantum equations ${\displaystyle {\hat {C}}_{I}\Psi =0}$. This is because it obviously solves the constraint at the quantum level and it simultaneously looks for states that are gauge invariant because ${\displaystyle {\hat {C}}_{I}}$ is the quantum generator of gauge transformations. At the classical level, solving the admissibility conditions and evolution equations are equivalent to solving all of Einstein's field equations, this underlines the central role of the quantum constraint equations in Dirac's approach to canonical quantum gravity.

Canonical quantization, Diffeomorphism invariance and Manifest Finiteness

A diffeomorphism can be thought of as simultaneously `dragging' the metric (gravitational field) and matter fields over the bare manifold while staying in the same coordinate system, and so are more radical than invariance under a mere coordinate transformation. This symmetry arises from the subtle requirement that the laws of general relativity cannot depend on any a-priori given space-time geometry. 

This diffeomorphism invariance has an important implication: canonical quantum gravity will be manifestly finite as the ability to `drag' the metric function over the bare manifold means that small and large `distances' between abstractly defined coordinate points are gauge-equivalent! A more rigorous argument has been provided by Lee Smolin: 

“A background independent operator must always be finite. This is because the regulator scale and the background metric are always introduced together in the regularization procedure. This is necessary, because the scale that the regularization parameter refers to must be described in terms of a background metric or coordinate chart introduced in the construction of the regulated operator. Because of this the dependence of the regulated operator on the cuttoff, or regulator parameter, is related to its dependence on the background metric. When one takes the limit of the regulator parameter going to zero one isolates the non-vanishing terms. If these have any dependence on the regulator parameter (which would be the case if the term is blowing up) then it must also have dependence on the background metric. Conversely, if the terms that are nonvanishing in the limit the regulator is removed have no dependence on the background metric, it must be finite.” 

In fact, as mentioned below, Thomas Thiemann has explicitly demonstrated that loop quantum gravity (a well developed version of canonical quantum gravity) is manifestly finite even in the presence of all forms of matter! So there is no need for renormalization and the elimination of infinities. 

In perturbative quantum gravity (from which the non-renormalization arguments originate), as with any perturbative scheme, one makes the assumption that the unperturbed starting point is qualitatively the same as the true quantum state – so perturbative quantum gravity makes the physically unwarranted assumption that the true structure of quantum space-time can be approximated by a smooth classical (usually Minkowski) spacetime. Canonical quantum gravity on the other hand makes no such assumption and instead allows the theory itself tell you, in principle, what the true structure of quantum space-time is. A long-held expectation is that in a theory of quantum geometry such as canonical quantum gravity that geometric quantities such as area and volume become quantum observables and take non-zero discrete values, providing a natural regulator which eliminates infinities from the theory including those coming from matter contributions. This `quantization' of geometric observables is in fact realized in loop quantum gravity (LQG).

Canonical quantization in metric variables

The quantization is based on decomposing the metric tensor as follows,

${\displaystyle g_{\mu \nu }dx^{\mu }\,dx^{\nu }=(-\,N^{2}+\beta _{k}\beta ^{k})dt^{2}+2\beta _{k}\,dx^{k}\,dt+\gamma _{ij}\,dx^{i}\,dx^{j}}$

where the summation over repeated indices is implied, the index 0 denotes time ${\displaystyle \tau =x^{0}}$, Greek indices run over all values 0, . . ., ,3 and Latin indices run over spatial values 1, . . ., 3. The function ${\displaystyle N}$ is called the lapse function and the functions ${\displaystyle \beta _{k}}$ are called the shift functions. The spatial indices are raised and lowered using the spatial metric ${\displaystyle \gamma _{ij}}$ and its inverse ${\displaystyle \gamma ^{ij}}$: ${\displaystyle \gamma _{ij}\gamma ^{jk}=\delta _{i}{}^{k}}$ and ${\displaystyle \beta ^{i}=\gamma ^{ij}\beta _{j}}$, ${\displaystyle \gamma =\det \gamma _{ij}}$, where ${\displaystyle \delta }$ is the Kronecker delta. Under this decomposition the Einstein–Hilbert Lagrangian becomes, up to total derivatives,

${\displaystyle L=\int d^{3}x\,N\gamma ^{1/2}(K_{ij}K^{ij}-K^{2}+{}^{(3)}R)}$

where ${\displaystyle {}^{(3)}R}$ is the spatial scalar curvature computed with respect to the Riemannian metric ${\displaystyle \gamma _{ij}}$ and ${\displaystyle K_{ij}}$ is the extrinsic curvature,

${\displaystyle K_{ij}=-{\frac {1}{2}}({\mathcal {L}}_{n}\gamma )_{ij}={\frac {1}{2}}N^{-1}\left(\nabla _{j}\beta _{i}+\nabla _{i}\beta _{j}-{\frac {\partial \gamma _{ij}}{\partial t}}\right),}$

where ${\displaystyle {\mathcal {L}}}$ denotes Lie-differentiation, ${\displaystyle n}$ is the unit normal to surfaces of constant ${\displaystyle t}$ and ${\displaystyle \nabla _{i}}$ denotes covariant differentiation with respect to the metric ${\displaystyle \gamma _{ij}}$. Note that ${\displaystyle \gamma _{\mu \nu }=g_{\mu \nu }+n_{\mu }n_{\nu }}$. DeWitt writes that the Lagrangian "has the classic form 'kinetic energy minus potential energy,' with the extrinsic curvature playing the role of kinetic energy and the negative of the intrinsic curvature that of potential energy." While this form of the Lagrangian is manifestly invariant under redefinition of the spatial coordinates, it makes general covariance opaque. 

Since the lapse function and shift functions may be eliminated by a gauge transformation, they do not represent physical degrees of freedom. This is indicated in moving to the Hamiltonian formalism by the fact that their conjugate momenta, respectively ${\displaystyle \pi }$ and ${\displaystyle \pi ^{i}}$, vanish identically (on shell and off shell). These are called primary constraints by Dirac. A popular choice of gauge, called synchronous gauge, is ${\displaystyle N=1}$ and ${\displaystyle \beta _{i}=0}$, although they can, in principle, be chosen to be any function of the coordinates. In this case, the Hamiltonian takes the form

${\displaystyle H=\int d^{3}x{\mathcal {H}},}$

where

${\displaystyle {\mathcal {H}}={\frac {1}{2}}\gamma ^{-1/2}(\gamma _{ik}\gamma _{jl}+\gamma _{il}\gamma _{jk}-\gamma _{ij}\gamma _{kl})\pi ^{ij}\pi ^{kl}-\gamma ^{1/2}{}^{(3)}R}$

and ${\displaystyle \pi ^{ij}}$ is the momentum conjugate to ${\displaystyle \gamma _{ij}}$. Einstein's equations may be recovered by taking Poisson brackets with the Hamiltonian. Additional on-shell constraints, called secondary constraints by Dirac, arise from the consistency of the Poisson bracket algebra. These are ${\displaystyle {\mathcal {H}}=0}$ and ${\displaystyle \nabla _{j}\pi ^{ij}=0}$. This is the theory which is being quantized in approaches to canonical quantum gravity. 

It can be shown that six Einstein equations describing time evolution (really a gauge transformation) can be obtained by calculating the Poisson brackets of the three-metric and its conjugate momentum with a linear combination of the spatial diffeomorphism and Hamiltonian constraint. The vanishing of the constraints, giving the physical phase space, are the four other Einstein equations. That is, we have: 

Spatial diffeomorphisms constraints 

${\displaystyle C_{a}(x)=0}$

of which there are an infinite number – one for value of ${\displaystyle x}$, can be smeared by the so-called shift functions ${\displaystyle {\vec {N}}(x)}$ to give an equivalent set of smeared spatial diffeomorphism constraints, 

${\displaystyle C({\vec {N}})=\int d^{3}xC_{a}(x)N^{a}(x)}$.

These generate spatial diffeomorphisms along orbits defined by the shift function ${\displaystyle N^{a}(x)}$.

Hamiltonian constraints 

${\displaystyle H(x)=0}$

of which there are an infinite number, can be smeared by the so-called lapse functions ${\displaystyle N(x)}$ to give an equivalent set of smeared Hamiltonian constraints, 

${\displaystyle H(N)=\int d^{3}xH(x)N(x)}$.

as mentioned above, the Poission bracket structure between the (smeared) constraints is important because they fully determine the classical theory, and must be reproduced in the semi-classical limit of any theory of quantum gravity.

The Wheeler-De-Witt equation

The Wheeler-De-Witt equation (sometimes called the Hamiltonian constraint, sometimes the Einstein-Schrödinger equation) is rather central as it encodes the dynamics at the quantum level. It is analogous to Schrödinger's equation, except as the time coordinate, ${\displaystyle t}$, is unphysical, a physical wavefunction can't depend on ${\displaystyle t}$ and hence `Schrödinger's equation' reduces to a constraint: 

${\displaystyle {\hat {H}}\Psi =0.}$

Using metric variables lead to seemingly un-summountable mathematical difficulties when trying to promote the classical expression to a well-defined quantum operator, and as such decades went by without making progress via this approach. This problem was circumvented and the formulation of a well-defined Wheeler-De-Witt equation was first accomplished with the introduction of Ashtekar-Barbero variables and the loop representation, this well defined operator formulated by Thomas Thiemann. 

Before this development the Wheeler-De-Witt equation had only been formulated in symmetry-reduced models, such as quantum cosmology.

Canonical quantization in Ashtekar-Barbero variables and LQG

Many of the technical problems in canonical quantum gravity revolve around the constraints. Canonical general relativity was originally formulated in terms of metric variables, but there seemed to be insurmountable mathematical difficulties in promoting the constraints to quantum operators because of their highly non-linear dependence on the canonical variables. The equations were much simplified with the introduction of Ashtekars new variables. Ashtekar variables describe canonical general relativity in terms of a new pair canonical variables closer to that of gauge theories. In doing so it introduced an additional constraint, on top of the spatial diffeomorphism and Hamiltonian constraint, the Gauss gauge constraint.

The loop representation is a quantum hamiltonian representation of gauge theories in terms of loops. The aim of the loop representation, in the context of Yang-Mills theories is to avoid the redundancy introduced by Gauss gauge symmetries allowing to work directly in the space of Gauss gauge invariant states. The use of this representation arose naturally from the Ashtekar-Barbero representation as it provides an exact non-perturbative description and also because the spatial diffeomorphism constraint is easily dealt with within this representation. 

Within the loop representation Thiemann has provided a well defined canonical theory in the presence of all forms of matter and explicitly demonstrated it to be manifestly finite! So there is no need for renormalization. However, as LQG approach is well suited to describe physics at the Planck scale, there are difficulties in making contact with familiar low energy physics and establishing it has the correct semi-classical limit.

The problem of time

All canonical theories of general relativity have to deal with the problem of time. In quantum gravity, the problem of time is a conceptual conflict between general relativity and quantum mechanics. In canonical general relativity, time is just another coordinate as a result of general covariance. In quantum field theories, especially in the Hamiltonian formulation, the formulation is split between three dimensions of space, and one dimension of time. Roughly speaking, the problem of time is that there is none in general relativity. This is because in general relativity the Hamiltonian is a constraint that must vanish. However, in any canonical theory, the Hamiltonian generates time translations. Therefore, we arrive at the conclusion that "nothing moves" ("there is no time") in general relativity. Since "there is no time", the usual interpretation of quantum mechanics measurements at given moments of time breaks down. This problem of time is the broad banner for all interpretational problems of the formalism.

The problem of quantum cosmology

The problem of quantum cosmology is that the physical states that solve the constraints of canonical quantum gravity represent quantum states of the entire universe and as such exclude an outside observer, however an outside observer is a crucial element in most interpretations of quantum mechanics.