Mt. Fuji (Fujisan) is the tallest mountain in Japan
and, with its classically symmetrical snow-capped cone, has long been
the symbol of that country. The volcano is regarded as a sacredkamior spirit in the Shintoreligion,
specifically that of Princess Konohanasakuya-hime (aka Fuji-hime or
Sengen), and climbing its slopes is considered an act of pilgrimage for
followers of that faith. The mountain also has several important sacred
shrines, caves, springs and a waterfall. As of 2013 CE Mt. Fuji is a
UNESCO World Heritage Site.
Geography
Mt. Fuji, although generally envisaged as a single mountain, actually
consists of three distinct volcanoes. Its classic symmetrical slopes
are made all the more impressive by the mountain's isolation from any
other peaks. Located on Honshu island and straddling the border between
the Shizuoka and Yamanashi prefectures, Mt. Fuji is just 100 kilometres
(60 miles) from Tokyo and can be seen from its tower blocks on a clear
day. The mountain rises to a height of 3,776 metres (12,389 ft) and
incorporates five lakes. The volcano is currently dormant with its last
eruption occurring in 1708 CE, although between that date and 781 CE
there were 17 eruptions recorded.
A Sacred Mountain
Considered sacred by the Ainu people, the indigenous inhabitants of
ancient Japan, Mt. Fuji's name may derive from Fuchi, the Ainu god of
fire and the hearth. Some Buddhist
sects considered the mountain a holy place, and from the 12th century
CE, it became a destination for those practising asceticism (shugendo) and seeking a 'rebirth' from their time on the mountain, in a mix of Buddhist, Taoist, and animist beliefs.
In the 15th century CE, a mythology developed which associated Mt. Fuji with Konohanasakuya-hime, the beautiful 'Flower-blossom Princess.'
However, it is in the Shinto religion that Mt. Fuji holds a
particularly special position. Eight major shrines were built around the
foot of the mountain and hundreds of smaller ones have since been
added. The most important Shinto shrine is the Fujisan Hongu Sengen
Taisha which was first constructed in 806 CE, although by tradition, it
was originally founded during the reign of Emperor Suinin (29 BCE - 70
CE) at another location at the foot of the mountain. The shrine's
current building, with its unusual two storeys, dates to 1604 CE and the
site is famous for its 500 cherry trees which blossom each April.
The Kawaguchi Asama Shrine was built in the 9th century CE in an effort to appease the anger of Fujisan after the eruptions of that century. In 1149 CE the Dainichiji temple was built near the peak by Matsudai Shonin where there is also a torii or sacred gate. Unlike most Shinto shrines, there is no honden building containing the goshintai, the sacred physical object which is the embodiment of the shrine's kami,
as the mountain itself is regarded as that embodiment. There is also
the shrine of Kitaguchi Hongu Fuji Sengen Jinja which was founded during
the reign of Emperor Keiko (71-130 CE) but rebuilt in 1718 CE following
the eruption eleven years earlier. It is one of the preferred starting
points to reach the top, and its torii is over 18 metres (60
ft.) high, making it the largest in Japan. It is likely to keep that
record as it is rebuilt every 60 years, each time a little taller than
its predecessor.
Climbing the mountain was and still is regarded as an important
pilgrimage, an act which became popular from the 15th century CE even
for those with no particular religious affiliation. Being a holy site,
though, climbers had to be accompanied by a monk or priest guide (Oshi). Prior to 1945 CE women, considered in Shinto to be a source of impurity (kegare),
were not permitted to climb the sacred mountain. There are five ascent
routes with a total of ten stations. The climb, usually undertaken in
July or August when the snow has melted from the peak, takes anywhere
from 4 to 8 hours. Some 400,000 people make the effort each year, a good
many of them doing so at night in order to catch the auspicious sunrise
while at the peak.
Sometime in the 15th century CE, a mythology developed which
associated Mt. Fuji with Konohanasakuya-hime, a princess renowned for
her beauty whose name translates as 'Flower-blossom Princess.' According
to the Kojiki('Record
of Ancient Things' compiled in 712 CE), the princess, daughter of the
mountain god Oyamatsumi no kami, descended to earth and became the wife
of Ninigi no mikoto, grandson of the sun goddess Amaterasu
and founder of Japan's imperial line. Konohanasakuya-hime became the
goddess of Mt. Fuji. There are two lava caves formed by ancient trees
being covered in a lava flow and then decomposing. Here adherents to the
cult of Konohanasakuya-hime, known to Buddhists as Sengen, go through a
ritual of 'rebirth.'
In the 17th century CE, Hasegawa Kakugyo founded a sect based on Mt.
Fuji as the divine source of all life, which was popular in the 18th
century CE and necessitated the building of 86 lodges. Kakugyo claimed
he had seen the spirit of the mountain in a vision while he was
practising asceticism in a cave on Fuji's slopes. Other sacred sites on
the mountain include the Oshino Hakkai Springs which have an underground
river as their source and the curved Shiraito Falls which are fed by
the mountain's annual snowmelt.
Mt. Fuji is also believed to be a gathering point for the spirits of
deceased ancestors, and prayers are offered to them as well as
(prudently) safety from volcanic eruption, fire, and childbirth (a
specific role of Konohanasakuya-hime). Even today, a dream in which Mt.
Fuji appears is considered a sign of coming good fortune. Finally, the
mountain not only has its own shrines but there are over 13,000 shrines
spread across Japan which are dedicated to Fujisan. Many of
these include small-scale replicas of the mountain which worshippers who
are unable to climb the real thing ascend in symbolic pilgrimage.
Mt. Fuji in the Arts
Mt. Fuji has long captured the imagination of writers and artists. The 8th-century CE poetry anthology Manyoshu has several poems dedicated to the mountain; it appears smoking at the end of the c. 909 CE 'Tale of the Bamboo Cutter' (Taketori Monogatari),
the oldest surviving work of Japanese fiction, and it is the setting of
many medieval folktales. The mountain features in several of the haiku poems by Matsuo Basho (1644-1694 CE) too.
In the land of Yamato,
It is our treasure, our tutelary god.
It never tires our eyes to look up
To the lofty peak of Mount Fuji Manyoshu (Dougill, 17)
More recently, Fuji was famously captured in the ukiyo-e
woodblock prints of the noted artist Katsushika Hokusai (1760-1849 CE)
who produced a series known as the 'Thirty-Six Views of Mount Fuji' (FugakuSanjurokkei). One of the set, Beneath the Wave off Kanagawa,
is perhaps the most famous of all Japanese artworks. The artist Utagawa
Hiroshige (1797-1858 CE) was even more enraptured and produced an even
bigger set of prints, his 'One Hundred Views of Mount Fuji' (Fuji Sanjurokkei) as well as his 'Fifty-Three Stations of the Tokaido Road' (Tokaido Gojusantsugi). The mountain also appears on the current 1000 Yen banknote.
Mark is a history writer based in Italy. Surrounded by
archaeological sites, his special interests include ancient ceramics,
architecture, and mythology. He holds an MA in Political Philosophy and
is the Publishing Director at AHE.
The
game results show that placing slightly “noisy” bots in a central
location (high-degree nodes) improves human coordination by reducing
same-color neighbor nodes (the goal of the game). Square nodes show the
bots and round nodes show human players; thick red lines show color
conflicts, which are reduced with bot participation (right). (credit:
Hirokazu Shirado and Nicholas A. Christakis/Nature)
It’s not about artificial intelligence (AI) taking over — it’s about AI improving human performance, a new study by Yale University researchers has shown.
“Much of the current conversation about artificial intelligence has
to do with whether AI is a substitute for human beings. We believe the
conversation should be about AI as a complement to human beings,” said Nicholas Christakis, Yale University
co-director of the Yale Institute for Network Science (YINS) and senior
author of a study by Yale Institute for Network Science.*
AI doesn’t even have to be super-sophisticated to make a difference
in people’s lives; even “dumb AI” can help human groups, based on the
study, which appears in the May 18, 2017 edition of the journal Nature.
How bots can boost human performance
In a series of experiments using teams of human players and
autonomous software agents (“bots”), the bots boosted the performance of
human groups and the individual players, the researchers found.
The experiment design involved an online color-coordination game that
required groups of people to coordinate their actions for a collective
goal. The collective goal was for every node to have a color different
than all of its neighbor nodes. The subjects were paid a US$2 show-up
fee and a declining bonus of up to US$3 depending on the speed of
reaching a global solution to the coordination problem (in which every
player in a group had chosen a different color from their connected
neighbors). When they did not reach a global solution within 5 min, the
game was stopped and the subjects earned no bonus.
The human players also interacted with anonymous bots that were
programmed with three levels of behavioral randomness — meaning the AI
bots sometimes deliberately made mistakes (introduced “noise”). In
addition, sometimes the bots were placed in different parts of the
social network to try different strategies.
The result: The bots reduced the median time for groups to solve
problems by 55.6%. The experiment also showed a cascade effect: People
whose performance improved when working with the bots then influenced
other human players to raise their game. More than 4,000 people
participated in the experiment, which used Yale-developed software
called breadboard.
The findings have implications for a variety of situations in which
people interact with AI technology, according to the researchers.
Examples include human drivers who share roadways with autonomous cars
and operations in which human soldiers work in tandem with AI.
“There are many ways in which the future is going to be like this,”
Christakis said. “The bots can help humans to help themselves.”
Practical business AI tools
One example: Salesforce CEO Marc Benioff uses a bot called Einstein to help him run his company, Business Intelligencereported Thursday (May 18, 2017).
“Powered by advanced machine learning, deep learning, predictive
analytics, natural language processing and smart data discovery,
Einstein’s models will be automatically customised for every single
customer,” according to the Salesforce blog. “It will learn, self-tune
and get smarter with every interaction and additional piece of data. And
most importantly, Einstein’s intelligence will be embedded within the
context of business, automatically discovering relevant insights,
predicting future behavior, proactively recommending best next actions
and even automating tasks.”
Benioff says he also uses a version called Einstein Guidance for
forecasting and modeling. It even helps end internal politics at
executive meetings, calling out under-performing executives.
“AI is the next platform. All future apps for all companies will be built on AI,” Benioff predicts.
* Christakis is a professor of sociology, ecology &
evolutionary biology, biomedical engineering, and medicine at Yale.
Grants from the Robert Wood Johnson Foundation and the National
Institute of Social Sciences supported the research.
Abstract of Locally noisy autonomous agents improve global human coordination in network experiments
Coordination in groups faces a sub-optimization problem and theory
suggests that some randomness may help to achieve global optima. Here we
performed experiments involving a networked colour coordination game in
which groups of humans interacted with autonomous software agents
(known as bots). Subjects (n = 4,000) were embedded in networks (n = 230)
of 20 nodes, to which we sometimes added 3 bots. The bots were
programmed with varying levels of behavioural randomness and different
geodesic locations. We show that bots acting with small levels of random
noise and placed in central locations meaningfully improve the
collective performance of human groups, accelerating the median solution
time by 55.6%. This is especially the case when the coordination
problem is hard. Behavioural randomness worked not only by making the
task of humans to whom the bots were connected easier, but also by
affecting the gameplay of the humans among themselves and hence creating
further cascades of benefit in global coordination in these
heterogeneous systems.
University of Minnesota Medical School”How Cues Drive Our Behavior.” NeuroscienceNews. NeuroscienceNews, 3 August 2018. Summary:
Researchers shed light on the role dopamine neurons play in assigning
values to transient environmental cues that drive behaviors.
Source: University of Minnesota Medical School.
Do
dopamine neurons have a role in causing cues in our environment to
acquire value? And, if so, do different groups of dopamine neurons serve
different functions within this process?
Those are the questions researchers at the University of Minnesota Medical School are looking to answer.
Recent research published in Nature Neuroscience
by University of Minnesota Medical School neuroscientist Benjamin
Saunders, PhD, uses a Pavlovian model of conditioning to see if turning
on a light – a simple cue – just before dopamine neurons were activated
could motivate action. The classic Pavlov model combined the ringing of a
bell with providing a tasty steak to a dog that, over time, conditioned
a dog to drool when the bell rang with or without a steak. In this
research, however, there was no “real” reward like food or water, in
order to allow researchers to isolate the function of dopamine neuron
activity.
“We wanted to know if dopamine neurons are actually
directly responsible for assigning a value to these transient
environmental cues, like signs,” said Saunders, who conducted some of
his research as a postdoctoral fellow in the laboratory of Patricia
Janak, PhD, at Johns Hopkins University.
Dopamine neurons, those
cells in the brain that turn on when experiencing a reward. They are
also the neurons that degenerate in Parkinson’s disease.
“We
learned that dopamine neurons are one way our brains give the cues
around us meaning,” said Saunders. “The activity of dopamine neurons
alone – even in the absence of food, drugs, or other innately rewarding
substances – can imbue cues with value, giving them the ability to
motivate actions.”
To answer the second core question, the
researchers targeted specific segments of dopamine neurons – those
located in the substantial nigra (SNc) and those located in the ventral
tegmental area (VTA). These two types of neurons have historically been
studied in different disease research fields – SNc neurons in
Parkinson’s disease, and VTA neurons in addiction studies.
Scientists
learned that cues predicting activation of the two types of neurons
drove very different responses – those predicting the SNc neurons led to
a sort of “get up and go” response of invigorated rapid movement. The
cue predicting VTA neuron activation, however, became enticing on it
own, driving approach to the cue’s location, a sort of “where do I go?”
response.
To
answer the second core question, the researchers targeted specific
segments of dopamine neurons – those located in the substantial nigra
(SNc) and those located in the ventral tegmental area (VTA). These two
types of neurons have historically been studied in different disease
research fields – SNc neurons in Parkinson’s disease, and VTA neurons in
addiction studies.NeuroscienceNews.com image is in the public domain.
“Our
results reveal parallel motivational roles for dopamine neurons in
response to cues. In a real world situation, both forms of motivation
are critical,” said Saunders. “You have to be motivated to move around
and behave, and you have to be motivated to go to the specific location
of things you want and need.”
These results provide important
understanding of the function of dopamine neurons related to motivations
triggered by environmental cues. And this work contributes to the
understanding of relapse for those struggling with addictions.
“If
a cue – a sign, an alley, a favorite bar – takes on this powerful
motivational value, they will be difficult to resist triggers for
relapse,” said Saunders. “We know dopamine is involved, but an essential
goal for future studies is to understand how normal, healthy
cue-triggered motivation differs from dysfunctional motivation that
occurs in humans with addiction and related diseases.”
About this neuroscience research article
Source: Krystle Barbour – University of Minnesota Medical School Publisher: Organized by NeuroscienceNews.com. Image Source: NeuroscienceNews.com image is in the public domain. Original Research:Abstract
for “Dopamine neurons create Pavlovian conditioned stimuli with
circuit-defined motivational properties” by Benjamin T. Saunders,
Jocelyn M. Richard, Elyssa B. Margolis & Patricia H. Janak in Nature Neuroscience. Published July 23 2018. doi:10.1038/s41593-018-0191-4
Abstract
Dopamine neurons create Pavlovian conditioned stimuli with circuit-defined motivational properties
Environmental
cues, through Pavlovian learning, become conditioned stimuli that guide
animals toward the acquisition of rewards (for example, food) that are
necessary for survival. We tested the fundamental role of midbrain
dopamine neurons in conferring predictive and motivational properties to
cues, independent of external rewards. We found that brief phasic
optogenetic excitation of dopamine neurons, when presented in temporal
association with discrete sensory cues, was sufficient to instantiate
those cues as conditioned stimuli that subsequently both evoked dopamine
neuron activity on their own and elicited cue-locked conditioned
behavior. Notably, we identified highly parcellated functions for
dopamine neuron subpopulations projecting to different regions of
striatum, revealing dissociable dopamine systems for the generation of
incentive value and conditioned movement invigoration. Our results
indicate that dopamine neurons orchestrate Pavlovian conditioning via
functionally heterogeneous, circuit-specific motivational signals to
create, gate, and shape cue-controlled behaviors.
John Locke (1632–1704), a leading philosopher of British empiricism
In philosophy, empiricism is a theory that states that knowledge comes only or primarily from sensory experience. It is one of several views of epistemology, the study of human knowledge, along with rationalism and skepticism. Empiricism emphasises the role of empirical evidence in the formation of ideas, over the idea of innate ideas or traditions. However, empiricists may argue that traditions (or customs) arise due to relations of previous sense experiences.
Empiricism in the philosophy of science emphasises evidence, especially as discovered in experiments. It is a fundamental part of the scientific method that all hypotheses and theories must be tested against observations of the natural world rather than resting solely on a priori reasoning, intuition, or revelation.
Empiricism, often used by natural scientists, says that
"knowledge is based on experience" and that "knowledge is tentative and
probabilistic, subject to continued revision and falsification".[4] Empirical research, including experiments and validated measurement tools, guides the scientific method.
Etymology
The English term empirical derives from the Ancient Greek word ἐμπειρία, empeiria, which is cognate with and translates to the Latin experientia, from which are derived the word experience and the related experiment.
History
Background
A central concept in science and the scientific method is that it must be empirically based on the evidence of the senses. Both natural and social sciences use working hypotheses that are testable by observation and experiment. The term semi-empirical is sometimes used to describe theoretical methods that make use of basic axioms,
established scientific laws, and previous experimental results in order
to engage in reasoned model building and theoretical inquiry.
Philosophical empiricists hold no knowledge to be properly
inferred or deduced unless it is derived from one's sense-based
experience.[5] This view is commonly contrasted with rationalism, which states that knowledge may be derived from reason independently of the senses. For example, John Locke held that some knowledge (e.g. knowledge of God's existence) could be arrived at through intuition and reasoning alone. Similarly Robert Boyle, a prominent advocate of the experimental method, held that we have innate ideas.[6][7] The main continental rationalists (Descartes, Spinoza, and Leibniz) were also advocates of the empirical "scientific method".[8][9]
Early empiricism
Vaisheshikadarsana, founded by the ancient Indian philosopher Kanada, accepted perception and inference as the only two reliable sources of knowledge. This is enumerated in his work Vaiśeṣika Sūtra.
The earliest Western proto-empiricists were the Empiric school of ancient Greek medical practitioners, who rejected the three doctrines of the Dogmatic school, preferring to rely on the observation of "phenomena".[10]
The notion of tabula rasa
("clean slate" or "blank tablet") connotes a view of mind as an
originally blank or empty recorder (Locke used the words "white paper")
on which experience leaves marks. This denies that humans have innate ideas. The image dates back to Aristotle:
What the mind (nous) thinks must be in it in the same sense as letters are on a tablet (grammateion) which bears no actual writing (grammenon); this is just what happens in the case of the mind. (Aristotle, On the Soul, 3.4.430a1).
Aristotle's explanation of how this was possible was not strictly
empiricist in a modern sense, but rather based on his theory of potentiality and actuality, and experience of sense perceptions still requires the help of the active nous. These notions contrasted with Platonic
notions of the human mind as an entity that pre-existed somewhere in
the heavens, before being sent down to join a body on Earth (see Plato's
Phaedo and Apology, as well as others). Aristotle was considered to give a more important position to sense perception than Plato, and commentators in the Middle Ages summarized one of his positions as "nihil in intellectu nisi prius fuerit in sensu" (Latin for "nothing in the intellect without first being in the senses").
This idea was later developed in ancient philosophy by the Stoic
school. Stoic epistemology generally emphasized that the mind starts
blank, but acquires knowledge as the outside world is impressed upon it.[11] The doxographer Aetius
summarizes this view as "When a man is born, the Stoics say, he has the
commanding part of his soul like a sheet of paper ready for writing
upon."[12]
During the Middle Ages Aristotle's theory of tabula rasa was developed by Islamic philosophers starting with Al Farabi, developing into an elaborate theory by Avicenna[13] and demonstrated as a thought experiment by Ibn Tufail.[14] For Avicenna (Ibn Sina), for example, the tabula rasa is a pure potentiality that is actualized through education,
and knowledge is attained through "empirical familiarity with objects
in this world from which one abstracts universal concepts" developed
through a "syllogistic method of reasoning in which observations lead to propositional statements which when compounded lead to further abstract concepts". The intellect itself develops from a material intellect (al-'aql al-hayulani), which is a potentiality "that can acquire knowledge to the active intellect (al-'aql al-fa'il), the state of the human intellect in conjunction with the perfect source of knowledge".[13] So the immaterial "active intellect", separate from any individual person, is still essential for understanding to occur.
A similar Islamic theological novel, Theologus Autodidactus, was written by the Arab theologian and physician Ibn al-Nafis
in the 13th century. It also dealt with the theme of empiricism through
the story of a feral child on a desert island, but departed from its
predecessor by depicting the development of the protagonist's mind
through contact with society rather than in isolation from society.[15]
During the 13th century Thomas Aquinas adopted the Aristotelian position that the senses are essential to mind into scholasticism. Bonaventure
(1221–1274), one of Aquinas' strongest intellectual opponents, offered
some of the strongest arguments in favour of the Platonic idea of the
mind.
Renaissance Italy
In the late renaissance various writers began to question the medieval and classical understanding of knowledge acquisition in a more fundamental way. In political and historical writing Niccolò Machiavelli and his friend Francesco Guicciardini
initiated a new realistic style of writing. Machiavelli in particular
was scornful of writers on politics who judged everything in comparison
to mental ideals and demanded that people should study the "effectual
truth" instead. Their contemporary, Leonardo da Vinci
(1452–1519) said, "If you find from your own experience that something
is a fact and it contradicts what some authority has written down, then
you must abandon the authority and base your reasoning on your own
findings."[16]
The decidedly anti-Aristotelian and anti-clerical music theorist Vincenzo Galilei (c. 1520 – 1591), father of Galileo and the inventor of monody,
made use of the method in successfully solving musical problems,
firstly, of tuning such as the relationship of pitch to string tension
and mass in stringed instruments, and to volume of air in wind
instruments; and secondly to composition, by his various suggestions to
composers in his Dialogo della musica antica e moderna (Florence, 1581). The Italian word he used for "experiment" was esperienza. It is known that he was the essential pedagogical influence upon the young Galileo, his eldest son (cf. Coelho, ed. Music and Science in the Age of Galileo Galilei),
arguably one of the most influential empiricists in history. Vincenzo,
through his tuning research, found the underlying truth at the heart of
the misunderstood myth of 'Pythagoras' hammers'
(the square of the numbers concerned yielded those musical intervals,
not the actual numbers, as believed), and through this and other
discoveries that demonstrated the fallibility of traditional
authorities, a radically empirical attitude developed, passed on to
Galileo, which regarded "experience and demonstration" as the sine qua non of valid rational enquiry.
British empiricism
British empiricism, though it was not a term used at the time, derives from the 17th century period of early modern philosophy and modern science. The term became useful in order to describe differences perceived between two of its founders Francis Bacon, described as empiricist, and René Descartes, who is described as a rationalist. Thomas Hobbes and Baruch Spinoza, in the next generation, are often also described as an empiricist and a rationalist respectively. John Locke, George Berkeley, and David Hume were the primary exponents of empiricism in the 18th century Enlightenment, with Locke being the person who is normally known as the founder of empiricism as such.
In response to the early-to-mid-17th century "continental rationalism" John Locke (1632–1704) proposed in An Essay Concerning Human Understanding (1689) a very influential view wherein the only knowledge humans can have is a posteriori, i.e., based upon experience. Locke is famously attributed with holding the proposition that the human mind is a tabula rasa,
a "blank tablet", in Locke's words "white paper", on which the
experiences derived from sense impressions as a person's life proceeds
are written. There are two sources of our ideas: sensation and
reflection. In both cases, a distinction is made between simple and
complex ideas. The former are unanalysable, and are broken down into
primary and secondary qualities. Primary qualities are essential for the
object in question to be what it is. Without specific primary
qualities, an object would not be what it is. For example, an apple is
an apple because of the arrangement of its atomic structure. If an apple
was structured differently, it would cease to be an apple. Secondary
qualities are the sensory information we can perceive from its primary
qualities. For example, an apple can be perceived in various colours,
sizes, and textures but it is still identified as an apple. Therefore,
its primary qualities dictate what the object essentially is, while its
secondary qualities define its attributes. Complex ideas combine simple
ones, and divide into substances, modes, and relations. According to
Locke, our knowledge of things is a perception of ideas that are in
accordance or discordance with each other, which is very different from
the quest for certainty of Descartes.
A generation later, the Irish Anglican bishop, George Berkeley (1685–1753), determined that Locke's view immediately opened a door that would lead to eventual atheism. In response to Locke, he put forth in his Treatise Concerning the Principles of Human Knowledge (1710) an important challenge to empiricism in which things only exist either as a result
of their being perceived, or by virtue of the fact that they are an
entity doing the perceiving. (For Berkeley, God fills in for humans by
doing the perceiving whenever humans are not around to do it.) In his
text Alciphron, Berkeley maintained that any order humans may see in nature is the language or handwriting of God.[17] Berkeley's approach to empiricism would later come to be called subjective idealism.[18][19]
The Scottish philosopher David Hume
(1711–1776) responded to Berkeley's criticisms of Locke, as well as
other differences between early modern philosophers, and moved
empiricism to a new level of skepticism.
Hume argued in keeping with the empiricist view that all knowledge
derives from sense experience, but he accepted that this has
implications not normally acceptable to philosophers. He wrote for
example, "Locke divides all arguments into demonstrative and probable.
On this view, we must say that it is only probable that all men must die
or that the sun will rise to-morrow, because neither of these can be
demonstrated. But to conform our language more to common use, we ought
to divide arguments into demonstrations, proofs, and probabilities—by
‘proofs’ meaning arguments from experience that leave no room for doubt
or opposition."[20] And,[21]
"I believe the most general and
most popular explication of this matter, is to say [See Mr. Locke,
chapter of power.], that finding from experience, that there are several
new productions in matter, such as the motions and variations of body,
and concluding that there must somewhere be a power capable of producing
them, we arrive at last by this reasoning at the idea of power and
efficacy. But to be convinced that this explication is more popular than
philosophical, we need but reflect on two very obvious principles.
First, That reason alone can never give rise to any original idea, and
secondly, that reason, as distinguished from experience, can never make
us conclude, that a cause or productive quality is absolutely requisite
to every beginning of existence. Both these considerations have been
sufficiently explained: and therefore shall not at present be any
farther insisted on."
— Hume Section XIV "of the idea of necessary connexion in A Treatise of Human Nature
Hume divided all of human knowledge into two categories: relations of ideas and matters of fact (see also Kant'sanalytic-synthetic distinction).
Mathematical and logical propositions (e.g. "that the square of the
hypotenuse is equal to the sum of the squares of the two sides") are
examples of the first, while propositions involving some contingent
observation of the world (e.g. "the sun rises in the East") are
examples of the second. All of people's "ideas", in turn, are derived
from their "impressions". For Hume, an "impression" corresponds roughly
with what we call a sensation. To remember or to imagine such
impressions is to have an "idea". Ideas are therefore the faint copies
of sensations.[22]
David Hume's empiricism led to numerous philosophical schools.
Hume maintained that no knowledge, even the most basic beliefs about
the natural world, can be conclusively established by reason. Rather,
he maintained, our beliefs are more a result of accumulated habits,
developed in response to accumulated sense experiences. Among his many
arguments Hume also added another important slant to the debate about scientific method—that of the problem of induction. Hume argued that it requires inductive reasoning
to arrive at the premises for the principle of inductive reasoning, and
therefore the justification for inductive reasoning is a circular
argument.[22]
Among Hume's conclusions regarding the problem of induction is that
there is no certainty that the future will resemble the past. Thus, as a
simple instance posed by Hume, we cannot know with certainty by inductive reasoning
that the sun will continue to rise in the East, but instead come to
expect it to do so because it has repeatedly done so in the past.[22]
Hume concluded that such things as belief in an external world
and belief in the existence of the self were not rationally justifiable.
According to Hume these beliefs were to be accepted nonetheless because
of their profound basis in instinct and custom. Hume's lasting legacy,
however, was the doubt that his skeptical arguments cast on the
legitimacy of inductive reasoning, allowing many skeptics who followed
to cast similar doubt.
Phenomenalism
Most of Hume's followers have disagreed with his conclusion that belief in an external world is rationally
unjustifiable, contending that Hume's own principles implicitly
contained the rational justification for such a belief, that is, beyond
being content to let the issue rest on human instinct, custom and habit.[23] According to an extreme empiricist theory known as phenomenalism,
anticipated by the arguments of both Hume and George Berkeley, a
physical object is a kind of construction out of our experiences.[24]
Phenomenalism is the view that physical objects, properties, events
(whatever is physical) are reducible to mental objects, properties,
events. Ultimately, only mental objects, properties, events, exist—hence
the closely related term subjective idealism.
By the phenomenalistic line of thinking, to have a visual experience of
a real physical thing is to have an experience of a certain kind of
group of experiences. This type of set of experiences possesses a
constancy and coherence that is lacking in the set of experiences of
which hallucinations, for example, are a part. As John Stuart Mill put it in the mid-19th century, matter is the "permanent possibility of sensation".[25]
Mill's empiricism went a significant step beyond Hume in still another respect: in maintaining that induction is necessary for all meaningful knowledge including mathematics. As summarized by D.W. Hamlin:
[Mill] claimed that mathematical
truths were merely very highly confirmed generalizations from
experience; mathematical inference, generally conceived as deductive
[and a priori] in nature, Mill set down as founded on induction.
Thus, in Mill's philosophy there was no real place for knowledge based
on relations of ideas. In his view logical and mathematical necessity
is psychological; we are merely unable to conceive any other
possibilities than those that logical and mathematical propositions
assert. This is perhaps the most extreme version of empiricism known,
but it has not found many defenders.[19]
Mill's empiricism thus held that knowledge of any kind is not from
direct experience but an inductive inference from direct experience.[26]
The problems other philosophers have had with Mill's position center
around the following issues: Firstly, Mill's formulation encounters
difficulty when it describes what direct experience is by
differentiating only between actual and possible sensations. This
misses some key discussion concerning conditions under which such
"groups of permanent possibilities of sensation" might exist in the
first place. Berkeley put God in that gap; the phenomenalists,
including Mill, essentially left the question unanswered. In the end,
lacking an acknowledgement of an aspect of "reality" that goes beyond
mere "possibilities of sensation", such a position leads to a version of
subjective idealism. Questions of how floor beams continue to support a
floor while unobserved, how trees continue to grow while unobserved and
untouched by human hands, etc., remain unanswered, and perhaps
unanswerable in these terms.[19][27]
Secondly, Mill's formulation leaves open the unsettling possibility
that the "gap-filling entities are purely possibilities and not
actualities at all".[27] Thirdly, Mill's position, by calling mathematics merely another
species of inductive inference, misapprehends mathematics. It fails to
fully consider the structure and method of mathematical science, the products of which are arrived at through an internally consistent deductive set of procedures which do not, either today or at the time Mill wrote, fall under the agreed meaning of induction.[19][27][28]
The phenomenalist phase of post-Humean empiricism ended by the
1940s, for by that time it had become obvious that statements about
physical things could not be translated into statements about actual and
possible sense data.[29]
If a physical object statement is to be translatable into a sense-data
statement, the former must be at least deducible from the latter. But
it came to be realized that there is no finite set of statements about
actual and possible sense-data from which we can deduce even a single
physical-object statement. The translating or paraphrasing statement
must be couched in terms of normal observers in normal conditions of
observation. There is, however, no finite set of statements that
are couched in purely sensory terms and can express the satisfaction of
the condition of the presence of a normal observer. According to
phenomenalism, to say that a normal observer is present is to make the
hypothetical statement that were a doctor to inspect the observer, the
observer would appear to the doctor to be normal. But, of course, the
doctor himself must be a normal observer. If we are to specify this
doctor's normality in sensory terms, we must make reference to a second
doctor who, when inspecting the sense organs of the first doctor, would
himself have to have the sense data a normal observer has when
inspecting the sense organs of a subject who is a normal observer. And
if we are to specify in sensory terms that the second doctor is a normal
observer, we must refer to a third doctor, and so on (also see the third man).[30][31]
Logical empiricism
Logical empiricism (also logical positivism or neopositivism)
was an early 20th-century attempt to synthesize the essential ideas of
British empiricism (e.g. a strong emphasis on sensory experience as the
basis for knowledge) with certain insights from mathematical logic that had been developed by Gottlob Frege and Ludwig Wittgenstein. Some of the key figures in this movement were Otto Neurath, Moritz Schlick and the rest of the Vienna Circle, along with A.J. Ayer, Rudolf Carnap and Hans Reichenbach.
The neopositivists subscribed to a notion of philosophy as the
conceptual clarification of the methods, insights and discoveries of
the sciences. They saw in the logical symbolism elaborated by Frege
(1848–1925) and Bertrand Russell
(1872–1970) a powerful instrument that could rationally reconstruct all
scientific discourse into an ideal, logically perfect, language that
would be free of the ambiguities and deformations of natural language.
This gave rise to what they saw as metaphysical pseudoproblems and other
conceptual confusions. By combining Frege's thesis that all
mathematical truths are logical with the early Wittgenstein's idea that
all logical truths are mere linguistic tautologies, they arrived at a twofold classification of all propositions: the analytic (a priori) and the synthetic (a posteriori).[32]
On this basis, they formulated a strong principle of demarcation
between sentences that have sense and those that do not: the so-called verification principle.
Any sentence that is not purely logical, or is unverifiable is devoid
of meaning. As a result, most metaphysical, ethical, aesthetic and other
traditional philosophical problems came to be considered
pseudoproblems.[33]
In the extreme empiricism of the neopositivists—at least before
the 1930s—any genuinely synthetic assertion must be reducible to an
ultimate assertion (or set of ultimate assertions) that expresses direct
observations or perceptions. In later years, Carnap and Neurath
abandoned this sort of phenomenalism in favor of a rational
reconstruction of knowledge into the language of an objective
spatio-temporal physics. That is, instead of translating sentences
about physical objects into sense-data, such sentences were to be
translated into so-called protocol sentences, for example, "X at location Y and at time T observes such and such."[34]
The central theses of logical positivism (verificationism, the
analytic–synthetic distinction, reductionism, etc.) came under sharp
attack after World War II by thinkers such as Nelson Goodman, W.V. Quine, Hilary Putnam, Karl Popper, and Richard Rorty.
By the late 1960s, it had become evident to most philosophers that the
movement had pretty much run its course, though its influence is still
significant among contemporary analytic philosophers such as Michael Dummett and other anti-realists.
Pragmatism
In the late 19th and early 20th century several forms of pragmatic philosophy arose. The ideas of pragmatism, in its various forms, developed mainly from discussions between Charles Sanders Peirce and William James
when both men were at Harvard in the 1870s. James popularized the term
"pragmatism", giving Peirce full credit for its patrimony, but Peirce
later demurred from the tangents that the movement was taking, and
redubbed what he regarded as the original idea with the name of
"pragmaticism". Along with its pragmatic theory of truth, this perspective integrates the basic insights of empirical (experience-based) and rational (concept-based) thinking.
Charles Peirce (1839–1914) was highly influential in laying the groundwork for today's empirical scientific method.[35]
Although Peirce severely criticized many elements of Descartes'
peculiar brand of rationalism, he did not reject rationalism outright.
Indeed, he concurred with the main ideas of rationalism, most
importantly the idea that rational concepts can be meaningful and the
idea that rational concepts necessarily go beyond the data given by
empirical observation. In later years he even emphasized the
concept-driven side of the then ongoing debate between strict empiricism
and strict rationalism, in part to counterbalance the excesses to which
some of his cohorts had taken pragmatism under the "data-driven"
strict-empiricist view.
Among Peirce's major contributions was to place inductive reasoning and deductive reasoning
in a complementary rather than competitive mode, the latter of which
had been the primary trend among the educated since David Hume wrote a
century before. To this, Peirce added the concept of abductive reasoning.
The combined three forms of reasoning serve as a primary conceptual
foundation for the empirically based scientific method today. Peirce's
approach "presupposes that (1) the objects of knowledge are real things,
(2) the characters (properties) of real things do not depend on our
perceptions of them, and (3) everyone who has sufficient experience of
real things will agree on the truth about them. According to Peirce's
doctrine of fallibilism,
the conclusions of science are always tentative. The rationality of the
scientific method does not depend on the certainty of its conclusions,
but on its self-corrective character: by continued application of the
method science can detect and correct its own mistakes, and thus
eventually lead to the discovery of truth".[36]
In his Harvard "Lectures on Pragmatism" (1903), Peirce enumerated what he called the "three cotary propositions of pragmatism" (L:cos, cotis whetstone), saying that they "put the edge on the maxim of pragmatism".
First among these he listed the peripatetic-thomist observation
mentioned above, but he further observed that this link between sensory
perception and intellectual conception is a two-way street. That is, it
can be taken to say that whatever we find in the intellect is also
incipiently in the senses. Hence, if theories are theory-laden then so
are the senses, and perception itself can be seen as a species of abductive inference,
its difference being that it is beyond control and hence beyond
critique—in a word, incorrigible. This in no way conflicts with the
fallibility and revisability of scientific concepts, since it is only
the immediate percept in its unique individuality or "thisness"—what the
Scholastics called its haecceity—that
stands beyond control and correction. Scientific concepts, on the
other hand, are general in nature, and transient sensations do in
another sense find correction within them. This notion of perception as
abduction has received periodic revivals in artificial intelligence and cognitive science research, most recently for instance with the work of Irvin Rock on indirect perception.[37][38]
Around the beginning of the 20th century, William James (1842–1910) coined the term "radical empiricism"
to describe an offshoot of his form of pragmatism, which he argued
could be dealt with separately from his pragmatism—though in fact the
two concepts are intertwined in James's published lectures. James
maintained that the empirically observed "directly apprehended universe
needs ... no extraneous trans-empirical connective support",[39] by which he meant to rule out the perception that there can be any value added by seeking supernatural explanations for naturalphenomena. James' "radical empiricism" is thus not radical in the context of the term "empiricism", but is instead fairly consistent with the modern use of the term "empirical". His method of argument in arriving at this view, however, still readily encounters debate within philosophy even today.
John Dewey (1859–1952) modified James' pragmatism to form a theory known as instrumentalism.
The role of sense experience in Dewey's theory is crucial, in that he
saw experience as unified totality of things through which everything
else is interrelated. Dewey's basic thought, in accordance with
empiricism was that reality
is determined by past experience. Therefore, humans adapt their past
experiences of things to perform experiments upon and test the pragmatic
values of such experience. The value of such experience is measured
experientially and scientifically, and the results of such tests
generate ideas that serve as instruments for future experimentation,[40] in physical sciences as in ethics.[41] Thus, ideas in Dewey's system retain their empiricist flavour in that they are only known a posteriori.
Could also be printed directly on human skin for pulse monitoring
or as a human-machine interface --- imagine a computer mouse built into
your fingertip
Schematic
of a new kind of 3D printer that can print touch sensors directly on a
model hand. (credit: Shuang-Zhuang Guo and Michael McAlpine/Advanced
Materials )
Engineering researchers at the University of Minnesota
have developed a process for 3D-printing stretchable, flexible, and
sensitive electronic sensory devices that could give robots or
prosthetic hands — or even real skin — the ability to mechanically sense
their environment.
One major use would be to give surgeons the ability to feel during
minimally invasive surgeries instead of using cameras, or to increase
the sensitivity of surgical robots. The process could also make it
easier for robots to walk and interact with their environment.
Printing electronics directly on human skin could be used for pulse
monitoring, energy harvesting (of movements), detection of finger
motions (on a keyboard or other devices), or chemical sensing (for
example, by soldiers in the field to detect dangerous chemicals or
explosives). Or imagine a future computer mouse built into your
fingertip, with haptic touch on any surface.
“While we haven’t printed on human skin yet, we were able to print on
the curved surface of a model hand using our technique,” said Michael McAlpine,
a University of Minnesota mechanical engineering associate professor
and lead researcher on the study.* “We also interfaced a printed device
with the skin and were surprised that the device was so sensitive that
it could detect your pulse in real time.”
The researchers also visualize use in “bionic organs.”
A unique skin-compatible 3D-printing process
(left)
Schematic of the tactile sensor. (center) Top view. (right) Optical
image showing the conformally printed 3D tactile sensor on a fingertip.
Scale bar = 4 mm. (credit: Shuang-Zhuang Guo et al./Advanced Materials)
McAlpine and his team made the sensing fabric with a one-of-a kind 3D
printer they built in the lab. The multifunctional printer has four
nozzles to print the various specialized “inks” that make up the layers
of the device — a base layer of silicone**, top and bottom electrodes
made of a silver-based piezoresistive
conducting ink, a coil-shaped pressure sensor, and a supporting layer
that holds the top layer in place while it sets (later washed away in
the final manufacturing process).
Surprisingly, all of the layers of “inks” used in the flexible
sensors can set at room temperature. Conventional 3D printing using
liquid plastic is too hot and too rigid to use on the skin. The sensors
can stretch up to three times their original size.
The researchers say the next step is to move toward semiconductor
inks and printing on a real surface. “The manufacturing is built right
into the process, so it is ready to go now,” McAlpine said.
The research was published online in the journal Advanced Materials. It was funded by the National Institute of Biomedical Imaging and Bioengineering of the National Institutes of Health.
* McAlpine integrated electronics and novel 3D-printed nanomaterials to create a “bionic ear” in 2013.
** The silicone rubber has a low modulus of elasticity of 150 kPa, similar to that of skin, and lower hardness (Shore A 10) than that of human skin, according to the Advanced Materials paper.
College of Science and Engineering, UMN | 3D Printed Stretchable Tactile Sensors
Abstract of 3D Printed Stretchable Tactile Sensors
The development of methods for the 3D printing of multifunctional
devices could impact areas ranging from wearable electronics and energy
harvesting devices to smart prosthetics and human–machine interfaces.
Recently, the development of stretchable electronic devices has
accelerated, concomitant with advances in functional materials and
fabrication processes. In particular, novel strategies have been
developed to enable the intimate biointegration of wearable electronic
devices with human skin in ways that bypass the mechanical and thermal
restrictions of traditional microfabrication technologies. Here, a
multimaterial, multiscale, and multifunctional 3D printing approach is
employed to fabricate 3D tactile sensors under ambient conditions
conformally onto freeform surfaces. The customized sensor is
demonstrated with the capabilities of detecting and differentiating
human movements, including pulse monitoring and finger motions. The
custom 3D printing of functional materials and devices opens new routes
for the biointegration of various sensors in wearable electronics
systems, and toward advanced bionic skin applications.
In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. In Feynman diagrams, which serve to calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the respective diagram. These may also be viewed as the inverse of the wave operator appropriate to the particle, and are, therefore, often called (causal) Green's functions (called "causal" to distinguish it from the elliptic Laplacian Green's function).
Non-relativistic propagators
In non-relativistic quantum mechanics, the propagator gives the probability amplitude for a particle to travel from one spatial point at one time to another spatial point at a later time. It is the Green's function (fundamental solution) for the Schrödinger equation. This means that, if a system has HamiltonianH, then the appropriate propagator is a function
satisfying
where Hx denotes the Hamiltonian written in terms of the x coordinates, δ(x) denotes the Dirac delta-function, Θ(t) is the Heaviside step function and K(x, t ;x′, t′) is the kernel of the differential operator in question, often referred to as the propagator instead of G in this context, and henceforth in this article. This propagator can also be written as
where Û(t, t′) is the unitary time-evolution operator for the system taking states at time t′ to states at time t.
The quantum mechanical propagator may also be found by using a path integral,
where the boundary conditions of the path integral include q(t) = x, q(t′) = x′. Here L denotes the Lagrangian of the system. The paths that are summed over move only forwards in time, and are integrated with the differential which follows the path in time.
In non-relativistic quantum mechanics,
the propagator lets you find the wave function of a system given an
initial wave function and a time interval. The new wave function is
given by the equation
If K(x,t;x′,t′) only depends on the difference x − x′, this is a convolution of the initial wave function and the propagator. This kernel is the kernel of integral transform.
Basic examples: propagator of free particle and harmonic oscillator
For a time-translationally invariant system, the propagator only depends on the time difference t − t′, so it may be rewritten as
In quantum field theory, the theory of a free (non-interacting) scalar field is a useful and simple example which serves to illustrate the concepts needed for more complicated theories. It describes spin zero particles. There are a number of possible propagators for free scalar field theory. We now describe the most common ones.
The different choices for how to deform the integration contour in the above expression lead to various forms for the propagator. The choice of contour is usually phrased in terms of the integral.
The integrand then has two poles at
so different choices of how to avoid these lead to different propagators.
Causal propagators
Retarded propagator
A contour going clockwise over both poles gives the causal retarded propagator. This is zero if x-y is spacelike or if x ⁰< y ⁰ (i.e. if y is to the future of x).
This choice of contour is equivalent to calculating the limit,
A contour going anti-clockwise under both poles gives the causal advanced propagator. This is zero if x-y is spacelike or if x ⁰> y ⁰ (i.e. if y is to the past of x).
This choice of contour is equivalent to calculating the limit
This expression can also be expressed in terms of the vacuum expectation value of the commutator of the free scalar field.
In this case,
Feynman propagator
A contour going under the left pole and over the right pole gives the Feynman propagator.
This choice of contour is equivalent to calculating the limit[5]
This expression can be derived directly from the field theory as the vacuum expectation value of the time-ordered product of the free scalar field, that is, the product always taken such that the time ordering of the spacetime points is the same,
This expression is Lorentz invariant, as long as the field operators commute with one another when the points x and y are separated by a spacelike interval.
The usual derivation is to insert a complete set of
single-particle momentum states between the fields with Lorentz
covariant normalization, and to then show that the Θ functions providing the causal time ordering may be obtained by a contour integral
along the energy axis, if the integrand is as above (hence the
infinitesimal imaginary part), to move the pole off the real line.
The Fourier transform of the position space propagators can be thought of as propagators in momentum space. These take a much simpler form than the position space propagators.
They are often written with an explicit ε term although this is understood to be a reminder about which integration contour is appropriate (see above). This ε term is included to incorporate boundary conditions and causality (see below).
For a 4-momentump the causal and Feynman propagators in momentum space are:
For purposes of Feynman diagram calculations, it is usually convenient to write these with an additional overall factor of −i (conventions vary).
Faster than light?
The Feynman propagator has some properties that seem baffling at first. In particular, unlike the commutator, the propagator is nonzero outside of the light cone,
though it falls off rapidly for spacelike intervals. Interpreted as an
amplitude for particle motion, this translates to the virtual particle
travelling faster than light. It is not immediately obvious how this can
be reconciled with causality: can we use faster-than-light virtual
particles to send faster-than-light messages?
The answer is no: while in classical mechanics
the intervals along which particles and causal effects can travel are
the same, this is no longer true in quantum field theory, where it is commutators that determine which operators can affect one another.
So what does the spacelike part of the propagator represent? In QFT the vacuum is an active participant, and particle numbers and field values are related by an uncertainty principle; field values are uncertain even for particle number zero. There is a nonzero probability amplitude to find a significant fluctuation in the vacuum value of the field Φ(x)
if one measures it locally (or, to be more precise, if one measures an
operator obtained by averaging the field over a small region). Furthermore, the dynamics of the fields tend to favor spatially
correlated fluctuations to some extent. The nonzero time-ordered product
for spacelike-separated fields then just measures the amplitude for a
nonlocal correlation in these vacuum fluctuations, analogous to an EPR correlation. Indeed, the propagator is often called a two-point correlation function for the free field.
Since, by the postulates of quantum field theory, all observable
operators commute with each other at spacelike separation, messages can
no more be sent through these correlations than they can through any
other EPR correlations; the correlations are in random variables.
Regarding virtual particles, the propagator at spacelike
separation can be thought of as a means of calculating the amplitude for
creating a virtual particle-antiparticle pair that eventually disappears into the vacuum, or for detecting a virtual pair emerging from the vacuum. In Feynman's
language, such creation and annihilation processes are equivalent to a
virtual particle wandering backward and forward through time, which can
take it outside of the light cone. However, no signaling back in time
is allowed.
Explanation using limits
This can be made clearer by writing the propagator in the following form for a massless photon,
This is the usual definition but normalised by a factor of . Then the rule is that one only takes the limit at the end of a calculation.
One sees that
if
and
if
Hence this means a single photon will always stay on the light cone.
It is also shown that the total probability for a photon at any time
must be normalised by the reciprocal of the following factor:
We see that the parts outside the light cone usually are zero in the limit and only are important in Feynman diagrams.
Propagators in Feynman diagrams
The most common use of the propagator is in calculating probability amplitudes for particle interactions using Feynman diagrams.
These calculations are usually carried out in momentum space. In
general, the amplitude gets a factor of the propagator for every internal line,
that is, every line that does not represent an incoming or outgoing
particle in the initial or final state. It will also get a factor
proportional to, and similar in form to, an interaction term in the
theory's Lagrangian for every internal vertex where lines meet. These prescriptions are known as Feynman rules.
Internal lines correspond to virtual particles. Since the
propagator does not vanish for combinations of energy and momentum
disallowed by the classical equations of motion, we say that the virtual
particles are allowed to be off shell. In fact, since the propagator is obtained by inverting the wave equation, in general, it will have singularities on the shell.
The energy carried by the particle in the propagator can even be negative. This can be interpreted simply as the case in which, instead of a particle going one way, its antiparticle is going the other
way, and therefore carrying an opposing flow of positive energy. The
propagator encompasses both possibilities. It does mean that one has to
be careful about minus signs for the case of fermions, whose propagators are not even functions in the energy and momentum (see below).
Virtual particles conserve energy and momentum. However, since
they can be off the shell, wherever the diagram contains a closed loop,
the energies and momenta of the virtual particles participating in the
loop will be partly unconstrained, since a change in a quantity for one
particle in the loop can be balanced by an equal and opposite change in
another. Therefore, every loop in a Feynman diagram requires an integral
over a continuum of possible energies and momenta. In general, these
integrals of products of propagators can diverge, a situation that must
be handled by the process of renormalization.
Other theories
Spin 1⁄2
If the particle possesses spin
then its propagator is in general somewhat more complicated, as it will
involve the particle's spin or polarization indices. The differential
equation satisfied by the propagator for a spin 1⁄2 particle is given by[6]
where I4 is the unit matrix in four dimensions, and employing the Feynman slash notation. This is the Dirac equation for a delta function source in spacetime. Using the momentum representation,
the equation becomes
where on the right-hand side an integral representation of the four-dimensional delta function is used. Thus
By multiplying from the left with
(dropping unit matrices from the notation) and using properties of the gamma matrices,
the momentum-space propagator used in Feynman diagrams for a Dirac field representing the electron in quantum electrodynamics is found to have form
The iε downstairs is a prescription for how to handle the poles in the complex p0-plane. It automatically yields the Feynman contour of integration by shifting the poles appropriately. It is sometimes written
for short. It should be remembered that this expression is just shorthand notation for (γμpμ − m)−1. "One over matrix" is otherwise nonsensical. In position space one has
This is related to the Feynman propagator by
where .
Spin 1
The propagator for a gauge boson in a gauge theory depends on the choice of convention to fix the gauge. For the gauge used by Feynman and Stueckelberg, the propagator for a photon is
The propagator for a massive vector field can be derived from the
Stueckelberg Lagrangian. The general form with gauge parameter λ reads
With this general form one obtains the propagator in unitary gauge for λ = 0, the propagator in Feynman or 't Hooft gauge for λ = 1 and in Landau or Lorenz gauge for λ = ∞. There are also other notations where the gauge parameter is the inverse of λ. The name of the propagator, however, refers to its final form and not necessarily to the value of the gauge parameter.
where is the Hubble constant. Note that upon taking the limit , the AdS propagator reduces to the Minkowski propagator.[7]
Related singular functions
The scalar propagators are Green's functions for the Klein–Gordon
equation. There are related singular functions which are important in quantum field theory. We follow the notation in Bjorken and Drell.[8] See also Bogolyubov and Shirkov (Appendix A). These functions are most simply defined in terms of the vacuum expectation value of products of field operators.
Solutions to the Klein–Gordon equation
Pauli–Jordan function
The commutator of two scalar field operators defines the Pauli–Jordan function by[8]
with
This satisfies
and is zero if .
Positive and negative frequency parts (cut propagators)
We can define the positive and negative frequency parts of , sometimes called cut propagators, in a relativistically invariant way.
This allows us to define the positive frequency part: