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Contents: v. A-B -- v. D-Feyman measure -- v. Fibonacci method-H -- v. I-Lituus -- v. Orbit-Rayleigh equation -- v. Reaction-diffusion equation-Stirling interpolation formula -- v. Stochastic approximation-Zygmund class of functions -- v. Subject index-Author index. Encyclopaedia of mathematics Hazewinkel, Michiel.

Vinogradov, Ivan Matveevich. Hazewinkel] D. Reidel , Kluwer Academic, cc : set v. M OPAC. Index OPAC. This additional volume contains nearly new entries written by experts and covers developments and topics not included in the already published volume set. These entries have been arranged alphabetically throughout.

A detailed index is included in the book. This Supplementary volume enhances the existing volume set. Together, these eleven volumes represent the most authoritative, comprehensive up-to-date Encyclopaedia of Mathematics available. This is the second supplementary volume to Kluwer's highly acclaimed eleven-volume Encyclopaedia of Mathematics. This additional volume contains nearly new entries written by experts and covers developments and topics not included in the previous volumes.

These entries are arranged alphabetically throughout and a detailed index is included. When the death of Queen Anne "changes" by moving farther into the past, this, too, is not ordinary change. If Boltzmann changes his mind from liking Loschmidt to disliking him, that is not an intrinsic change in Loschmidt; it is an ordinary, intrinsic change in Boltzmann, but not in Loschmidt.

For the relational theory, the term "property" is intended to exclude what Nelson Goodman called grue-like properties. Let us define an object to be grue if and only if, during the time that it exists, it is green before the beginning of the year but is blue thereafter. Classical substantival theories are also called "absolute theories. One sense of "to be absolute" is to be immutable, or changeless. Another sense is to be independent of reference frame. Centuries ago, the commonsense image of time was relationist, but due to the influence of Newton on the teaching of science in subsequent centuries plus this influence on the average person who is not a scientist, the commonsense image is now substantivalist.

The first advocate of a relational theory was Aristotle. The ancient Greek atomists such as Democritus spoke of there being an existing space within which matter's atoms move, implying space is substance-like rather than relational, but the atomists were not as influential as Aristotle on this topic. The battle lines between substantivalism and relationism were drawn more clearly in the early 18th century when Leibniz argued for relationism and Newton argued against it.

Leibniz claimed that space is nothing but the "order of co-existing things," so without objects there is no space. In other terms, we can say Leibniz's relational world is one in which spatial relationships are ontologically prior to space itself, and relationships among events are ontologically prior to time itself. Opposing Leibniz, Newton returned to a Democritus-like view of space as existing independent of material things, and he similarly accepted a substantival theory of time, with time being independent of all motions and other events.

Newton's actual equations of motion and his law of gravity are consistent with both relationism and substantivalism, although this point was not clear at the time to either Leibniz or Newton. In in his Lectiones Geometricae , the English physicist Isaac Barrow rejected any necessary linkage between time and change:. Whether things run or stand still, whether we sleep or wake, time flows in its even tenor. Barrow said time existed even before God created the matter in the universe. Newton believed time is not a primary substance, but is like the primary substances in not being dependent on anything except God.

For Newton, God chose some instant of pre-existing time at which to create the physical world. From these initial conditions, including the forces acting on the material objects the timeless scientific laws took over and guided the material objects, with God intervening only occasionally to perform miracles.

If it were not for God's intervention, one might properly think of the future as a logical consequence of the present. Leibniz objected. He was suspicious of Newton's absolute, substance-like time because it seemed to him to be undetectable. He argued that time is not a kind of stuff; and it is not an entity existing independently of actual events.

He insisted that Newton had under-emphasized the fact that time necessarily involves an ordering of events, the "successive order of things. Leibniz added that this overall order is time. So, he advocated relationism and rejected Newton's substantivalism.

Leibniz asked what is different about the new, shifted world. There is nothing to distinguish one point from another in absolute space, nor one instant of absolute time from another, so there would be no discernible difference in Newton's two worlds, the one before and the one after the shift. His point about time could have been expressed by saying Newton's two universes differ in their absolute times but not in their relative times, yet only relative times are discernible.

Leibniz offered another criticism. Newton's theory violates Leibniz's Law of Sufficient Reason: that there is a reason why any aspect of the universe is one way but might have been another. Leibniz complained that, if God shifted the world in time or space but made no other changes, then He would have no reason to do so.

Newton responded that Leibniz is correct to accept the Principle of Sufficient Reason, yet Newton pointed out that the Principle does not require there to be sufficient reasons for humans ; God might have had His own reason for creating the universe at a given absolute place and time even though mere mortals cannot comprehend His reasons.

Newton later admitted to friends that his two-part theological response to Leibniz was weak. Historians of philosophy generally agree that if Newton had said no more, he would have lost the debate. However, Newton found a much better argument. If you spin around and feel dizzy, then you are detecting absolute space. He didn't put it in exactly those terms, though. Partially fill the bucket with water, grasp the bucket, and, without spilling any water, rotate it many times until the rope is twisted.

When everything quiets down, the water surface is flat and there is no relative motion between the bucket and its water. That is situation 1. Now let go of the bucket, and let it spin until there is once again no relative motion between the bucket and its water. At this time, the bucket is spinning, and there is a concave curvature of the water surface.

That is situation 2. How can a relational theory explain the difference in the shape of the water's surface in the two situations? It cannot, said Newton. If we ignore our hands, the rope, the tree, and the rest of the universe, says Newton, each situation is simply a bucket with still water; the situations appear to differ only in the shape of the water surface. A relationist such as Leibniz cannot account for the difference in shape.

He said that when the bucket is not spinning, there is no motion relative to space itself, that is, to absolute space; but, when it is spinning, there is motion relative to space itself, and space itself must be exerting a force on the water to make the concave shape. This force pushing away from the center of the bucket is called "centrifugal force," and its presence is a way to detect absolute space. One hundred years later, Kant entered the arena on the side of Newton.

Consider two nearly identical gloves except that one is right-handed and the other is left-handed. In a world containing only a right-hand glove, said Kant, Leibniz's theory could not account for its handedness because all the internal relationships would be the same as in a world containing only a left-hand glove.

This indirectly suggests that the absolute theory of time is better, too. In , the philosopher Sydney Shoemaker presented a thought experiment attempting to establish that time's existing without change is at least conceivable, despite Aristotle's declaration that it is not. With the following scenario, we all can conceive "empty time," says Shoemaker. Divide all space into three disjoint regions, called region 3, region 4, and region 5. Region 3's change ceases everywhere every third year for one year.

There's Plenty of Room at the Bottom

People in regions 4 and 5 can verify this and then convince the people in region 3 of it after they come back to life at the end of their "frozen" year. Suppose people in region 3 become convinced by these reports that the universe periodically freezes in their region every three years for just one year.

Similarly, change ceases in region 4 every fourth year for a year; and change ceases in region 5 every fifth year for a year, and the inhabitants can be convinced that their region behaves this way. Every sixty years—that is, every 3 x 4 x 5 years—all three regions freeze simultaneously for a year. At the beginning of year sixty-one, everyone comes back to life, and they are justified in believing time has marched on for the previous year with no change anywhere in the universe. This year would be a year of empty time.

Yes, there is no person available to observe the freezing in year sixty, but we all believe in things that we don't directly observe, don't we? Because this is a merely possible world, there need be no explanation of how the freezing and thawing is implemented, and it is conceivable that the freezing occurs after sixty years even if there is no person available to measure the freezing.

Many philosophers accept this argument about conceivability, but would say the issue remains as to whether time would exist without change. In the 19th century, a vast majority of physicists believed in time without change. They not only believed in absolute space and time, but also had a favorite candidate for a large substance that is stationary in absolute space, the ether.

They believed the ether was needed for an adequate explanation of what waves when there is a light wave. In , James Clerk Maxwell proposed his theory of light. The theory was quickly and universally accepted. So, they believed Maxwell when he said the ether was needed as a medium for the propagation of light and that it did exist even if it had never been directly detected. The experimental physicist A. Michelson set out to detect the ether. Although his clever interferometer experiment failed to detect the ether, he believed he should have detected it if it existed.

Most physicists disagreed with Michelson's conclusion because they believed A. Fresnel who had said the Earth might drag the ether with it. If so, this would make the ether undetectable by Michelson's experimental apparatus, as long as the apparatus were used on Earth and not in outer space. However, this rescue of the ether hypothesis, and substantivalism along with it, did not last long.

In , Einstein proposed his special theory of relativity that implied there should be no ether. When his theory was experimentally confirmed, beliefs in the ether, substantival space, and substantival time were largely abandoned. Einstein and the philosopher Hans Reichenbach declared the special theory of relativity to be a victory for relationism. Till now it was believed that time and space existed by themselves, even if there was nothing—no Sun, no Earth, no stars—while now we know that time and space are not the vessel for the Universe, but could not exist at all if there were no contents, namely, no Sun, no Earth, and other celestial bodies.

According to Einstein's and Reichenbach's interpretation of relativity theory, Leibniz's notion of shifting the whole universe a specific distance or shifting it in time simply was not a coherent notion. Most other experts agreed. So, assuming inertia causes the rotating bucket's water to take its concave shape, if Newton's bucket were to hang still with no rotation while the background stars spun around the bucket, then the water would creep up the side of the bucket and form a concave surface.

If that is correct, then Newton's substantival space is not needed to explain the concave shape. Presumably substantival time is not needed either. By Ockham's Razor, if substantival space and time are not needed for successful explanations of the bucket situation or of any other situation, then substantival space and time should be rejected. Even Einstein himself changed his mind about the coherence of substantivalism. In his Nobel Prize acceptance speech on December 10, , he said relativity theory does rule out Maxwell's ether, but it does not rule out some other underlying substance that is pervasive in space; all that is required is that if such a substance exists, then it must obey the principles of relativity.

Another defense of substantivalism says that what physicists call empty space is an energetic and active field. Because of its continual activity quantum mechanically, there is no region of the field where there could be empty time in the relationist sense, so the spacetime field, the gravitational field, serves the role of being the underlying substance. The field does not go away even if the field's values reach a minimum everywhere. This sort of substance, however, is quite different from what was imagined by Newton and other early substantivalists, perhaps so different that they would not recognize it as being a substance at all.

Here is a related defense of substantivalism. Relativity theory implies there is a four-dimensional continuous manifold of point-events having a metric field and matter fields. In this vein, another kind of substantivalism says that spacetime is not just the manifold but rather is a combination of the manifold plus a single, essential metrical structure. Yet another position is that the debate between substantivalism and relationism no longer makes sense given the new terminology of the general theory of relativity and quantum mechanics because the very distinction between spacetime and event in spacetime, and between space itself and matter in space, has broken down.

In the early 20 th century, the appearance of the theories of relativity, quantum mechanics and the big bang transformed the investigation of time from a primarily speculative and metaphysical investigation into one that occupied scientists in their professional journals. The scientific community trusts their implications for time. For one example, the big bang theory places demands on the amount of past time there must be.

The past needs to have a duration of at least Einstein's theories of special relativity and general relativity have had the biggest impact on our understanding of time. Perhaps most significantly, they imply that two synchronized clocks will disagree on their readings once they move relative to each other or undergo different gravitational forces. Because of this, a clock in a car parked near your apartment building runs slower than the stationary clock in your upper floor apartment because the upper floor feels less gravitational force from the Earth.

Effects on time by speed and gravitation are called "time dilation effects. If I am walking along the road and you drive by me toward the traffic signal ahead, then we can agree on the time at which the traffic signal changed color, but if we want to know what event on a planet in the Andromeda Galaxy is simultaneous with the traffic signal's color change, we will correctly choose Andromeda events that differ from each other by several weeks. This is yet another example of how relativistic effects usually do not arise in our everyday experience but only in extreme situations involving extremely high speeds, extremely large masses, or, in this example, extreme distances.

This situation with the Andromeda Galaxy is also an example of how, for a pair of events that are extremely distant from each other so that neither event could have had a causal effect upon the other, the theory of relativity does not put any time order structure on the pair; one could happen first, the other could happen first, or they could be simultaneous, and only the imposition of a reference frame on the universe will force a decision on their temporal order. But since this order depends on the reference frame, the time order of the pair is not objective but only frame-relative.

Relativity theory demands a cosmic speed limit: light speed. This means no object can increase its speed enough to catch up with a particle of light that had a head start. So, if it takes , years for light to cross the Milky Way Galaxy, could some human being ever cross it in less time?

By exploiting the principles of time dilation and length contraction in the theory of relativity, the time limits on human exploration of the universe can be removed, at least in principle. Assuming you can travel safely at any high speed under light speed, then as your spaceship approaches light speed, the trip's distance contracts toward an infinitesimal length and the trip time approaches an infinitesimal duration as measured by your personal clock. So, you do have time to cross the Milky Way Galaxy, even though such a trip takes light itself , years as measured by an Earth clock.

In , the mathematician Hermann Minkowski had an original idea in metaphysics regarding space and time. He was the first person to claim that spacetime is more fundamental than either time or space alone. The above claims about spacetime are challenged by, among others, the pragmatist and physicist Lee Smolin. He says,. By succumbing to the temptation to conflate the representation with the reality and identify the graph of the records of the motion with the motion itself, these scientists have taken a big step toward the expulsion of time from our conception of nature.

The confusion worsens when we represent time as an axis on a graph This can be called spatializing time. And the mathematical conjunction of the representations of space and time, with each having its own axis, can be called spacetime. The pragmatisist will insist that this spacetime is not the real world. It's entirely a human invention, just another representation If we confuse spacetime with reality, we are committing a fallacy, which can be called the fallacy of the spatialization of time.

It is a consequence of forgetting the distinction between reording motion in time and time itself. Once you commit this fallacy, you're free to fantasize about the universe being timeless, and even being nothing but mathematics. But, the pragmatist says, timelessness and mathematics are properties of representations of records of motion--and only that. According to special relativity, there is no curvature to time, space or spacetime. According to general relativity they all curve, and the curvature is not relative to the chosen reference frame.

The "curvature" of time can be detected by noticing that synchronized clocks become unsynchronized. Spacetime is dynamic in the sense that any change in the amount and distribution of matter-energy will change the curvature. This change is propagated at the speed of light, not instantaneously. All the most fundamental physical laws have symmetry under time-translation. If a theory has this time-translation symmetry, then its laws do not change as time goes by. It follows that the law of gravitation in the 21 st century is the same law of gravitation that held one thousand centuries ago.

All those fundamental physical laws also have symmetry under time-reversal or at least CPT reversal. More informally, the point can be expressed by saying time-reversal symmetry implies that if you make a film of any process allowed by the laws of science, such as unbroken eggs being turned into an omelette, then show the film backward, what is shown is a process also allowed by the laws.

Science places special requirements on the micro-structure of time. For instance, to express laws using calculus, the physicists need to speak of one instantaneous event happening pi seconds after another, and of one event happening the square root of three seconds after another. In ordinary discourse outside of science, we would never need this kind of detail. This need requires time to be a linear continuum. Any linear continuum has the same structure as the real numbers in their natural order.

It follows from this that physical theories treat time as being somewhat like a single spatial dimension. Time has this structure in all well-accepted fundamental theories. Quantum theory does not quantize time. Science requires time to be one-dimensional. Time's one-dimensional structure might be like an unbounded straight line, or like a segment of that line, or like a ray, or even a circle. Two-dimensional time has been studied by mathematical physicists, but no theories implying time has more than one dimension have acquired a significant number of supporters among the experts.

Professional cosmologists say with a low degree of confidence that the past is infinite and, with more confidence, the future will be infinite. The most likely scenario for the end of time is that the remaining particles will get ever farther from each other, with no end to the dilution. This scenario depends upon a guess about the total energy of the universe.

Energy can be positive or negative. Mutually gravitating objects have gravitational potential energy that is negative. Their kinetic energy of motion is positive. Stephen Hawking, James Hartle, and other cosmologists, say the difficulty of knowing whether the past and future are infinite turns on our ignorance of whether the universe's positive energy is exactly canceled out by its negative energy so the kinetic energy of expansion is canceled by the mutual gravitational attraction , and whether the universe's positive charge is exactly canceled out by negative charge.

If the total is exactly zero, then time is infinite. There is no evidence that the totals are non-zero, so the best guess is that time is infinite; but the experts' confidence in this is not strong. Regarding the beginning of time instead of the end, the cosmologists' currently well-accepted theory of past time requires an explosion of all space when the universe had a very small volume. This caused all material in the space to expand, too.

Theories that imply this phenomenon are called big bang theories, the classical version of which says time began from a singularity a finite time ago. The controversy is whether there was unlimited time before that. The mathematical physicist Stephen Hawking once famously quipped that asking for what happened before the big bang is like asking what is north of the north pole. He later retracted that remark and said it is an open question whether there was time before the big bang, but he slightly favored a "yes" answer.

Even if "yes" were established as the correct answer, the question would remain as to whether this prior time was finite or infinite. Instead, the small, expanding volume of the universe before Perhaps there have been a cycle of bounces, a repetition of compression followed by expansion, and perhaps the cycles will continue forever and have been occurring forever. Much depends on whether the universe's total negative energy exactly balances its positive energy.

There has been much speculation over the centuries about the extent of the past and the future, although almost all remarks have contained serious ambiguities. For example, regarding the end of time, is this a the end of humanity, or b the end of life, or c the end of the universe that was created by God, but not counting God, or d the end of all natural and supernatural change?

Intimately related to these questions are two others: Is it being assumed that time exists without change, and just what is meant by the term "change"? With these cautions in mind, here is a brief summary of speculations throughout the centuries about whether time has a beginning or end. Regarding the beginning of time, the Greek atomist Lucretius in about 50 B. For surely the atoms did not hold council, assigning order to each, flexing their keen minds with questions of place and motion and who goes where. But shuffled and jumbled in many ways, in the course of endless time they are buffeted, driven along chancing upon all motions, combinations.

At last they fall into such an arrangement as would create this universe.

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The Feynman Lectures on Physics Vol. II Ch. 2: Differential Calculus of Vector Fields

The implication is that time has always existed, but that an organized universe began a finite time ago with a random fluctuation. Plato and Aristotle, both of whom were opponents of the atomists, agreed with them that the past is infinite eternal. Aristotle offered two reasons. Time had no beginning because, for any time, we always can imagine an earlier time. In addition, time had no beginning because everything in the world has a prior, efficient cause. In the fifth century, Augustine disagreed with Aristotle and said time itself came into existence by an act of God a finite time ago.

Martin Luther estimated the universe to have begun in 4, B. Then Johannes Kepler estimated that it began in 4, B.

Math Antics - Volume

In the early seventeenth century, the Calvinist James Ussher calculated from the Bible that the world began in 4, B. In about , Isaac Newton claimed future time is infinite and that, although God created the material world some finite time ago, there was an infinite period of past time before that. Advances in geology eventually refuted the low estimates that the universe was created in about 4, B. A much better estimate for a lower limit on the age of the universe as a whole comes not from geology but from the big bang theory which implies our universe is at least It is also difficult to understand St.

What do current cosmologists say about whether future time ends? It definitely ends for any object that falls into a black hole. Similarly, if the current expansion of our universe stops, then reverses, and eventually collapses to a point, then our future time stops, too. Cosmologists consider this to be an unlikely scenario because it violates quantum mechanics and is likely to violate the principles of any future theory of quantum gravity.

Here is a summary of some serious, competing suggestions by twenty-first century cosmologists about our universe's future, beginning with the most popular one:. Heat emerges from molecular motion, but no molecule is hot. Heat is not an autonomous property, though, because there can be no change in the heat without a corresponding change in the underlying molecular motion. Emergence is about useful, new features concepts, components or properties being supervenient upon more basic features but not existing at that more basic level. For another example, causation is not part of the most fundamental physics.

Causation emerges as a very useful characteristic of the universe at the higher, coarser level, where we properly and usefully speak of an atom's radioactive decay causing the release of a neutron, or hunger causing a person to visit the supermarket. Which level is fundamental? The answer is relative to the user's purpose. Biologists and physicists have different purposes.

To a biologist the hunger causing you to visit the supermarket emerges from the fundamental level of cellular activity. But to a physicist, the level of cellular activity is not fundamental but emerges from the more fundamental level of molecular motions. Does time emerge from spacetime? Some physicists speculate that once there were four dimensions of space and none of time, but that eventually one of the space dimensions disappeared as the time dimension appeared.

Could it be, instead, that time is fundamental, but spacetime is what emerges? In , after winning the Nobel Prize in physics, David Gross expressed that viewpoint. Speaking about string theory, his favored theory for reconciling the conflicts between quantum mechanics and the general theory of relativity, he said. Everyone in string theory is convinced We have an enormous amount of evidence that space is doomed.

We even have examples, mathematically well-defined examples, where space is an emergent concept They have emergent space but not time. It is very hard for me to imagine a formulation of physics without time as a primary concept because physics is typically thought of as predicting the future given the past. We have unitary time evolution. How could we have a theory of physics where we start with something in which time is never mentioned?

Many physicists working in the field of quantum gravity suspect that resolving the contradiction between quantum theory and gravitational theory will require forcing both spacetime and time to emerge from some more basic timeless substrate at or below the level of the Planck length and the Planck time. However, there is no experimental evidence yet to back up this suspicion, nor any agreed-upon theory of what the substrate is.

The relation of this substrate to the spacetime itself cannot be analogous to the relation of a brick to a brick wall because the brick's having a definite size would violate special relativity's requirement that any "brick" of time has a size that must change depending upon which reference frame is chosen.

Thus the emphasis on "covariant" entities, namely entities that are reference-frame independent. The physicist Carlo Rovelli, an advocate of loop quantum gravity, said: "At the fundamental level, the world is a collection of events not ordered in time" Rovelli , p. Nevertheless, at the macroscopic level, he would say time does exist. Eliminativism is the theory in ontology that emergent entities are unreal.

If time is emergent, it is not real. If pain is emergent, it is not real. The theory is also called strong emergentism. The opposite and more popular position in ontology, anti-eliminativism or weak emergence, is that emergent entities are real despite being emergent. The English physicist Julian Barbour is an eliminativist. In Barbour , p. Nothing happens; there is being but no becoming.

The flow of time and motion are illusions. There is just a vast, jumbled heap of moments. Each moment is an instantaneous configuration relative to one observer's reference frame of all the objects in space. Like a photograph, a moment or configuration contains information about change, but it, itself, does not change. If the universe is as Barbour describes, then space the relative spatial relationships within a configuration is ontologically fundamental, but time is not, and neither is spacetime.

In this way, time is removed from the foundations of physics and emerges as some general measure of the differences among the existing spatial configurations. For more on Barbour's position, see Smolin , pp. For a description of the different, detailed speculations on what the ultimate constituents of spacetime are, see Merali, Time has both conventional and non-conventional aspects. If event 1 happens before event 2, and event 2 happens before event 3, then event 1 also happens before event 3. No exceptions. This transitivity is a general feature of time, not a convention. In the philosophical literature, there is a philosophical dispute regarding why the positive direction of time is always toward the future rather than toward the past.

Is it just a convention that follows from the definitions of "future" and "positive," or does it have an objective basis? The temporal unit called a "second" is clearly conventional. This is because our society could have chosen to make the second be longer or shorter than it now is. It is a convention that there are sixty-seconds in a minute rather than sixty-six, that there are twenty-four hours in a day instead of twenty-three, and that no week fails to contain a Tuesday.

The issue here is conventional vs. Although the term "convention" is somewhat vague, conventions are up to us to freely adopt and are not objective features of the external world that we are forced to accept if we seek the truth. Conventions are inventions or artificial features as opposed to being natural or mandatory or factual. It is a fact that the color of normal, healthy leaves is green; this is not conventional or a matter of the custom of language usage.

What is conventional here is that "green" means green. Conventions need not be arbitrary; they can be useful or have other pragmatic virtues. Nevertheless, if a feature is conventional, then there must in some sense be reasonable alternative conventions that could have been adopted. Also, conventions can be explicit or implicit. For one last caution, conventions can become recognized as having been facts.

The assumption that matter is composed of atoms was a useful convention in late 19th century physics; but, after Einstein's explanation of Brownian motion in terms of atoms, the convention was recognized as having been a fact all along. It is a useful convention that, in order to keep future noons from occuring during the night, clocks are re-set by one hour as one moves across a time-zone on the Earth's surface, and that leap days and leap seconds are used.

The minor adjustments with leap seconds are required because the Earth's rotations and revolutions are not exactly regular. For political and social reasons, time zones do not always have longitudes for boundaries. For similar reasons, some geographical regions use daylight savings time instead of standard time. Consider the ordinary way a clock is used to measure how long an event lasts. We adopt the following metric , or method: Take the time at which the event ends, and subtract the time it starts.

For example, to find how long an event lasts that starts at and ends at , take the absolute value of the difference of the two numbers and get the answer of two hours. Is the use of this method merely a convention, or in some objective sense is it the only way that a clock could and should be used?

The nonrelativistic analysis shows that with this form the antiparticle still has positive energy. In the Fourier transform, this means shifting the pole in p 0 slightly, so that the inverse Fourier transform will pick up a small decay factor in one of the time directions:. Without these terms, the pole contribution could not be unambiguously evaluated when taking the inverse Fourier transform of p 0.

The terms can be recombined:. This is the mathematically precise form of the relativistic particle propagator, free of any ambiguities. So in the relativistic case, the Feynman path-integral representation of the propagator includes paths going backwards in time, which describe antiparticles. The paths that contribute to the relativistic propagator go forward and backwards in time, and the interpretation of this is that the amplitude for a free particle to travel between two points includes amplitudes for the particle to fluctuate into an antiparticle, travel back in time, then forward again.

Unlike the nonrelativistic case, it is impossible to produce a relativistic theory of local particle propagation without including antiparticles. All local differential operators have inverses that are nonzero outside the light cone, meaning that it is impossible to keep a particle from travelling faster than light.

Such a particle cannot have a Green's function which is only nonzero in the future in a relativistically invariant theory. However, the path integral formulation is also extremely important in direct application to quantum field theory, in which the "paths" or histories being considered are not the motions of a single particle, but the possible time evolutions of a field over all space.

In principle, one integrates Feynman's amplitude over the class of all possible field configurations. Much of the formal study of QFT is devoted to the properties of the resulting functional integral, and much effort not yet entirely successful has been made toward making these functional integrals mathematically precise. Such a functional integral is extremely similar to the partition function in statistical mechanics. Indeed, it is sometimes called a partition function , and the two are essentially mathematically identical except for the factor of i in the exponent in Feynman's postulate 3.

Analytically continuing the integral to an imaginary time variable called a Wick rotation makes the functional integral even more like a statistical partition function and also tames some of the mathematical difficulties of working with these integrals. As stated above, the unadorned path integral in the denominator ensures proper normalization.

Strictly speaking, the only question that can be asked in physics is: What fraction of states satisfying condition A also satisfy condition B? The answer to this is a number between 0 and 1, which can be interpreted as a conditional probability , written as P B A. In particular, this could be a state corresponding to the state of the Universe just after the Big Bang , although for actual calculation this can be simplified using heuristic methods.

Since this expression is a quotient of path integrals, it is naturally normalised.

Journal of the Mathematical Society of Japan

Since this formulation of quantum mechanics is analogous to classical action principle, one might expect that identities concerning the action in classical mechanics would have quantum counterparts derivable from a functional integral. This is often the case. In the language of functional analysis, we can write the Euler—Lagrange equations as. The quantum analogues of these equations are called the Schwinger—Dyson equations. In the deWitt notation this looks like [15]. These equations are the analog of the on-shell EL equations. The time ordering is taken before the time derivatives inside the S , i.

If J called the source field is an element of the dual space of the field configurations which has at least an affine structure because of the assumption of the translational invariance for the functional measure , then the generating functional Z of the source fields is defined to be. For example, if. Then, from the properties of the functional integrals. This is true, for example, for nonlinear sigma models where the target space is diffeomorphic to R n. However, if the target manifold is some topologically nontrivial space, the concept of a translation does not even make any sense.

In that case, we would have to replace the S in this equation by another functional. The path integrals are usually thought of as being the sum of all paths through an infinite space—time. However, in local quantum field theory we would restrict everything to lie within a finite causally complete region, for example inside a double light-cone. This gives a more mathematically precise and physically rigorous definition of quantum field theory.

Now how about the on shell Noether's theorem for the classical case?


  • Richard Feynman.
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Does it have a quantum analog as well? Yes, but with a caveat. The functional measure would have to be invariant under the one parameter group of symmetry transformation as well. Let's just assume for simplicity here that the symmetry in question is local not local in the sense of a gauge symmetry , but in the sense that the transformed value of the field at any given point under an infinitesimal transformation would only depend on the field configuration over an arbitrarily small neighborhood of the point in question. Let's also assume that the action is local in the sense that it is the integral over spacetime of a Lagrangian , and that.

Here, Q is a derivation which generates the one parameter group in question. We could have antiderivations as well, such as BRST and supersymmetry.

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This property is called the invariance of the measure. And this does not hold in general. See anomaly physics for more details. Now, let's assume even further that Q is a local integral. More general Lagrangians would require a modification to this definition! We're not insisting that q x is the generator of a symmetry i. And we also assume the even stronger assumption that the functional measure is locally invariant:. We'd simply have.

Path integrals as they are defined here require the introduction of regulators. Changing the scale of the regulator leads to the renormalization group. In fact, renormalization is the major obstruction to making path integrals well-defined. Regardless of whether one works in configuration space or phase space, when equating the operator formalism and the path integral formulation, an ordering prescription is required to resolve the ambiguity in the correspondence between non-commutative operators and the commutative functions that appear in path integrands. In one interpretation of quantum mechanics , the "sum over histories" interpretation, the path integral is taken to be fundamental, and reality is viewed as a single indistinguishable "class" of paths that all share the same events.

For this interpretation, it is crucial to understand what exactly an event is. The sum-over-histories method gives identical results to canonical quantum mechanics, and Sinha and Sorkin [16] claim the interpretation explains the Einstein—Podolsky—Rosen paradox without resorting to nonlocality. Some [ who? Whereas in quantum mechanics the path integral formulation is fully equivalent to other formulations, it may be that it can be extended to quantum gravity, which would make it different from the Hilbert space model.

Feynman had some success in this direction, and his work has been extended by Hawking and others. Quantum tunnelling can be modeled by using the path integral formation to determine the action of the trajectory through a potential barrier. This form is specifically useful in a dissipative system , in which the systems and surroundings must be modeled together.

Using the Langevin equation to model Brownian motion , the path integral formation can be used to determine an effective action and pre-exponential model to see the effect of dissipation on tunnelling.


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From Wikipedia, the free encyclopedia. This article is about a formulation of quantum mechanics. For integrals along a path, also known as line or contour integrals, see line integral. Classical mechanics Old quantum theory Bra—ket notation Hamiltonian Interference. Advanced topics. Quantum annealing Quantum chaos Quantum computing Density matrix Quantum field theory Fractional quantum mechanics Quantum gravity Quantum information science Quantum machine learning Perturbation theory quantum mechanics Relativistic quantum mechanics Scattering theory Spontaneous parametric down-conversion Quantum statistical mechanics.

Let us now picture one of the intermediate q s, say q k , as varying continuously while the other ones are fixed. The only important part in the domain of integration of q k is thus that for which a comparatively large variation in q k produces only a very small variation in F.

This part is the neighbourhood of a point for which F is stationary with respect to small variations in q k. We can apply this argument to each of the variables of integration This shows the way in which equation 11 goes over into classical results when h becomes extremely small. Play media. Feynman diagram. Standard Model. Quantum electrodynamics Electroweak interaction Quantum chromodynamics Higgs mechanism.

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