title: Communist Party of Ukraine

text: In the 2014 Ukrainian presidential election, Symonenko initially again ran as a candidate of his party, but he withdrew from the race on 16 May. The Central Election Commission was unable to remove from the ballot his name because he withdrew from the race after the deadline of 1 May 2014. In the election, he still received 1.51% of the vote.


title: Communist Party of Ukraine

text: On 1 July, six MPs left the Communist Party faction in parliament, reducing it to 23 members. On 22 July, a vote supported by 232 MPs gave the Chairman of the Verkhovna Rada (the speaker of Ukraine's parliament) the power to dissolve a faction that has lost some of its members compared to the number it had while it was formed during the first parliamentary session after the previous election, pending a signature from President Petro Poroshenko. Later that day, Poroshenko signed this bill, giving effect to this new parliamentary regulation. The next day, speaker and former Acting President Turchynov announced the party's impending dissolution and added to MPs: "We only have to tolerate this party for another day". The party's faction in parliament was dissolved on 24 July by Turchynov. That same day, it was announced that at the time 308 criminal proceedings against members of the party had been opened. Communists were accused of openly supporting the annexation of Crimea by Russia, supporting the creation of self-proclaimed Donetsk People's Republic and Lugansk People's Republic and agitating for annexation of the Dnipropetrovsk Oblast to Russia. The party leadership at the time stated its support for Ukrainian territorial integrity and excluded separatist dissenters from its membership.


title: Communist Party of Ukraine

text: On 4 September, the Kyiv District Administrative Court indefinitely postponed the hearing about the ban of the party.


title: Communist Party of Ukraine

text: The October 2014 parliamentary election further marginalized the party as it won no constituency seats and came 1.12% short of reaching the 5% election threshold. Since prior to its independence, Ukraine was a constituent republic of the Communist Soviet Union and this meant that for the first time since 1918 Communists were not represented in Ukrainian national politics. The first ten members on the party list for the 2014 Ukrainian parliamentary election were Petro Symonenko (MP), Adam Martynyuk (MP), Kateryna Samoylyk (senior), Vasyl Sirenko (Koretsky Institute of State and Law, non-partisan), Petro Tsybenko (MP), Ihor Aleksyeyev (MP), Serhiy Hordiyenko (MP), Yevhen Marmazov (MP), Spiridon Kilinkarov (MP) and Serhiy Khrapov (unemployed).


title: Communist Party of Ukraine

text: In May 2015, laws that banned communist symbols (the so-called "decommunization laws") came into effect in Ukraine, meaning that the party could not use communist symbols or sing the Soviet national hymn or "The Internationale". In a 24 July decree based on these laws, the Ukrainian Interior Ministry stripped the party of its right to participate in elections and stated that it was continuing the court actions (which started in July 2014) to end the registration of Ukraine's communist parties.


title: Communist Party of Ukraine

text: On 30 September, the District Administrative Court in Kyiv banned two smaller communist parties, the Communist Party of Workers and Peasants and the Communist Party of Ukraine (renewed). However, the Communist Party was not banned because it had filed an appeal against the Justice Ministry's decree on its activity termination.


title: Communist Party of Ukraine

text: The party decided to take part in the October 2015 local elections as part of the umbrella party Left Opposition. According to the Interior Ministry, this was legal as long as the new party did not use communist symbols. Other party members took part in this election as Nova Derzhava. The political party Nova Derzhava was established in 2012. On 1 August, it elected a new leader Oleh Melnyk. Formally along with the Communist Party, it is also a member of the Left Opposition Association.


title: Communist Party of Ukraine

text: In late 2015, 19 local party leaders from the party's South and East Ukraine organizations resigned from the central committee to protest against the repression of internal dissent they blamed on Symonenko.


title: Communist Party of Ukraine

text: On 16 December, the District Administrative Court in Kyiv satisfied the claim of the Ministry of Justice and banned the activities of the Communist Party. This ban was criticized by John Dalhuisen of Amnesty International, who stated that the ban was "the same style of draconian measures used to stifle dissent” as used by the Soviet Union. On 28 December 2015, the party appealed. On 25 January 2016, the Supreme Administrative Court of Ukraine denied the party in the consideration of the cassation of the (16 December 2015) ban. The court suspended the appeal for the time being until the Constitutional Court determines the legitimacy of the law on decommunization. This resulted in the court's decision to ban the Communist Party did not come into force. Nevertheless, the party appealed its ban at the European Court of Human Rights. The April 2015 decommunization law does contain a norm that allows the Ministry of Justice to prohibit the Communist Party from participating in elections. The attempts to ban the party never did forbid individual members of the party to take part in elections as an independent candidate.


title: Communist Party of Ukraine

text: In January 2017, the National Agency for Prevention of Corruption stated that since the Communist Party is not officially banned, it must report on its property and finances. Hence, the party still sends in its required financial reports and is still listed on the website of the Ministry of Justice and the website of the Department of State Registration and Notary. Due to the decommunization laws, the party changed its logo and the name which is since been written in a shortened version. In the summer of 2018, the parties website was closed by the police "due to the demonstration of communist symbols". According to party leader Symonenko, the reason for the website closure was a photo on the website of the first secretary of the Central Committee of the Communist Party of Ukraine Volodymyr Shcherbytsky.


title: Communist Party of Ukraine

text: In February 2019, the Central Election Commission of Ukraine refused to register the candidacy of Symonenko for the 2019 Ukrainian presidential election due to the fact that the statute, name and symbolism of the Communist Party did not comply with the 2015 decommunization laws.


title: Communist Party of Ukraine

text: In June 2019, the Central Election refused party members to register for the July 2019 Ukrainian parliamentary election.


title: Communist Party of Ukraine

text: According to a Kyiv Polytechnic professor, who published an article in "The Guardian", the party came into conflict with the Ukrainian government after the Revolution of Dignity due to prominent displays of support for ousted Ukrainian President Viktor Yanukovych during the Euromaidan protests and alleged involvement with the separatist movement in Donbas as well as the party's pro-Russian government agenda. However, the party did vote in favour of the impeachment of Yanukovych. Two days after the Ukrainian parliament changed its regulations regarding the required size of parliamentary groups, the Communist Party faction was dissolved on 24 July 2014.


title: Communist Party of Ukraine

text: According to political scientist Tadeusz A. Olszański, the party "effectively supports the separatist rebellion" during the Russo-Ukrainian War.


title: Communist Party of Ukraine

text: Explaining the withdrawal of the status of political party from the KPU and 2 of its satellites, the secretary of state security and defense Oleksandr Turchynov stated in July 2015 that the communist party took a treacherous position from the very first days of Russian aggression and acted as its Fifth Column.


title: Communist Party of Ukraine

text: On 6 July 2022, the KPU was again banned after a Lviv court ruling which turned over all its assets, including party buildings and funds, to the Ukrainian state. In a statement, the Eighth Administrative Appeal Court said that it had satisfied the claims of the Ministry of Justice and ordered the party's closure. "The activity of the Communist Party of Ukraine is prohibited; the property, funds and other assets of the party, its regional, city, district organisations, primary centres and other structural entities have been transferred to the state." During the 2022 Russian invasion of Ukraine, the party was reported to have taken a pro-Russian stance, and the party's leader Petro Symonenko in March had fled to Belarus with the assistance of Russian forces during the Kyiv offensive.


title: Communist Party of Ukraine

text: In its statute, the Communist Party claims that "on voluntary basis it unites citizens of Ukraine who are supporters of the Communist idea". The party considers itself a successor of the Communist Party of Ukraine of the Soviet Union and calls itself a "battle detachment of RKP(b)–VKP(b)–KPSS". The party claims that prohibition of that party in August 1991 was unlawful, which was confirmed by the decision of the Constitutional Court of Ukraine on 27 December 2001. The party sets itself in an opposition to any government and seeks a full restoration of the socialist state in the country without any particular association with any other political parties.


title: Communist Party of Ukraine

text: The KPU was established as "the inheritor of the ideas and traditions of the KPU, as it existed until its banning in August 1991". In general, the party has laid weight on nostalgia for the Soviet Union to gain votes. In contrast to many parts of the former Soviet Union where leftist conservatives have tried to win votes by promoting local nationalism, the KPU supports a form of Soviet nationalism, considering the establishment of an independent Ukraine as illegal. The party has remained loyal to the legacy of the Soviet Union. In 1998, to celebrate the would-be 80th anniversary of the Soviet Union it published "Historical Thesis", a text which painted a rosy picture of the former state. The Soviet Union is barely criticized and controversial events such as the Great Purge and Holodomor are not mentioned in the party press. There are some who are favorable to Joseph Stalin's legacy, giving the impression that things "only began to go wrong with [Nikita] Khrushchev's 'adventurism'". Despite all this, when the Soviet Union and the Communist Party of the Soviet Union (CPSU) is criticized at all, the favored line is that the party and state lost their belief in key Leninist principles. Vladimir Lenin, the founder of the Soviet Union, "is still considered sacrosanct" by the party and official pronouncements talk of the "Leninist Communist Party of Ukraine" and more precisely that the KPU continues "speaking in the words of Lenin".


title: Communist Party of Ukraine

text: Symonenko has criticized the label of conservative on the KPU, stating that the party is not willing to abandon its own history. He has referred to the dissolution of the Soviet Union as "the tragic events of the recent past". Further, the KPU believes the Soviet Union "was criminally destroyed". The party believes that Ukraine has been living off the legacy of the Soviet Union since its independence. However, certain concessions to the present have been made and at the 2nd KPU Congress it was stated that "it would be utopian to try and revive a socio-economic system of different relations, which existed in different conditions, under different principles and different organizations of production and distribution, different social-class structures of society, a different level of consciousness".


title: Communist Party of Ukraine

text: The party adheres and believes in the Marxian concepts of class struggle and historical materialism. Their ongoing belief in historical materialism cements their views that the socialist mode of production will still be the society of the future. It could be said that the party believed stronger than ever in the possibility of a socialist future since the "careerists", symbolized by Mikhail Gorbachev, Boris Yeltsin and Leonid Kravchuk, were gone.


title: Communist Party of Ukraine

text: The KPU believes that since the West has developed into a post-industrial society, capitalism through globalization was actively "de-modernizing" Ukraine. This was in their favor since de-modernization would lead to the reestablishment of a dominant proletarian class. As Vasyl Tereshchuk, a former party theoretician expelled in 2005, noted: "People are surviving on what they accumulated in the years of Soviet power: that is, they are not yet a classic proletariat as they still have much to lose (a flat, a car, a dacha, etc.). But their full proletarianization will come sooner or later". Secondly, the dissolution of the Soviet Union directly led to the reestablishment of class antagonism in society. This antagonism led to the exploitation of the proletariat by "a comprador bourgeoisie ... behind which stands world imperialism headed by the USA". According to Symonenko, on this basis there was no chance for a social democratic movement ever to develop in Ukraine. The "softening of class antagonism in the West" which had led to the establishment of social democratic parties "was only possible because the local working class, as part of the 'golden billion', lived 'as parasites on the labour of the countries of the world periphery' to which Ukraine was rapidly being consigned. Ukraine could not expect any 'lessening of class


title: Communist Party of Ukraine

text: antagonism, only the reverse". Symonenko appreciates the economical aid and partnership with China and calls to use Communist Party of China as the example, giving the country back to the working people, and "build our country into a strong country like China".


title: Communist Party of Ukraine

text: At least in the beginning, the party is best described as Soviet patriotic. As Yurii Solomatin, a member of parliament, noted in 2000, "we are Soviet communists; we are Soviet people; we are Soviet patriots". The party continues speaking about the existence of a "Soviet people" and "Soviet homeland" and at the beginning no concessions were given to local Ukrainian nationalism. There has been no talk of establishing a national communism unique to Ukraine and the 1st KPU Congress even criticized the notion of establishing a unique "Ukrainian communism". Instead, the KPU has opted promoting Ukraine as a "bi-cultural state". At the 1st KPU Congress, Symonenko told the delegates that "'the interests, rights and specific traits of one nation above those of other nations and nationalities', and in which 'the Ukrainian language' should not be 'over'-privileged, but left alone to enjoy 'its natural development, purged of the imposed language of the diaspora. The Russian language, as the native language of half the population of Ukraine, [should be given] the status of a state language alongside Ukrainian". Their views on patriotism is highly nostalgic. When the Union of Communist Parties – Communist Party of the Soviet Union (UCP–CPSU), a loose organization of post-Soviet parties was formed, it was met with open arms. However, when the Communist Party of the Russian Federation proposed in 1995 to transform the organization into a modern-day Comintern, the KPU opposed because of their Soviet patriotic views.


title: Communist Party of Ukraine

text: In recent years, their commitment to Soviet patriotism has been partially replaced with a vaguer Eurasianism. Wishing not to reestablish a union with Russia "'as a protectorate of the Russian bourgeoisie", "the Ukrainian Communists have rediscovered the natural link from Soviet to East Slavic or Eurasian nationalism in the supposed common 'economic civilization' and proclivity for collective labour of all the East Slavic peoples". As noted in the party journal "Communist", the "'Soviet man ... did not emerge from nothing before him stood the courageous Slavic-Rusich, the labour-loving Ukrainian peasant, the self-sacrificing Cossack". At the 4th KPU Congress, the party conceded that Ukraine would not join any particular union as long as it weakened the country's sovereignty. At the same time, Petro Symonenko publicly backed Ukraine's membership in the Eurasian Customs Union.


title: Communist Party of Ukraine

text: Symonenko has often been referred to as a Ukrainophobe. Symonenko made controversy in 2007 when he accused the Ukrainian nationalist figure Roman Shukhevych of receiving two Iron Crosses from Adolf Hitler. Shukhevych's kids submitted a lawsuit against Symonenko in response. The Pechersk District Court of Kyiv city declared that Symonenko failed to present any proof of his claim and obligated "to refute the false information he spread about Roman Shukhevych at the next plenary session of the Verkhovna Rada of Ukraine after the court's decision enters into force".


title: Communist Party of Ukraine

text: Writing on "The Guardian", Ukrainian sociologist Volodymyr Ishchenko described the KPU as a "conservative and pro-Russian group", whose leaders "became a part of the bourgeois elite and invited business support for their cause", pointing out that the richest deputy of the 7th Ukrainian Verkhovna Rada (Oksana Kaletnik) was a member of the Communist faction. Thus, according to Ishchenko "the only things the party has in common with the determined Bolshevik revolutionaries of the past who spared neither themselves nor others are devotion to the Soviet symbols and appeals to empty “Marxist-Leninist” phrases".


title: Communist Party of Ukraine

text: After the start of Euromaidan and the Revolution of Dignity, the party newspaper "Komunist" published an article comparing the protests to riots in Black ghettoes in the United States during the 1960s; the article, titled "White on the outside, black on the inside", stated that "at least in New York, Los Angeles and San Francisco the police sometimes make raids on such places and simply kill a few rabid Negroes. […] Even the dark-skinned vendors in Kyiv second hand shops seem a bit more civilized than our ‘light-skinned brothers’ from the western regions of the country, who have gathered on the Maidan". The article was widely condemned as racist.


title: Path integral formulation

text: The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.


title: Path integral formulation

text: This formulation has proven crucial to the subsequent development of theoretical physics, because manifest Lorentz covariance (time and space components of quantities enter equations in the same way) is easier to achieve than in the operator formalism of canonical quantization. Unlike previous methods, the path integral allows one to easily change coordinates between very different canonical descriptions of the same quantum system. Another advantage is that it is in practice easier to guess the correct form of the Lagrangian of a theory, which naturally enters the path integrals (for interactions of a certain type, these are "coordinate space" or "Feynman path integrals"), than the Hamiltonian. Possible downsides of the approach include that unitarity (this is related to conservation of probability; the probabilities of all physically possible outcomes must add up to one) of the S-matrix is obscure in the formulation. The path-integral approach has proven to be equivalent to the other formalisms of quantum mechanics and quantum field theory. Thus, by "deriving" either approach from the other, problems associated with one or the other approach (as exemplified by Lorentz covariance or unitarity) go away.


title: Path integral formulation

text: The path integral also relates quantum and stochastic processes, and this provided the basis for the grand synthesis of the 1970s, which unified quantum field theory with the statistical field theory of a fluctuating field near a second-order phase transition. The Schrödinger equation is a diffusion equation with an imaginary diffusion constant, and the path integral is an analytic continuation of a method for summing up all possible random walks.


title: Path integral formulation

text: The basic idea of the path integral formulation can be traced back to Norbert Wiener, who introduced the Wiener integral for solving problems in diffusion and Brownian motion. This idea was extended to the use of the Lagrangian in quantum mechanics by Paul Dirac in his 1933 article. The complete method was developed in 1948 by Richard Feynman. Some preliminaries were worked out earlier in his doctoral work under the supervision of John Archibald Wheeler. The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the Wheeler–Feynman absorber theory using a Lagrangian (rather than a Hamiltonian) as a starting point.


title: Path integral formulation

text: In quantum mechanics, as in classical mechanics, the Hamiltonian is the generator of time translations. This means that the state at a slightly later time differs from the state at the current time by the result of acting with the Hamiltonian operator (multiplied by the negative imaginary unit, ). For states with a definite energy, this is a statement of the de Broglie relation between frequency and energy, and the general relation is consistent with that plus the superposition principle.


title: Path integral formulation

text: The Hamiltonian in classical mechanics is derived from a Lagrangian, which is a more fundamental quantity relative to special relativity. The Hamiltonian indicates how to march forward in time, but the time is different in different reference frames. The Lagrangian is a Lorentz scalar, while the Hamiltonian is the time component of a four-vector. So the Hamiltonian is different in different frames, and this type of symmetry is not apparent in the original formulation of quantum mechanics.


title: Path integral formulation

text: The Hamiltonian is a function of the position and momentum at one time, and it determines the position and momentum a little later. The Lagrangian is a function of the position now and the position a little later (or, equivalently for infinitesimal time separations, it is a function of the position and velocity). The relation between the two is by a Legendre transformation, and the condition that determines the classical equations of motion (the Euler–Lagrange equations) is that the action has an extremum.


title: Path integral formulation

text: In quantum mechanics, the Legendre transform is hard to interpret, because the motion is not over a definite trajectory. In classical mechanics, with discretization in time, the Legendre transform becomes


title: Path integral formulation

text: where the partial derivative with respect to formula_3 holds fixed. The inverse Legendre transform is


title: Path integral formulation

text: In quantum mechanics, the state is a superposition of different states with different values of , or different values of , and the quantities and can be interpreted as noncommuting operators. The operator is only definite on states that are indefinite with respect to . So consider two states separated in time and act with the operator corresponding to the Lagrangian:


title: Path integral formulation

text: If the multiplications implicit in this formula are reinterpreted as "matrix" multiplications, the first factor is


title: Path integral formulation

text: and if this is also interpreted as a matrix multiplication, the sum over all states integrates over all , and so it takes the Fourier transform in to change basis to . That is the action on the Hilbert space – change basis to at time .


title: Path integral formulation

text: This is not very different from just ordinary time evolution: the factor contains all the dynamical information – it pushes the state forward in time. The first part and the last part are just Fourier transforms to change to a pure basis from an intermediate basis.


title: Path integral formulation

text: Another way of saying this is that since the Hamiltonian is naturally a function of and , exponentiating this quantity and changing basis from to at each step allows the matrix element of to be expressed as a simple function along each path. This function is the quantum analog of the classical action. This observation is due to Paul Dirac.


title: Path integral formulation

text: and this gives the time-evolution operator between time and time . While in the representation the quantity that is being summed over the intermediate states is an obscure matrix element, in the representation it is reinterpreted as a quantity associated to the path. In the limit that one takes a large power of this operator, one reconstructs the full quantum evolution between two states, the early one with a fixed value of and the later one with a fixed value of . The result is a sum over paths with a phase, which is the quantum action. Crucially, Dirac identified in this article the deep quantum-mechanical reason for the principle of least action controlling the classical limit (see quotation box).


title: Path integral formulation

text: Dirac's work did not provide a precise prescription to calculate the sum over paths, and he did not show that one could recover the Schrödinger equation or the canonical commutation relations from this rule. This was done by Feynman. That is, the classical path arises naturally in the classical limit.


title: Path integral formulation

text: Feynman showed that Dirac's quantum action was, for most cases of interest, simply equal to the classical action, appropriately discretized. This means that the classical action is the phase acquired by quantum evolution between two fixed endpoints. He proposed to recover all of quantum mechanics from the following postulates:


title: Path integral formulation

text: In order to find the overall probability amplitude for a given process, then, one adds up, or integrates, the amplitude of the 3rd postulate over the space of "all" possible paths of the system in between the initial and final states, including those that are absurd by classical standards. In calculating the probability amplitude for a single particle to go from one space-time coordinate to another, it is correct to include paths in which the particle describes elaborate curlicues, curves in which the particle shoots off into outer space and flies back again, and so forth. The path integral assigns to all these amplitudes "equal weight" but varying phase, or argument of the complex number. Contributions from paths wildly different from the classical trajectory may be suppressed by interference (see below).


title: Path integral formulation

text: Feynman showed that this formulation of quantum mechanics is equivalent to the canonical approach to quantum mechanics when the Hamiltonian is at most quadratic in the momentum. An amplitude computed according to Feynman's principles will also obey the Schrödinger equation for the Hamiltonian corresponding to the given action.


title: Path integral formulation

text: The path integral formulation of quantum field theory represents the transition amplitude (corresponding to the classical correlation function) as a weighted sum of all possible histories of the system from the initial to the final state. A Feynman diagram is a graphical representation of a perturbative contribution to the transition amplitude.


title: Path integral formulation

text: One common approach to deriving the path integral formula is to divide the time interval into small pieces. Once this is done, the Trotter product formula tells us that the noncommutativity of the kinetic and potential energy operators can be ignored.


title: Path integral formulation

text: For a particle in a smooth potential, the path integral is approximated by zigzag paths, which in one dimension is a product of ordinary integrals. For the motion of the particle from position at time to at time , the time sequence


title: Path integral formulation

text: where is the Lagrangian of the one-dimensional system with position variable and velocity considered (see below), and corresponds to the position at the th time step, if the time integral is approximated by a sum of terms.


title: Path integral formulation

text: In the limit , this becomes a functional integral, which, apart from a nonessential factor, is directly the product of the probability amplitudes (more precisely, since one must work with a continuous spectrum, the respective densities) to find the quantum mechanical particle at in the initial state and at in the final state .


title: Path integral formulation

text: in the Riemann sum approximating the time integral, which are finally integrated over to with the integration measure , is an arbitrary value of the interval corresponding to , e.g. its center, .


title: Path integral formulation

text: Thus, in contrast to classical mechanics, not only does the stationary path contribute, but actually all virtual paths between the initial and the final point also contribute.


title: Path integral formulation

text: In terms of the wave function in the position representation, the path integral formula reads as follows:


title: Path integral formulation

text: where formula_17 denotes integration over all paths formula_18 with formula_19 and where formula_20 is a normalization factor. Here formula_21 is the action, given by


title: Path integral formulation

text: The path integral representation gives the quantum amplitude to go from point to point as an integral over all paths. For a free-particle action (for simplicity let , )


title: Path integral formulation

text: To do this, it is convenient to start without the factor in the exponential, so that large deviations are suppressed by small numbers, not by cancelling oscillatory contributions. The amplitude (or Kernel) reads:


title: Path integral formulation

text: where the is interpreted as a finite collection of integrations at each integer multiple of . Each factor in the product is a Gaussian as a function of centered at with variance . The multiple integrals are a repeated convolution of this Gaussian with copies of itself at adjacent times:


title: Path integral formulation

text: where the number of convolutions is . The result is easy to evaluate by taking the Fourier transform of both sides, so that the convolutions become multiplications:


title: Path integral formulation

text: The proportionality constant is not really determined by the time-slicing approach, only the ratio of values for different endpoint choices is determined. The proportionality constant should be chosen to ensure that between each two time slices the time evolution is quantum-mechanically unitary, but a more illuminating way to fix the normalization is to consider the path integral as a description of a stochastic process.


title: Path integral formulation

text: The result has a probability interpretation. The sum over all paths of the exponential factor can be seen as the sum over each path of the probability of selecting that path. The probability is the product over each segment of the probability of selecting that segment, so that each segment is probabilistically independently chosen. The fact that the answer is a Gaussian spreading linearly in time is the central limit theorem, which can be interpreted as the first historical evaluation of a statistical path integral.


title: Path integral formulation

text: The probability interpretation gives a natural normalization choice. The path integral should be defined so that


title: Path integral formulation

text: For oscillatory path integrals, ones with an in the numerator, the time slicing produces convolved Gaussians, just as before. Now, however, the convolution product is marginally singular, since it requires careful limits to evaluate the oscillating integrals. To make the factors well defined, the easiest way is to add a small imaginary part to the time increment . This is closely related to Wick rotation. Then the same convolution argument as before gives the propagation kernel:


title: Path integral formulation

text: which, with the same normalization as before (not the sum-squares normalization – this function has a divergent norm), obeys a free Schrödinger equation:


title: Path integral formulation

text: Write its trajectory as the classical trajectory plus some perturbation, and the action as . The classical trajectory can be written as


title: Path integral formulation

text: Next, expand the deviation from the classical path as a Fourier series, and calculate the contribution to the action , which gives


title: Path integral formulation

text: One may finally expand in powers of : All terms in this expansion get multiplied by the factor in the front, yielding terms of the form


title: Path integral formulation

text: Comparison to the above eigenstate expansion yields the standard energy spectrum for the simple harmonic oscillator,


title: Path integral formulation

text: Feynman's time-sliced approximation does not, however, exist for the most important quantum-mechanical path integrals of atoms, due to the singularity of the Coulomb potential at the origin. Only after replacing the time by another path-dependent pseudo-time parameter


title: Path integral formulation

text: the singularity is removed and a time-sliced approximation exists, which is exactly integrable, since it can be made harmonic by a simple coordinate transformation, as discovered in 1979 by İsmail Hakkı Duru and Hagen Kleinert. The combination of a path-dependent time transformation and a coordinate transformation is an important tool to solve many path integrals and is called generically the Duru–Kleinert transformation.


title: Path integral formulation

text: The path integral reproduces the Schrödinger equation for the initial and final state even when a potential is present. This is easiest to see by taking a path-integral over infinitesimally separated times.


title: Path integral formulation

text: Since the time separation is infinitesimal and the cancelling oscillations become severe for large values of , the path integral has most weight for close to . In this case, to lowest order the potential energy is constant, and only the kinetic energy contribution is nontrivial. (This separation of the kinetic and potential energy terms in the exponent is essentially the Trotter product formula.) The exponential of the action is


title: Path integral formulation

text: The first term rotates the phase of locally by an amount proportional to the potential energy. The second term is the free particle propagator, corresponding to times a diffusion process. To lowest order in they are additive; in any case one has with (1):


title: Path integral formulation

text: As mentioned, the spread in is diffusive from the free particle propagation, with an extra infinitesimal rotation in phase which slowly varies from point to point from the potential:


title: Path integral formulation

text: and this is the Schrödinger equation. The normalization of the path integral needs to be fixed in exactly the same way as in the free particle case. An arbitrary continuous potential does not affect the normalization, although singular potentials require careful treatment.


title: Path integral formulation

text: Since the states obey the Schrödinger equation, the path integral must reproduce the Heisenberg equations of motion for the averages of and variables, but it is instructive to see this directly. The direct approach shows that the expectation values calculated from the path integral reproduce the usual ones of quantum mechanics.


title: Path integral formulation

text: Now at each separate time is a separate integration variable. So it is legitimate to change variables in the integral by shifting: where is a different shift at each time but , since the endpoints are not integrated:


title: Path integral formulation

text: But this was just a shift of integration variables, which doesn't change the value of the integral for any choice of . The conclusion is that this first order variation is zero for an arbitrary initial state and at any arbitrary point in time:


title: Path integral formulation

text: If the action contains terms which multiply and , at the same moment in time, the manipulations above are only heuristic, because the multiplication rules for these quantities is just as noncommuting in the path integral as it is in the operator formalism.


title: Path integral formulation

text: If the variation in the action exceeds by many orders of magnitude, we typically have destructive interference other than in the vicinity of those trajectories satisfying the Euler–Lagrange equation, which is now reinterpreted as the condition for constructive interference. This can be shown using the method of stationary phase applied to the propagator. As decreases, the exponential in the integral oscillates rapidly in the complex domain for any change in the action. Thus, in the limit that goes to zero, only points where the classical action does not vary contribute to the propagator.


title: Path integral formulation

text: The formulation of the path integral does not make it clear at first sight that the quantities and do not commute. In the path integral, these are just integration variables and they have no obvious ordering. Feynman discovered that the non-commutativity is still present.


title: Path integral formulation

text: To see this, consider the simplest path integral, the brownian walk. This is not yet quantum mechanics, so in the path-integral the action is not multiplied by :


title: Path integral formulation

text: This shows that the random walk is not differentiable, since the ratio that defines the derivative diverges with probability one.


title: Path integral formulation

text: In elementary calculus, the two are only different by an amount which goes to 0 as goes to 0. But in this case, the difference between the two is not 0:


title: Path integral formulation

text: Then is a rapidly fluctuating statistical quantity, whose average value is 1, i.e. a normalized "Gaussian process". The fluctuations of such a quantity can be described by a statistical Lagrangian


title: Path integral formulation

text: and the equations of motion for derived from extremizing the action corresponding to just set it equal to 1. In physics, such a quantity is "equal to 1 as an operator identity". In mathematics, it "weakly converges to 1". In either case, it is 1 in any expectation value, or when averaged over any interval, or for all practical purpose.


title: Path integral formulation

text: This is called the Itō lemma in stochastic calculus, and the (euclideanized) canonical commutation relations in physics.


title: Path integral formulation

text: and in quantum mechanics, the extra imaginary unit in the action converts this to the canonical commutation relation,


title: Path integral formulation

text: For a particle in curved space the kinetic term depends on the position, and the above time slicing cannot be applied, this being a manifestation of the notorious operator ordering problem in Schrödinger quantum mechanics. One may, however, solve this problem by transforming the time-sliced flat-space path integral to curved space using a multivalued coordinate transformation (nonholonomic mapping explained here).


title: Path integral formulation

text: Sometimes (e.g. a particle moving in curved space) we also have measure-theoretic factors in the functional integral:


title: Path integral formulation

text: then it means that each spatial slice is multiplied by the measure . This measure cannot be expressed as a functional multiplying the measure because they belong to entirely different classes.


title: Path integral formulation

text: It is very common in path integrals to perform a Wick rotation from real to imaginary times. In the setting of quantum field theory, the Wick rotation changes the geometry of space-time from Lorentzian to Euclidean; as a result, Wick-rotated path integrals are often called Euclidean path integrals.


title: Path integral formulation

text: If we replace formula_77 by formula_78, the time-evolution operator formula_79 is replaced by formula_80. (This change is known as a Wick rotation.) If we repeat the derivation of the path-integral formula in this setting, we obtain


title: Path integral formulation

text: Note the sign change between this and the normal action, where the potential energy term is negative. (The term "Euclidean" is from the context of quantum field theory, where the change from real to imaginary time changes the space-time geometry from Lorentzian to Euclidean.)


title: Path integral formulation

text: where formula_85 includes all the remaining dependence of the integrand on the path. This integral has a rigorous mathematical interpretation as integration against the Wiener measure, denoted formula_86. The Wiener measure, constructed by Norbert Wiener gives a rigorous foundation to Einstein's mathematical model of Brownian motion. The subscript formula_87 indicates that the measure formula_88 is supported on paths formula_18 with formula_19.


title: Path integral formulation

text: Although the Wick-rotated Schrödinger equation does not have a direct physical meaning, interesting properties of the Schrödinger operator formula_94 can be extracted by studying it.


title: Path integral formulation

text: Much of the study of quantum field theories from the path-integral perspective, in both the mathematics and physics literatures, is done in the Euclidean setting, that is, after a Wick rotation. In particular, there are various results showing that if a Euclidean field theory with suitable properties can be constructed, one can then undo the Wick rotation to recover the physical, Lorentzian theory. On the other hand, it is much more difficult to give a meaning to path integrals (even Euclidean path integrals) in quantum field theory than in quantum mechanics.


title: Path integral formulation

text: The path integral is just the generalization of the integral above to all quantum mechanical problems—


title: Path integral formulation

text: is the action of the classical problem in which one investigates the path starting at time and ending at time , and formula_17 denotes the integration measure over all paths. In the classical limit, formula_97, the path of minimum action dominates the integral, because the phase of any path away from this fluctuates rapidly and different contributions cancel.


title: Path integral formulation

text: The connection with statistical mechanics follows. Considering only paths which begin and end in the same configuration, perform the Wick rotation , i.e., make time imaginary, and integrate over all possible beginning-ending configurations. The Wick-rotated path integral—described in the previous subsection, with the ordinary action replaced by its "Euclidean" counterpart—now resembles the partition function of statistical mechanics defined in a canonical ensemble with inverse temperature proportional to imaginary time, . Strictly speaking, though, this is the partition function for a statistical field theory.


title: Path integral formulation

text: Clearly, such a deep analogy between quantum mechanics and statistical mechanics cannot be dependent on the formulation. In the canonical formulation, one sees that the unitary evolution operator of a state is given by


title: Path integral formulation

text: where the state is evolved from time . If one makes a Wick rotation here, and finds the amplitude to go from any state, back to the same state in (imaginary) time is given by


title: Path integral formulation

text: which is precisely the partition function of statistical mechanics for the same system at the temperature quoted earlier. One aspect of this equivalence was also known to Erwin Schrödinger who remarked that the equation named after him looked like the diffusion equation after Wick rotation. Note, however, that the Euclidean path integral is actually in the form of a "classical" statistical mechanics model.


title: Path integral formulation

text: Both the Schrödinger and Heisenberg approaches to quantum mechanics single out time and are not in the spirit of relativity. For example, the Heisenberg approach requires that scalar field operators obey the commutation relation


title: Path integral formulation

text: for two simultaneous spatial positions and , and this is not a relativistically invariant concept. The results of a calculation "are" covariant, but the symmetry is not apparent in intermediate stages. If naive field-theory calculations did not produce infinite answers in the continuum limit, this would not have been such a big problem – it would just have been a bad choice of coordinates. But the lack of symmetry means that the infinite quantities must be cut off, and the bad coordinates make it nearly impossible to cut off the theory without spoiling the symmetry. This makes it difficult to extract the physical predictions, which require a careful limiting procedure.


title: Path integral formulation

text: The problem of lost symmetry also appears in classical mechanics, where the Hamiltonian formulation also superficially singles out time. The Lagrangian formulation makes the relativistic invariance apparent. In the same way, the path integral is manifestly relativistic. It reproduces the Schrödinger equation, the Heisenberg equations of motion, and the canonical commutation relations and shows that they are compatible with relativity. It extends the Heisenberg-type operator algebra to operator product rules, which are new relations difficult to see in the old formalism.


title: Path integral formulation

text: Further, different choices of canonical variables lead to very different-seeming formulations of the same theory. The transformations between the variables can be very complicated, but the path integral makes them into reasonably straightforward changes of integration variables. For these reasons, the Feynman path integral has made earlier formalisms largely obsolete.


title: Path integral formulation

text: The price of a path integral representation is that the unitarity of a theory is no longer self-evident, but it can be proven by changing variables to some canonical representation. The path integral itself also deals with larger mathematical spaces than is usual, which requires more careful mathematics, not all of which has been fully worked out. The path integral historically was not immediately accepted, partly because it took many years to incorporate fermions properly. This required physicists to invent an entirely new mathematical object – the Grassmann variable – which also allowed changes of variables to be done naturally, as well as allowing constrained quantization.


title: Path integral formulation

text: The integration variables in the path integral are subtly non-commuting. The value of the product of two field operators at what looks like the same point depends on how the two points are ordered in space and time. This makes some naive identities fail.


title: Path integral formulation

text: In relativistic theories, there is both a particle and field representation for every theory. The field representation is a sum over all field configurations, and the particle representation is a sum over different particle paths.


title: Path integral formulation

text: The nonrelativistic formulation is traditionally given in terms of particle paths, not fields. There, the path integral in the usual variables, with fixed boundary conditions, gives the probability amplitude for a particle to go from point to point in time :


title: Path integral formulation

text: This is called the propagator. Superposing different values of the initial position with an arbitrary initial state constructs the final state:


title: Path integral formulation

text: For a spatially homogeneous system, where is only a function of , the integral is a convolution, the final state is the initial state convolved with the propagator:


title: Path integral formulation

text: For a free particle of mass , the propagator can be evaluated either explicitly from the path integral or by noting that the Schrödinger equation is a diffusion equation in imaginary time, and the solution must be a normalized Gaussian:


title: Path integral formulation

text: and in -space the proportionality factor here is constant in time, as will be verified in a moment. The Fourier transform in time, extending to be zero for negative times, gives Green's function, or the frequency-space propagator:


title: Path integral formulation

text: which is the reciprocal of the operator that annihilates the wavefunction in the Schrödinger equation, which wouldn't have come out right if the proportionality factor weren't constant in the -space representation.


title: Path integral formulation

text: The infinitesimal term in the denominator is a small positive number, which guarantees that the inverse Fourier transform in will be nonzero only for future times. For past times, the inverse Fourier transform contour closes toward values of where there is no singularity. This guarantees that propagates the particle into the future and is the reason for the subscript "F" on . The infinitesimal term can be interpreted as an infinitesimal rotation toward imaginary time.


title: Path integral formulation

text: It is also possible to reexpress the nonrelativistic time evolution in terms of propagators going toward the past, since the Schrödinger equation is time-reversible. The past propagator is the same as the future propagator except for the obvious difference that it vanishes in the future, and in the Gaussian is replaced by . In this case, the interpretation is that these are the quantities to convolve the final wavefunction so as to get the initial wavefunction:


title: Path integral formulation

text: Given the nearly identical only change is the sign of and , the parameter in Green's function can either be the energy if the paths are going toward the future, or the negative of the energy if the paths are going toward the past.


title: Path integral formulation

text: For a nonrelativistic theory, the time as measured along the path of a moving particle and the time as measured by an outside observer are the same. In relativity, this is no longer true. For a relativistic theory the propagator should be defined as the sum over all paths that travel between two points in a fixed proper time, as measured along the path (these paths describe the trajectory of a particle in space and in time):


title: Path integral formulation

text: The usual definition of the relativistic propagator only asks for the amplitude is to travel from to , after summing over all the possible proper times it could take:


title: Path integral formulation

text: where is a weight factor, the relative importance of paths of different proper time. By the translation symmetry in proper time, this weight can only be an exponential factor and can be absorbed into the constant :


title: Path integral formulation

text: This is the Schwinger representation. Taking a Fourier transform over the variable can be done for each value of separately, and because each separate contribution is a Gaussian, gives whose Fourier transform is another Gaussian with reciprocal width. So in -space, the propagator can be reexpressed simply:


title: Path integral formulation

text: which is the Euclidean propagator for a scalar particle. Rotating to be imaginary gives the usual relativistic propagator, up to a factor of and an ambiguity, which will be clarified below:


title: Path integral formulation

text: This expression can be interpreted in the nonrelativistic limit, where it is convenient to split it by partial fractions:


title: Path integral formulation

text: For states where one nonrelativistic particle is present, the initial wavefunction has a frequency distribution concentrated near . When convolving with the propagator, which in space just means multiplying by the propagator, the second term is suppressed and the first term is enhanced. For frequencies near , the dominant first term has the form


title: Path integral formulation

text: The second term has a nonrelativistic limit also, but this limit is concentrated on frequencies that are negative. The second pole is dominated by contributions from paths where the proper time and the coordinate time are ticking in an opposite sense, which means that the second term is to be interpreted as the antiparticle. The nonrelativistic analysis shows that with this form the antiparticle still has positive energy.


title: Path integral formulation

text: The proper way to express this mathematically is that, adding a small suppression factor in proper time, the limit where of the first term must vanish, while the limit of the second term must vanish. In the Fourier transform, this means shifting the pole in slightly, so that the inverse Fourier transform will pick up a small decay factor in one of the time directions:


title: Path integral formulation

text: Without these terms, the pole contribution could not be unambiguously evaluated when taking the inverse Fourier transform of . The terms can be recombined:


title: Path integral formulation

text: which when factored, produces opposite-sign infinitesimal terms in each factor. This is the mathematically precise form of the relativistic particle propagator, free of any ambiguities. The term introduces a small imaginary part to the , which in the Minkowski version is a small exponential suppression of long paths.


title: Path integral formulation

text: 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.


title: Path integral formulation

text: 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.


title: Path integral formulation

text: 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. The action is referred to technically as a functional of the field: , where the field is itself a function of space and time, and the square brackets are a reminder that the action depends on all the field's values everywhere, not just some particular value. "One" such given function of spacetime is called a "field configuration". In principle, one integrates Feynman's amplitude over the class of all possible field configurations.


title: Path integral formulation

text: 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.


title: Path integral formulation

text: 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 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.


title: Path integral formulation

text: In quantum field theory, if the action is given by the functional of field configurations (which only depends locally on the fields), then the time-ordered vacuum expectation value of polynomially bounded functional , , is given by


title: Path integral formulation

text: The symbol here is a concise way to represent the infinite-dimensional integral over all possible field configurations on all of space-time. As stated above, the unadorned path integral in the denominator ensures proper normalization.


title: Path integral formulation

text: Strictly speaking, the only question that can be asked in physics is: "What fraction of states satisfying condition also satisfy condition ?" The answer to this is a number between 0 and 1, which can be interpreted as a conditional probability, written as . In terms of path integration, since , this means


title: Path integral formulation

text: where the functional is the superposition of all incoming states that could lead to the states we are interested in. 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.


title: Path integral formulation

text: 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.


title: Path integral formulation

text: (the left-hand side is a functional derivative; the equation means that the action is stationary under small changes in the field configuration). The quantum analogues of these equations are called the Schwinger–Dyson equations.


title: Path integral formulation

text: If the functional measure turns out to be translationally invariant (we'll assume this for the rest of this article, although this does not hold for, let's say nonlinear sigma models), and if we assume that after a Wick rotation


title: Path integral formulation

text: for some , it goes to zero faster than a reciprocal of any polynomial for large values of , then we can integrate by parts (after a Wick rotation, followed by a Wick rotation back) to get the following Schwinger–Dyson equations for the expectation:


title: Path integral formulation

text: These equations are the analog of the on-shell EL equations. The time ordering is taken before the time derivatives inside the .


title: Path integral formulation

text: If (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 of the source fields is defined to be


title: Path integral formulation

text: Basically, if is viewed as a functional distribution (this shouldn't be taken too literally as an interpretation of QFT, unlike its Wick-rotated statistical mechanics analogue, because we have time ordering complications here!), then are its moments, and is its Fourier transform.


title: Path integral formulation

text: If is a functional of , then for an operator , is defined to be the operator that substitutes for . For example, if


title: Path integral formulation

text: If the functional measure is not translationally invariant, it might be possible to express it as the product , where is a functional and is a translationally invariant measure. This is true, for example, for nonlinear sigma models where the target space is diffeomorphic to . However, if the target manifold is some topologically nontrivial space, the concept of a translation does not even make any sense.


title: Path integral formulation

text: If we expand this equation as a Taylor series about "J" 0, we get the entire set of Schwinger–Dyson equations.


title: Path integral formulation

text: 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.


title: Path integral formulation

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


title: Path integral formulation

text: 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


title: Path integral formulation

text: If we don't assume any special boundary conditions, this would not be a "true" symmetry in the true sense of the term in general unless or something. Here, is a derivation which generates the one parameter group in question. We could have antiderivations as well, such as BRST and supersymmetry.


title: Path integral formulation

text: for any polynomially-bounded functional . This property is called the invariance of the measure. And this does not hold in general. See anomaly (physics) for more details.


title: Path integral formulation

text: (this is assuming the Lagrangian only depends on and its first partial derivatives! More general Lagrangians would require a modification to this definition!). We're not insisting that is the generator of a symmetry (i.e. we are "not" insisting upon the gauge principle), but just that is. And we also assume the even stronger assumption that the functional measure is locally invariant:


title: Path integral formulation

text: Now for the case where , we can forget about all the boundary conditions and locality assumptions. We'd simply have


title: Path integral formulation

text: 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.


title: Path integral formulation

text: 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. For example, the operator formula_147 can be translated back as either formula_148, formula_149, or formula_150 depending on whether one chooses the formula_151, formula_152, or Weyl ordering prescription; conversely, formula_150 can be translated to either formula_151, formula_152, or formula_147 for the same respective choice of ordering prescription.


title: Path integral formulation

text: 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 claim the interpretation explains the Einstein–Podolsky–Rosen paradox without resorting to nonlocality.


title: Path integral formulation

text: Some advocates of interpretations of quantum mechanics emphasizing decoherence have attempted to make more rigorous the notion of extracting a classical-like "coarse-grained" history from the space of all possible histories.


title: Path integral formulation

text: 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. Approaches that use this method include causal dynamical triangulations and spinfoam models.


title: Path integral formulation

text: Quantum tunnelling can be modeled by using the path integral formation to determine the action of the trajectory through a potential barrier. Using the WKB approximation, the tunneling rate () can be determined to be of the form


title: Path integral formulation

text: with the effective action and pre-exponential factor . 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. From this model, tunneling rates of macroscopic systems (at finite temperatures) can be predicted.


title: Benedict Taylor (musician)

text: Taylor works in contemporary composition, modern string performance and improvised music in the British and European new music world. The central focus of his work is in new composition for live performance, film, theatre, contemporary dance, art installation, and electroacoustic composition. In performance, he predominantly works within improvisation, new composition and 20th/21st century music.


title: Benedict Taylor (musician)

text: Within his performance work there is a focus on solo performance as a creative and investigative process. In 2013 he initiated an ongoing commission series and solo label Subverten (For Viola Solo). The first works premiered in autumn 2013 and continue several times each year.


title: Benedict Taylor (musician)

text: He has worked with a number of music organisations and ensembles including; clapTON Ensemble, re:sound, London Improvisers Orchestra, Berlin Improvisers Orchestra, Project Instrumental, Tokyo Improvisers Orchestra, Kammer Klang Players, Squib-Box, ARCO, London Sinfonietta Collective, Regular Music II.


title: Benedict Taylor (musician)

text: Notable festival, venue and radio appearances include; Jazz en Nord Festival France, Tete a Tete Opera Festival, Aldeburgh Festival, Cantiere Internazionale d'Arte – Montepulciano, Huddersfield Contemporary Music Festival, Fete de la Musique Berlin, Manchester International Festival, Manchester Jazz Festival, Cheltenham Festival, BBC Radio 3, Resonance FM, Radio Libertaire/Epsilonia (Paris).


title: Octavia the Younger

text: Octavia the Younger (; c. 66 BC – 11 BC) was the elder sister of the first Roman Emperor, Augustus (known also as Octavian), the half-sister of Octavia the Elder, and the fourth wife of Mark Antony. She was also the great-grandmother of the Emperor Caligula and Empress Agrippina the Younger, maternal grandmother of the Emperor Claudius, and paternal great-grandmother and maternal great-great-grandmother of the Emperor Nero.


title: Octavia the Younger

text: One of the most prominent women in Roman history, Octavia was respected and admired by contemporaries for her loyalty, nobility and humanity, as well as for maintaining traditional Roman feminine virtues.


title: Octavia the Younger

text: Octavia was born around 66 BC. Full sister to Augustus, Octavia was the only daughter born of Gaius Octavius' second marriage to Atia, niece of Julius Caesar. Octavia was born in Nola, present-day Italy; her father, a Roman governor and senator, died in 59 BC from natural causes. Her mother later remarried, to the consul Lucius Marcius Philippus. Octavia spent much of her childhood travelling with her parents. Marcius was in charge of educating Octavia and her brother Octavian, later known as Augustus.


title: Octavia the Younger

text: Some time before 54 BC, her stepfather arranged for her to marry Gaius Claudius Marcellus. Marcellus was a man of consular rank, a man who was considered worthy of her and was consul in 50 BC. He was also a member of the influential Claudian family and descended from Marcus Claudius Marcellus, a famous general in the Second Punic War. In 54 BC, Octavia's great-uncle Julius Caesar is said to have been anxious for her to divorce her husband so that she could marry Pompey, who had just lost his wife Julia (Caesar's daughter, and thus Octavia's cousin once removed). The couple did not want to get a divorce, so instead Pompey declined the proposal and married Cornelia Metella. Thus, Octavia's husband continued to oppose Julius Caesar, including in the crucial year of his consulship, 50 BC. Civil war broke out when Caesar invaded Italy from Gaul in 49 BC.


title: Octavia the Younger

text: Marcellus, a friend of Cicero, was an initial opponent of Julius Caesar when Caesar invaded Italy, but did not take up arms against his wife's great uncle at the Battle of Pharsalus, and was eventually pardoned by him. In 47 BC he was able to intercede with Caesar for his cousin and namesake, also a former consul, then living in exile. Presumably, Octavia continued to live with her husband from the time of their marriage (she would have been between 12 and 15 when they married) to her husband's death. They had three surviving children: Claudia Marcella Major, Claudia Marcella Minor and Marcus Claudius Marcellus. All three were born in Italy. Although according to the anonymous "Περὶ τοῦ καισαρείου γένους" Octavia bore Marcellus four sons and four daughters. Her husband Marcellus died in May 40 BC.


title: Octavia the Younger

text: By a Senatorial decree, Octavia married Mark Antony in October 40 BC, as his fourth wife (his third wife Fulvia having died shortly before). This marriage had to be approved by the Senate, as she was pregnant with her first husband's child, and was a politically motivated attempt to cement the uneasy alliance between her brother Octavian and Mark Antony; however, Octavia does appear to have been a loyal and faithful wife to Antony. Between 40 and 36 BC, she travelled with Antony to various provinces and lived with him in his Athenian mansion. There she raised her children by Marcellus as well as Antony's two sons; Antyllus and Iullus, as well as the two daughters of her marriage to Antony, Antonia Major and Antonia Minor who were born there.


title: Octavia the Younger

text: The alliance was severely tested by Antony's abandonment of Octavia and their children in favor of his former lover Queen Cleopatra VII of Egypt (Antony and Cleopatra had met in 41 BC, an interaction that resulted in Cleopatra bearing twins, Alexander Helios, a boy, and Cleopatra Selene, a girl). After 36 BC, Octavia returned to Rome with the daughters of her second marriage. On several occasions she acted as a political advisor and negotiator between her husband and brother. For example, in the spring of 37 BC, while pregnant with her daughter Antonia Minor, she was considered essential to an arms deal held at Tarentum, in which Antony and Augustus agreed to aid each other in their Parthian and Sicilian campaigns. She had won over her brother's advisers Agrippa and Maecenas and convinced him to renew their aliance. She was hailed as a "marvel of womankind." In 35 BC, after Antony suffered a disastrous campaign in Parthia, she brought fresh troops, provisions, and funds to Athens. There Antony had left a letter for her, instructing her to go no further. Mark Antony divorced Octavia in late 33 BC. In 33, Antony sent men to eject her from his house in Rome. She became sole caretaker of their children, except for Antyllus who was already with his father in the East. After Antony's suicide in 30 BC, her brother executed Antyllus but allowed Octavia to raise Antony's younger son Iullus by Fulvia as well as his children by Cleopatra (the two sons Alexander Helios and Ptolemy Philadelphus, and one daughter, Cleopatra Selene II).


title: Octavia the Younger

text: In 35 BC, Augustus accorded a number of honours and privileges to Octavia, and Augustus' wife Livia, previously unheard of for women in Rome. They were granted "sacrosanctitas," meaning it was illegal to verbally insult them. Previously, this had been only granted to tribunes. Livia and Octavia were made immune from "tutela," the male guardianship which all women in Rome except for the Vestal Virgins were required to have. This meant they could freely manage their own finances. Finally, they were the first women in Rome to have statues and portraits displayed en masse in public places. Previously, only one woman, Cornelia, mother of the Gracchi, had been part of the public statues displayed in Rome. In Augustus' rebuilding of Rome as a city of marble, Octavia was featured. In all her representations she wore the "nodus" hairstyle, which at the time was considered conservative and dignified, and worn by women from many classes.


title: Octavia the Younger

text: Augustus adored, but never adopted, her son Marcellus. When Marcellus died of illness in 23 BC unexpectedly, Augustus was thunderstruck, Octavia disconsolate almost beyond recovery.


title: Octavia the Younger

text: recited three whole books [of his Aeneid] for Augustus: the second, fourth, and sixth—this last out of his well-known affection for Octavia, who (being present at the recitation) is said to have fainted at the lines about her son, "… You shall be Marcellus" ["Aen". 6.884]. Revived only with difficulty, she sent Virgil ten thousand sesterces for each of the verses."


title: Octavia the Younger

text: She may have never fully recovered from the death of her son and retired from public life, except on important occasions. The major source that Octavia never recovered is Seneca ("De Consolatione ad Marciam", II.) but Seneca may wish to show off his rhetorical skill with hyperbole, rather than adhere to fact. Some dispute Seneca's version, as Octavia publicly opened the Library of Marcellus, dedicated in his memory, while her brother completed the Marcellus's theatre in his honor. Undoubtedly Octavia attended both ceremonies, as well as the Ara Pacis ceremony to welcome her brother's return in 13BC from the provinces. She was also consulted in regard to, and in some versions advised, that Augustus's daughter Julia marry Agrippa after her mourning for Marcellus ended. Agrippa had to divorce Octavia's daughter Claudia Marcella Major in order to marry Julia, so Augustus wanted Octavia's endorsement very much.


title: Octavia the Younger

text: Octavia died of natural causes. Suetonius says she died in Augustus' 54th year, thus 11 BC with Roman inclusive counting. Her funeral was a public one, with her sons-in-law (Drusus, Ahenobarbus, Iullus Antony, and possibly Paullus Aemillius Lepidus) carrying her to the grave in the Mausoleum of Augustus. Drusus delivered one funeral oration from the rostra and Augustus gave her the highest posthumous honors (building the Gate of Octavia and Porticus Octaviae in her memory). Augustus also had the Roman senate declare his sister to be a goddess. Augustus declined some other honors decreed to her by the senate, for reasons unknown. She was one of the first Roman women to have coins minted bearing her image; only Antony's previous wife Fulvia pre-empted her.


title: Octavia the Younger

text: Octavia and Mark Antony had two surviving daughters by their marriage (her second, his fourth), and both were the ancestors of later Roman emperors.


title: Octavia the Younger

text: A famous anecdote, recorded in the late fourth-century "vita" of Virgil by Aelius Donatus, in which the poet read the passage in Book VI in praise of Octavia's late son Marcellus and Octavia fainted with grief, has inspired several works of art. The most famous example is Jean-Auguste-Dominique Ingres's 1812 painting "Virgil reading The Aeneid before Augustus, Livia and Octavia" but other artists, including Jean-Joseph Taillasson, Antonio Zucchi, Jean-Baptiste Wicar, Jean-Bruno Gassies and Angelica Kaufmann, have also been inspired to depict this scene.


title: Octavia the Younger

text: Octavia's later life, around the time of the death of Marcellus, is depicted in the 1976 television adaptation of Robert Graves's novel "I, Claudius". The role was played by Angela Morant, and should not be confused with her great-granddaughter Claudia Octavia (also referred to as "Octavia" in the series), Claudius's daughter and wife of the future emperor Nero, who was played by Cheryl Johnson.


title: Octavia the Younger

text: A highly fictionalized version of Octavia's early life is depicted in the 2005 television series "Rome", in which Octavia of the Julii (Kerry Condon) seduces and sleeps with her younger brother, Gaius Octavian, has a lesbian affair with Servilia of the Junii (the series' version of Servilia) and a romantic relationship with Marcus Agrippa (based on the historical Marcus Vipsanius Agrippa), none of which has any historical basis.


title: Axa

text: Axa S.A. (styled as AXA or GIG in the Middle East) is a French multinational insurance company. The head office is in the 8th arrondissement of Paris, France. It also provides investment management and other financial services.


title: Axa

text: The Axa Group operates primarily in Western Europe, North America, the India Pacific region and the Middle East, with a presence also in Africa. Axa is a conglomerate of independently run businesses, operated according to the laws and regulations of many countries. It is a component of the Euro Stoxx 50 stock market index.


title: Axa

text: The company was founded in 1816 as Mutuelle de L'assurance contre L'incendie (the Ancienne Mutuelle). It acquired Compagnie Parisienne de Garantie in 1978 and became Mutuelles Unies. In 1982, it merged with the Drouot Group, owned by the Hottinguer family, becoming Mutuelles Unies/Drouot.


title: Axa

text: The firm adopted the Axa name in 1985. Axa took over The Equitable in 1991 and bought Union des Assurances De Paris (UAP), France's largest insurer, in 1996 to become Axa-UAP. It reverted to the name Axa in 1999. In February 1999, Axa acquired Guardian Royal Exchange. In May 2000, it acquired all shares it did not already own in Sun Life & Provincial Holdings. On 14 June 2006, Axa acquired Winterthur Group from Credit Suisse for approximately €9 billion. As of 2011, Axa was the second most powerful transnational corporation in terms of corporate control over global financial stability.


title: Axa

text: In May 2016, the firm announced that it would stop investing in tobacco shares and bonds and allow its portfolio of tobacco-related bonds to run off. On 12 September 2018 Axa acquired XL Group Ltd, a Bermuda-based property and casualty commercial lines insurer and reinsurer, for $15.3 billion. In October 2019, the company sold Axa Bank Belgium to Crelan for €620 million (US$688.51 million).


title: Axa

text: Despite being written by the company in upper case, "AXA" is not an acronym, but was chosen because its name can be pronounced easily by people who speak any language. After acquiring the Drouot Group in 1982, chairman and CEO Claude Bébéar hired an outside consultant to conduct a computer-aided search for a new name. Bébéar wanted a short and snappy name to convey vitality and could be pronounced the same way in every language, consistent with the group's desire for an international presence. Initially, "Elan" was the top choice, but Canadian executives balked because "elan" is the French word for a moose or elk. In 1985, Bébéar chose the name Axa.


title: Axa

text: Axa trades in the United Kingdom as Axa UK, with subsidiaries including Axa Insurance, Axa Wealth and Axa Health. AXA PPP International was the trading name for AXA PPP healthcare’s international health insurance division, which later rebranded as AXA - Global Healthcare on 1 January 2017. The company also owns the online insurer Swiftcover and fund manager Architas. In January 2007 Axa was reorganised into "strategic business units" aimed at competing within their specific markets.


title: Axa

text: In September 2013, Axa Wealth was fined £1.8 million by the FCA for failing to ensure it gave suitable investment advice to its customers. The regulator says it found "serious defects" in the way Axa advisers in Clydesdale Bank, Yorkshire Bank and the West Bromwich Building Society advised customers on investments.


title: Axa

text: In 2018, they became the 'Official Global Insurance' partner of Premier League team Liverpool F.C. and in 2019, began sponsoring the training kits of the team. In 2020, the partnership was cemented further in 2020, when they purchased the naming rights to the club's newly built training centre, which became officially known as 'The AXA Training Centre.'


title: Axa

text: Axa Sun Life was created following the merger between Axa Equity & Law and Sun Life Assurance Society. In 2006 Winterthur Life in the UK was absorbed although Axa continues to use the Winterthur brand for high-net-worth wealth management business.


title: Axa

text: Axa sold Axa Sun Life Holdings to Resolution Limited in autumn 2010, whilst retaining Axa Wealth (including Architas and the Elevate wrap platform), SunLife and Bancassurance business units. Closure of the Bancassurance arm was announced in April 2013. In 2016, Axa Wealth was sold to the Phoenix Group.


title: Axa

text: AXA Health sells private medical insurance in the UK and was known as AXA PPP Healthcare until 2020. It was previously the London Association for Hospital Services, set up in 1938 as a private healthcare scheme for people of middle income in London. It was incorporated in 1940 with assistance from the British Medical Association, the King's Fund, and the medical royal colleges.


title: Axa

text: Guardian Royal Exchange Assurance bought it in 1998 for £435 million; a year later it was bought by Sun Life & Provincial Holdings, an Axa subsidiary.


title: Axa

text: Axa - Global Healthcare has previously been known as Axa PPP International, Axa Global Protect or Axa Healthcare Management. In 2017, the subsidiary Axa - Global Healthcare became an independent Managing General Agent, selling and administering international health insurance products. While Axa Health provides insurance across the UK, Axa - Global Healthcare caters to those needing health insurance around the world.


title: Axa

text: In 2018, Axa - Global Healthcare launched a Virtual Doctor service for its customers with outpatient cover and in 2020 offered it across all plans. Provided by Teladoc Health, it offers anytime access to medical advice by phone or video by doctors located around the world who speak more than 15 languages and is targeted at expatriates.


title: Axa

text: Axa established a presence in Ireland in 1999 when it bought British-based Guardian Royal Exchange, which had previously acquired PMPA. PMPA, Private Motorists Protection Association, was at the time one of Ireland's biggest insurance companies. It is the third-largest general insurer in the Republic of Ireland.


title: Axa

text: Axa Canada marketed insurance in Quebec, Ontario, Western Canada and Atlantic Canada. In 2009 it had a total of approximately 2300 employees and 4000 brokers and advisors. The head office was in Montreal, Quebec. In 2011 it was sold to Intact Financial Corp. for C$2.6 billion.


title: Axa

text: The American arm of Axa is "Axa Financial, Inc.", which is known mainly through its subsidiaries such as "Axa Advisors", "Axa Network", "MONY" (formerly Mutual of New York), "US Financial Life", and "AllianceBernstein". The Equitable was acquired in 1991; the Mutual Life Insurance Company of New York (MONY) was acquired in 2004.


title: Axa

text: On 15 December 2006 Axa Advisors entered into agreements with LPL (Linsco Private Ledger), the country's largest independent broker-dealer. LPL agreed to provide brokerage, clearing and custody services on a fully disclosed basis. The terms of the agreements are five years, subject to additional 24-month extensions. Services began in August 2007.


title: Axa

text: In 2018, Axa spun off most of its US operations in an initial public offering. the resulting company was Equitable Holdings.


title: Axa

text: Present in the region for more than 60 years, Axa is the largest insurer in the GCC with branches across Bahrain, Oman, UAE and Turkey. In Saudi Arabia, Axa is known as Axa Cooperative Insurance. It offers insurance for individuals and businesses with Health, Motor, Home, Travel, Personal and Group Accident, Fire, Engineering, Life and various other products.


title: Axa

text: Axa is present in Lebanon as Axa Middle East. In 2000, Axa Group acquired 51% of SLF's capital and the Nasnas & Hanemoglou's Group the remaining 49% to start the operations in the Middle East.


title: Axa

text: In July 2008, Axa acquired ING Insurance Mexico, offering essentially the same services that ING offered. Axa then sued an ING Group subsidiary over alleged misrepresentations in the $1.5 billion sale stating it suffered "tens (if not hundreds) of millions of dollars in damages."


title: Axa

text: In 1995 the Axa Group purchased a 51 per cent controlling interest in the Australian life insurer, National Mutual Life Association of Australia for NF4 billion, that was claimed at the time to contribute about 15 percent of the Axa Group's worldwide insurance-premium income. In 2011, [AMP Limited] and Axa SA agreed to acquire Axa Asia pacific, merging Australian operations under the AMP brand.


title: Axa

text: In April 2016 Axa unveiled a partnership with Lloyd's of London insurer Chaucer Holdings to enter the growing market of speciality insurance in Africa. Axa Africa Specialty Risks helps to mitigate risks through its coverholder, Chaucer Syndicate 1084 and Axa Africa Specialty Risks 6130. Axa ASR focuses on specialty and corporate lines, with coverage and an on-the-ground presence across Africa. Axa and Lloyd's both have excellent international financial strength ratings (AA- by Fitch and A+ by Standard & Poor's). The lines of business include: Political Risk, Political Violence and Terrorism, Energy, Construction, Property, Liability, Marine and Aviation. Since 2014 Axa has made several acquisitions of other insurers in an attempt to expand. Axa already operates in Cameroon, Egypt, Gabon, Ivory Coast, Morocco, Nigeria, Senegal and Algeria. Axa Africa also has a role in the trades between Asia and some of African countries.


title: Axa

text: In January 2016, Axa created Kamet Ventures, an insurtech incubator, to build disruptive businesses in insurance, healthcare and assistance space with an initial funding of €100 million. Some of Kamet's investments include insurance advice platform, Anorak Technologies and elderly care platform, Birdie


title: Axa

text: Axa Investment Managers (Axa IM), is a global investment —management firm with offices in over 22 locations worldwide. As of 31 December 2020, it manages over €858 billion in assets on behalf of institutional and retail clients. It operates as the investment arm for Axa.


title: Axa

text: AXA Asia Pacific Holdings Limited acquired MLC (Hong Kong) Limited and re-branded it as AXA (Hong Kong) Life Insurance Company Limited in 2006.


title: Axa

text: Axa headquarters is located in the 8th arrondissement of Paris. Axa, which already owned 23 Avenue Matignon, acquired the former Hotel de La Vaupalière, an 18th-century building, in the late 1990s. Architect Ricardo Bofill integrated the facade of the hotel with a modern glass building that covers the courtyard that the hotel also occupies. The complex serves as Axa's head office.


title: Axa

text: Besides its artistic and social philanthropic initiatives (Axa Heart in Action), Axa created the Axa Research Fund in 2008. Endowed with €100 million, it provides support for research focused on understanding and preventing the risks threatening the environment, human life and society.


title: Axa

text: 256 basic-research projects were funded within three years, including 230 young researchers, PhDs or in postdoctoral year, working in laboratories all around the world. Axa also gathers every year the young fellows to meet a world-class researcher, exchange best practices and network.


title: Axa

text: The Fund has also awarded eleven endowments for several million euros supporting research institutions of excellence (HEC Paris, National University of Singapore, University of Bristol, London School of Economics, Met Office, INSERM, IHES). These research and education chairs intend to attract the best scientists. For example, the Axa Polytechnique Chair in Cellular Cardiovascular Engineering, held by Abdul Barakat, aims to promote research on cardiovascular diseases, but also to train and develop young researchers through extended educational programs.


title: Axa

text: After joining the Access to Medicine Foundation, Axa allocates a donation of 500,000 euros to the Foundation's activities relating to the prevention of infectious diseases and the response to pandemics such as COVID-19.


title: Seventeen Seconds

text: Seventeen Seconds is the second studio album by English rock band the Cure, released on 18 April 1980 by Fiction Records. The album marked the first time frontman Robert Smith co-produced with Mike Hedges. After the departure of original bassist Michael Dempsey, Simon Gallup became an official member along with keyboardist Matthieu Hartley. The single "A Forest" was the band's first entry in the top 40 of the UK Singles Chart.


title: Seventeen Seconds

text: At the end of the Cure's 1979 UK tour supporting Siouxsie and the Banshees, Robert Smith spoke less and less with bassist Michael Dempsey. Early versions of "Play for Today" and "M" had been performed at a few concerts, but Dempsey did not like the new musical direction that Smith wanted to take. Smith commented: "I think the final straw came when I played Michael the demos for the next album and he hated them. He wanted us to be XTC part 2 and – if anything – I wanted us to be the Banshees part 2. So he left". Playing guitar with the Banshees for two months and learning their songs opened up another horizon to Smith. "It allowed me to think beyond what we were doing. I wanted to have a band that does what Steven Severin and Budgie do, where they just get a bassline and the drum part and Siouxsie wails". The records to which Smith was frequently listening during the composition of the album were "Five Leaves Left" by Nick Drake, "Isle of Wight" by Jimi Hendrix, "Astral Weeks" by Van Morrison and "Low" by David Bowie. Smith wrote the lyrics and music for most of the record at his parents' home, on a Hammond organ with a built-in tape recorder. Interviewed in 2004, producer Mike Hedges did not recall any demo tracks, with the band generally playing the track in the studio before laying down a backing track to which overdubs were added.


title: Seventeen Seconds

text: Two members of the Magazine Spies, bass guitarist Simon Gallup and keyboardist Matthieu Hartley, were added to the band's lineup. Gallup replaced Dempsey, which relieved Smith as he felt Dempsey's basslines were too ornate. Hartley's synth work added a new dimension to the band's newly ethereal sound, although he would later clash with Smith over complexity; Hartley enjoyed complex chords but Smith wanted single notes.


title: Seventeen Seconds

text: Money was short, so the album was recorded and mixed in seven days on a budget of between £2,000 and £3,000, which resulted in the band working 16 or 17 hours a day. Smith stated that as a result, the track "The Final Sound", which was planned to be much longer, was cut down to 53 seconds because the tape ran out while recording and band could not record it again. The album, mostly a collection of downbeat tracks, features ambient echoing vocals with the sonic direction driven by its drum sound.


title: Seventeen Seconds

text: Retrospectively, "Seventeen Seconds" has been considered an early example of gothic rock. Its "gloomscapes" are considered to be "a sonic touchstone" for the forthcoming movement. The track "The Final Sound" is "so positively gothic you could almost be fooled into believing that it was lifted from the soundtrack of some Hammer horror gorefest". The album has also been described as new wave.


title: Seventeen Seconds

text: "Seventeen Seconds" was released on 18 April 1980. It reached No. 20 on the British album charts. The record was repackaged in the US in 1981 (on the A&M label) with "Faith" as "Happily Ever After", available as a double LP. In 2005, the album was remastered as part of Universal's Deluxe Edition series, featuring bonus live tracks and demos as well as studio material by Cult Hero, a 1970s-style progressive rock band along the lines of Easy Cure that featured Smith's postman Frank Bell as lead singer. "Seventeen Seconds" charted in the United States for the first time in September 2020, when it debuted at No. 186 on the "Billboard" 200 album chart.


title: Seventeen Seconds

text: The album's songs have been described by critics as featuring vague, often unsettling lyrics and dark, spare, minimalistic melodies. Some reviewers, such as Nick Kent of "NME", felt that "Seventeen Seconds" represented a far more mature Cure, who had come very far musically in less than one year. Ian Cranna of "Smash Hits" wrote that the band were creating more ambitious music, while still retaining their "powerful melodic intensity". Chris Westwood of "Record Mirror" was less enthusiastic, viewing the album as "a sidewards step" rather than a progression; he found the material "biteless, a bit distant", showcasing a "reclusive, disturbed Cure, sitting in cold, dark, empty rooms, watching clocks".


title: Seventeen Seconds

text: Simon Reynolds said the album was "translucent-sounding", with shades of the Durutti Column, Young Marble Giants and "Another Green World" by Brian Eno. AllMusic writer Chris True said that while "Seventeen Seconds" had come to be largely overlooked in later years apart from its single "A Forest", it nonetheless represented an important development for the Cure, capturing them becoming "more rigid in sound, and more disciplined in attitude", and anticipating the bleak lyrical themes that would become more apparent on subsequent Cure albums.


title: Seventeen Seconds

text: "Seventeen Seconds" was included in the book "1001 Albums You Must Hear Before You Die". The authors note: "Like the album's cover art, which is little more than an abstract blur, the bleak, minimalist sound of "Seventeen Seconds"-era Cure is subtly suggestive." Attention is drawn to the "beguiling bleakness, both in its brief instrumentals and the more pop-oriented tracks (such as the sharp, hook-laden 'Play for Today') that hark back to their earlier work." In 2020, "Rolling Stone" included "Seventeen Seconds" in their "80 Greatest albums of 1980" list.


title: Christopher Daniel Barnes

text: Christopher Daniel Barnes (born November 7, 1972), also known professionally as C. D. Barnes, C. B. Barnes and Chris Barnes, is an American actor and writer. He is best known for his voice role as the title superhero of the 1990s animated series "Spider-Man" (1994–1998) and as Prince Eric in "The Little Mermaid" (1989), as well as his on-screen portrayal of Greg Brady in the films "The Brady Bunch Movie" (1995) and "A Very Brady Sequel" (1996).


title: Christopher Daniel Barnes

text: He starred in 22 episodes of the science fiction series "Starman" (1986–1987) as the 14-year-old son of the title character played by Robert Hays. He played Leonard on "Malcolm and Eddie" (1998–2000).


title: Christopher Daniel Barnes

text: In the 1986–1987 series "Starman", he played the teenage boy Scott, the son of the Starman. He starred in the NBC sitcom "Day by Day" for two seasons. In one episode's dream sequence which included many guest stars from "The Brady Bunch", he played a fourth Brady son, "Chuck Brady". He later played Greg Brady in the Brady Bunch parody movies. He starred as Leonard Rickets in the sitcom "Malcolm & Eddie".


title: Christopher Daniel Barnes

text: He appeared in "Beverly Hills, 90210", "Clueless", "Wings", "Blossom", "JAG", "American Dreamer", "The Golden Girls", "Touched by an Angel", "Herman's Head", "Time Trax", "7th Heaven", "Empty Nest", "Girlfriends", "CBS Schoolbreak Special" and "ABC Weekend Specials".


title: Christopher Daniel Barnes

text: At age 16 in 1989, Barnes voiced Prince Eric in "The Little Mermaid" and reprised the role in 2006 for "Kingdom Hearts II". He was unable to return for the sequel "" in 2000 and was replaced by Rob Paulsen. He voiced Prince Charming in the "Cinderella" sequels "" and "".


title: Christopher Daniel Barnes

text: Vocally, Barnes is best known for his main role in the 1994–1998 "Spider-Man" series. He later provided the voice of Spider-Man Noir in the 2010 video game "", Spider-Man 2099 in the 2011 video game "", Electro, Spyder-Knight and Wolf Spider in the 2012 cartoon series "Ultimate Spider-Man", and also voiced two special costumes, Symbiote Spider-Man and Superior Spider-Man 2.0, in the 2013 video game "Marvel Heroes". He reprised his role of Electro and voiced Vulture in the Mobile game "Spider-Man Unlimited".


title: Christopher Daniel Barnes

text: He has done voices for other animated series including several characters in "Captain Planet and the Planeteers", Tagert McStone in "Jackie Chan Adventures", Stripes the Tiger in "Sonic Underground", and also voiced adult Speckles from "".


title: Christopher Daniel Barnes

text: He has done voice acting in video games including "Return to Krondor", as Scott Donovan in the "Law & Order" titles "" and "".


title: Christopher Daniel Barnes

text: Barnes is a writer of short stories which he shares freely on his website at christopherdanielbarnes.com.


title: Christopher Daniel Barnes

text: Barnes earned his BA in 2004 and his MA in 2009 and enjoys reading, writing short stories, playing the guitar, and practicing yoga.