Quantum spacetime

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In mathematical physics, the concept of quantum spacetime is a generalization of the usual concept of spacetime in which some variables that ordinarily commute are assumed not to commute and form a different Lie algebra. The choice of that algebra still varies from theory to theory. As a result of this change some variables that are usually continuous may become discrete. Often only such discrete variables are called "quantized"; usage varies.

The idea of quantum spacetime was proposed in the early days of quantum theory by Heisenberg and Ivanenko as a way to eliminate infinities from quantum field theory. The germ of the idea passed from Heisenberg to Rudolf Peierls, who noted that electrons in a magnetic field can be regarded as moving in a quantum spacetime, and to Robert Oppenheimer, who carried it to Hartland Snyder, who published the first concrete example. Snyder's Lie algebra was made simple by C. N. Yang in the same year.

Overview

Physical spacetime is a quantum spacetime when in quantum mechanics position and momentum variables ${\displaystyle x,p}$ are already noncommutative, obey the Heisenberg uncertainty principle, and are continuous. Because of the Heisenberg uncertainty relations, greater energy is needed to probe smaller distances. Ultimately, according to gravity theory, the probing particles form black holes that destroy what was to be measured. The process cannot be repeated, so it cannot be counted as a measurement. This limited measurability led many to expect that our usual picture of continuous commutative spacetime breaks down at Planck scale distances, if not sooner.

Again, physical spacetime is expected to be quantum because physical coordinates are already slightly noncommutative. The astronomical coordinates of a star are modified by gravitational fields between us and the star, as in the deflection of light by the sun, one of the classic tests of general relativity. Therefore, the coordinates actually depend on gravitational field variables. According to quantum theories of gravity these field variables do not commute; therefore coordinates that depend on them likely do not commute.

Both arguments are based on pure gravity and quantum theory, and they limit the measurement of time by the only time constant in pure quantum gravity, the Planck time. Our instruments, however, are not purely gravitational but are made of particles. They may set a more severe, larger, limit than the Planck time.

Criteria

Quantum spacetimes are often described mathematically using the noncommutative geometry of Connes, quantum geometry, or quantum groups.

Any noncommutative algebra with at least four generators could be interpreted as a quantum spacetime, but the following desiderata have been suggested:

• Local Lorentz group and Poincaré group symmetries should be retained, possibly in a generalised form. Their generalisation often takes the form of a quantum group acting on the quantum spacetime algebra.
• The algebra might plausibly arise in an effective description of quantum gravity effects in some regime of that theory. For example, a physical parameter ${\displaystyle \lambda }$, perhaps the Planck length, might control the deviation from commutative classical spacetime, so that ordinary Lorentzian spacetime arises as ${\displaystyle \lambda \to 0}$.
• There might be a notion of quantum differential calculus on the quantum spacetime algebra, compatible with the (quantum) symmetry and preferably reducing to the usual differential calculus as ${\displaystyle \lambda \to 0}$.

This would permit wave equations for particles and fields and facilitate predictions for experimental deviations from classical spacetime physics that can then be tested experimentally.

• The Lie algebra should be semisimple. This makes it easier to formulate a finite theory.

Models

Several models were found in the 1990s more or less meeting most of the above criteria.

Bicrossproduct model spacetime

The bicrossproduct model spacetime was introduced by Shahn Majid and Henri Ruegg and has Lie algebra relations

${\displaystyle [x_i,x_j]=0,[x_i,t]=i\lambda x_i}$

for the spatial variables ${\displaystyle x_i}$ and the time variable ${\displaystyle t}$. Here ${\displaystyle \lambda }$ has dimensions of time and is therefore expected to be something like the Planck time. The Poincaré group here is correspondingly deformed, now to a certain bicrossproduct quantum group with the following characteristic features.

Orbits for the action of the Lorentz group on momentum space in the construction of the bicrossproduct model in units of ${\displaystyle {\lambda }^{-1}}$. Mass-shell hyperboloids are 'squashed' into a cylinder.

The momentum generators ${\displaystyle p_i}$ commute among themselves but addition of momenta, reflected in the quantum group structure, is deformed (momentum space becomes a non-abelian group). Meanwhile, the Lorentz group generators enjoy their usual relations among themselves but act non-linearly on the momentum space. The orbits for this action are depicted in the figure as a cross-section of ${\displaystyle p_0}$ against one of the ${\displaystyle p_i}$. The on-shell region describing particles in the upper center of the image would normally be hyperboloids but these are now 'squashed' into the cylinder

${\displaystyle \sqrt{p_1^2+p_2^2+p_3^2}<{\lambda }^{-1}}$

in simplified units. The upshot is that Lorentz-boosting a momentum will never increase it above the Planck momentum. The existence of a highest momentum scale or lowest distance scale fits the physical picture. This squashing comes from the non-linearity of the Lorentz boost and is an endemic feature of bicrossproduct quantum groups known since their introduction in 1988. Some physicists dub the bicrossproduct model doubly special relativity, since it sets an upper limit to both speed and momentum.

Another consequence of the squashing is that the propagation of particles is deformed, even of light, leading to a variable speed of light. This prediction requires the particular ${\displaystyle p_0,p_i}$ to be the physical energy and spatial momentum (as opposed to some other function of them). Arguments for this identification were provided in 1999 by Giovanni Amelino-Camelia and Majid through a study of plane waves for a quantum differential calculus in the model. They take the form

${\displaystyle e^{i\sum _i{p}_i{x}_i}{e}^{i{p}_{0}t}}$

in other words a form which is sufficiently close to classical that one might plausibly believe the interpretation. At the moment such wave analysis represents the best hope to obtain physically testable predictions from the model.

Prior to this work there were a number of unsupported claims to make predictions from the model based solely on the form of the Poincaré quantum group. There were also claims based on an earlier ${\displaystyle \kappa }$-Poincaré quantum group introduced by Jurek Lukierski and co-workers which should be viewed as an important precursor to the bicrossproduct one, albeit without the actual quantum spacetime and with different proposed generators for which the above picture does not apply. The bicrossproduct model spacetime has also been called ${\displaystyle \kappa }$-deformed spacetime with ${\displaystyle \kappa ={\lambda }^{-1}}$.

q-Deformed spacetime

This model was introduced independently by a team working under Julius Wess in 1990 and by Shahn Majid and coworkers in a series of papers on braided matrices starting a year later. The point of view in the second approach is that usual Minkowski spacetime has a nice description via Pauli matrices as the space of 2 x 2 hermitian matrices. In quantum group theory and using braided monoidal category methods one has a natural q-version of this defined here for real values of ${\displaystyle q}$ as a 'braided hermitian matrix' of generators and relations

${\displaystyle \left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)={\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)}^{†},\beta \alpha =q^2\alpha \beta ,[\alpha ,\delta ]=0,[\beta ,\gamma ]=(1-q^{-2})\alpha (\delta -\alpha ),[\delta ,\beta ]=(1-q^{-2})\alpha \beta }$

These relations say that the generators commute as ${\displaystyle q\to 1}$ thereby recovering usual Minkowski space. One can work with more familiar variables ${\displaystyle x,y,z,t}$ as linear combinations of these. In particular, time

${\displaystyle t={\text{Trace}}_q\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)=q\delta +q^{-1}\alpha }$

is given by a natural braided trace of the matrix and commutes with the other generators (so this model has a very different flavour from the bicrossproduct one). The braided-matrix picture also leads naturally to a quantity

${\displaystyle {det}_q\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)=\alpha \delta -q^2\gamma \beta }$

which as ${\displaystyle q\to 1}$ returns us the usual Minkowski distance (this translates to a metric in the quantum differential geometry). The parameter ${\displaystyle q=e^{\lambda }}$ or ${\displaystyle q=e^{i\lambda }}$ is dimensionless and ${\displaystyle \lambda }$ is thought to be a ratio of the Planck scale and the cosmological length. That is, there are indications that this model relates to quantum gravity with non-zero cosmological constant, the choice of ${\displaystyle q}$ depending on whether this is positive or negative. We have described the mathematically better understood but perhaps less physically justified positive case here.

A full understanding of this model requires (and was concurrent with the development of) a full theory of 'braided linear algebra' for such spaces. The momentum space for the theory is another copy of the same algebra and there is a certain 'braided addition' of momentum on it expressed as the structure of a braided Hopf algebra or quantum group in a certain braided monoidal category). This theory by 1993 had provided the corresponding ${\displaystyle q}$-deformed Poincaré group as generated by such translations and ${\displaystyle q}$-Lorentz transformations, completing the interpretation as a quantum spacetime.

In the process it was discovered that the Poincaré group not only had to be deformed but had to be extended to include dilations of the quantum spacetime. For such a theory to be exact we would need all particles in the theory to be massless, which is consistent with experiment as masses of elementary particles are indeed vanishingly small compared to the Planck mass. If current thinking in cosmology is correct then this model is more appropriate, but it is significantly more complicated and for this reason its physical predictions have yet to be worked out.

Fuzzy or spin model spacetime

This refers in modern usage to the angular momentum algebra

${\displaystyle [x_1,x_2]=2i\lambda x_3,[x_2,x_3]=2i\lambda x_1,[x_3,x_1]=2i\lambda x_2}$

familiar from quantum mechanics but interpreted in this context as coordinates of a quantum space or spacetime. These relations were proposed by Roger Penrose in his earliest spin network theory of space. It is a toy model of quantum gravity in 3 spacetime dimensions (not the physical 4) with a Euclidean (not the physical Minkowskian) signature. It was again proposed in this context by Gerardus 't Hooft. A further development including a quantum differential calculus and an action of a certain 'quantum double' quantum group as deformed Euclidean group of motions was given by Majid and E. Batista.

A striking feature of the noncommutative geometry here is that the smallest covariant quantum differential calculus has one dimension higher than expected, namely 4, suggesting that the above can also be viewed as the spatial part of a 4-dimensional quantum spacetime. The model should not be confused with fuzzy spheres which are finite-dimensional matrix algebras which one can think of as spheres in the spin model spacetime of fixed radius.

Heisenberg model spacetimes

The quantum spacetime of Hartland Snyder proposes that

${\displaystyle [x_{\mu },x_{\nu }]=i{M}_{\mu \nu }}$

where the ${\displaystyle M_{\mu \nu }}$ generate the Lorentz group. This quantum spacetime and that of C. N. Yang entail a radical unification of spacetime, energy-momentum, and angular momentum.

The idea was revived in a modern context by Sergio Doplicher, Klaus Fredenhagen and John Roberts in 1995  by letting ${\displaystyle M_{\mu \nu }}$ simply be viewed as some function of ${\displaystyle x_{\mu }}$ as defined by the above relation, and any relations involving it viewed as higher order relations among the ${\displaystyle x_{\mu }}$. The Lorentz symmetry is arranged so as to transform the indices as usual and without being deformed.

An even simpler variant of this model is to let ${\displaystyle M}$ here be a numerical antisymmetric tensor, in which context it is usually denoted ${\displaystyle \theta }$, so the relations are ${\displaystyle [x_{\mu },x_{\nu }]=i{\theta }_{\mu \nu }}$. In even dimensions ${\displaystyle D}$, any nondegenerate such theta can be transformed to a normal form in which this really is just the Heisenberg algebra but the difference that the variables are being proposed as those of spacetime. This proposal was for a time quite popular because of its familiar form of relations and because it has been argued that it emerges from the theory of open strings landing on D-branes, see noncommutative quantum field theory and Moyal plane. However, this D-brane lives in some of the higher spacetime dimensions in the theory and hence it is not our physical spacetime that string theory suggests to be effectively quantum in this way. You also have to subscribe to D-branes as an approach to quantum gravity in the first place. Even when posited as quantum spacetime it is hard to obtain physical predictions and one reason for this is that if ${\displaystyle \theta }$ is a tensor then by dimensional analysis it should have dimensions of length${\displaystyle {}^2}$, and if this length is speculated to be the Planck length then the effects would be even harder to ever detect than for other models.

Noncommutative extensions to spacetime

Although not quantum spacetime in the sense above, another use of noncommutative geometry is to tack on 'noncommutative extra dimensions' at each point of ordinary spacetime. Instead of invisible curled up extra dimensions as in string theory, Alain Connes and coworkers have argued that the coordinate algebra of this extra part should be replaced by a finite-dimensional noncommutative algebra. For a certain reasonable choice of this algebra, its representation and extended Dirac operator, one is able to recover the Standard Model of elementary particles. In this point of view the different kinds of matter particles are manifestations of geometry in these extra noncommutative directions. Connes's first works here date from 1989 but has been developed considerably since then. Such an approach can theoretically be combined with quantum spacetime as above.

Period 4 element

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Period_4_element

A period 4 element is one of the chemical elements in the fourth row (or period) of the periodic table of the chemical elements. The periodic table is laid out in rows to illustrate recurring (periodic) trends in the chemical behaviour of the elements as their atomic number increases: a new row is begun when chemical behaviour begins to repeat, meaning that elements with similar behaviour fall into the same vertical columns. The fourth period contains 18 elements beginning with potassium and ending with krypton – one element for each of the eighteen groups. It sees the first appearance of d-block (which includes transition metals) in the table.

Properties

Every single one of these elements is stable, and many are extremely common in the Earth's crust and/or core; it is the last period with no unstable elements at all. Many of the transition metals in period 4 are very strong, and therefore commonly used in industry, especially iron. Three adjacent elements are known to be toxic, with arsenic one of the most well-known poisons, selenium being toxic to humans in large quantities, and bromine, a toxic liquid. Many elements are essential to humans' survival, such as calcium being what forms bones.

Atomic structure

Progressing towards increase of atomic number, the Aufbau principle causes elements of the period to put electrons onto 4s, 3d, and 4p subshells, in that order. However, there are exceptions, such as chromium. The first twelve elements—K, Ca, and transition metals—have from 1 to 12 valence electrons respectively, which are placed on 4s and 3d.

Twelve electrons over the electron configuration of argon reach the configuration of zinc, namely 3d¹⁰ 4s². After this element, the filled 3d subshell effectively withdraws from chemistry and the subsequent trend looks much like trends in the periods 2 and 3. The p-block elements of period 4 have their valence shell composed of 4s and 4p subshells of the fourth (n = 4) shell and obey the octet rule.

For quantum chemistry namely this period sees transition from the simplified electron shell paradigm to research of many differently-shaped subshells. The relative disposition of their energy levels is governed by the interplay of various physical effects. The period's s-block metals put their differentiating electrons onto 4s despite having vacancies among nominally lower n = 3 states – a phenomenon unseen in lighter elements. Contrariwise, the six elements from gallium to krypton are the heaviest where all electron shells below the valence shell are filled completely. This is no longer possible in further periods due to the existence of f-subshells starting from n = 4.

List of elements

Chemical element			Block	Electron configuration
19	K	Potassium	s-block	[Ar] 4s1
20	Ca	Calcium	s-block	[Ar] 4s2
21	Sc	Scandium	d-block	[Ar] 3d1 4s2
22	Ti	Titanium	d-block	[Ar] 3d2 4s2
23	V	Vanadium	d-block	[Ar] 3d3 4s2
24	Cr	Chromium	d-block	[Ar] 3d5 4s1 (*)
25	Mn	Manganese	d-block	[Ar] 3d5 4s2
26	Fe	Iron	d-block	[Ar] 3d6 4s2
27	Co	Cobalt	d-block	[Ar] 3d7 4s2
28	Ni	Nickel	d-block	[Ar] 3d8 4s2
29	Cu	Copper	d-block	[Ar] 3d10 4s1 (*)
30	Zn	Zinc	d-block	[Ar] 3d10 4s2
31	Ga	Gallium	p-block	[Ar] 3d10 4s2 4p1
32	Ge	Germanium	p-block	[Ar] 3d10 4s2 4p2
33	As	Arsenic	p-block	[Ar] 3d10 4s2 4p3
34	Se	Selenium	p-block	[Ar] 3d10 4s2 4p4
35	Br	Bromine	p-block	[Ar] 3d10 4s2 4p5
36	Kr	Krypton	p-block	[Ar] 3d10 4s2 4p6

(*) Exception to the Madelung rule

s-block elements

Potassium

Potassium (K) is an alkali metal, placed under sodium and over rubidium, and is the first element of period 4. It is one of the most reactive elements in the periodic table, therefore usually only found in compounds. It tends to oxidize in air very rapidly, thus accounting for its rapid reaction with oxygen when freshly exposed to air. When freshly exposed, it is rather silvery, but it quickly begins to tarnish as it reacts with air. It is soft enough to be cut with a knife and it is the second least dense element. Potassium has a relatively low melting point; it will melt just by putting it under a small open flame. It also is less dense than water, and can, in turn, float.

Calcium

Calcium (Ca) is the second element in the period. An alkali earth metal, calcium is almost never found in nature due to its high reactivity with water. It has one of the most widely known and acknowledged biological roles in all animals and some plants, making up bones and teeth, and used in some applications in cells, such as signals for cellular processeses. It is regarded as the most abundant mineral in the body's mass.

d-block elements

Scandium

Scandium (Sc) is the third element in the period, and is the first transition metal in the periodic table. Scandium is quite common in nature, but difficult to isolate because it is most prevalent in rare earth compounds, which are difficult to isolate elements from. Scandium has very few commercial applications because of the aforementioned facts, and currently its only major application is in aluminium alloys.

Titanium

Titanium (Ti) is an element in group 4. Titanium is both one of the least dense metals and one of the strongest and most corrosion-resistant, and as such has many applications, especially in alloys with other elements, such as iron. Due to its aforementioned properties, it is commonly used in airplanes, golf clubs, and other objects that must be strong, but lightweight.

Vanadium

Vanadium (V) is an element in group 5. Vanadium is never found in pure form in nature, but is commonly found in compounds. Vanadium is similar to titanium in many ways, such as being very corrosion-resistant, however, unlike titanium, it oxidizes in air even at room temperature. All vanadium compounds have at least some level of toxicity, with some of them being extremely toxic.

Chromium

Chromium (Cr) is an element in group 6. Chromium is, like titanium and vanadium before it, extremely resistant to corrosion, and is indeed one of the main components of stainless steel. Chromium also has many colorful compounds, and as such is very commonly used in pigments, such as chrome green.

Manganese

Manganese (Mn) is an element in group 7. Manganese is often found in combination with iron. Manganese, like chromium before it, is an important component in stainless steel, preventing the iron from rusting. Manganese is also often used in pigments, again like chromium. Manganese is also poisonous; if enough is inhaled, it can cause irreversible neurological damage.

Iron

Iron (Fe) is an element in group 8. Iron is the most common on Earth among elements of the period, and probably the most well-known of them. It is the principal component of steel. Iron-56 has the lowest energy density of any isotope of any element, meaning that it is the most massive element that can be produced in supergiant stars. Iron also has some applications in the human body; hemoglobin is partly iron.

Cobalt

Cobalt (Co) is an element in group 9. Cobalt is commonly used in pigments, as many compounds of cobalt are blue in color. Cobalt is also a core component of many magnetic and high-strength alloys. The only stable isotope, cobalt-59, is an important component of vitamin B-12, while cobalt-60 is a component of nuclear fallout and can be dangerous in large enough quantities due to its radioactivity.

Nickel

Nickel (Ni) is an element in group 10. Nickel is rare in the Earth's crust, mainly due to the fact that it reacts with oxygen in the air, with most of the nickel on Earth coming from nickel iron meteorites. However, nickel is very abundant in the Earth's core; along with iron it is one of the two main components. Nickel is an important component of stainless steel, and in many superalloys.

Copper

Copper (Cu) is an element in group 11. Copper is one of the few metals that is not white or gray in color, the only others being gold, osmium and caesium. Copper has been used by humans for thousands of years to provide a reddish tint to many objects, and is even an essential nutrient to humans, although too much is poisonous. Copper is also commonly used as a wood preservative or fungicides.

Zinc

Zinc (Zn) is an element in group 12. Zinc is one of the main components of brass, being used since the 10th century BCE. Zinc is also incredibly important to humans; almost 2 billion people in the world suffer from zinc deficiency. However, too much zinc can cause copper deficiency. Zinc is often used in batteries, aptly named carbon-zinc batteries, and is important in many platings, as zinc is very corrosion resistant.

p-block elements

Gallium

Gallium (Ga) is an element in group 13, under aluminium. Gallium is noteworthy because it has a melting point at about 303 kelvins, right around room temperature. For example, it will be solid on a typical spring day, but will be liquid on a hot summer day. Gallium is an important component in the alloy galinstan, along with tin. Gallium can also be found in semiconductors.

Germanium

Germanium (Ge) is an element in group 14. Germanium, like silicon above it, is an important semiconductor and is commonly used in diodes and transistors, often in combination with arsenic. Germanium is fairly rare on Earth, leading to its comparatively late discovery. Germanium, in compounds, can sometimes irritate the eyes, skin, or lungs.

Arsenic

Arsenic (As) is an element in group 15. Arsenic, as mentioned above, is often used in semiconductors in alloys with germanium. Arsenic, in pure form and some alloys, is incredibly poisonous to all multicellular life, and as such is a common component in pesticides. Arsenic was also used in some pigments before its toxicity was discovered.

Selenium

Selenium (Se) is an element in group 16. Selenium is the first nonmetal in period 4, with properties similar to sulfur. Selenium is quite rare in pure form in nature, mostly being found in minerals such as pyrite, and even then it is quite rare. Selenium is necessary for humans in trace amounts, but is toxic in larger quantities. Selenium is a chalcogen. Selenium is red in monomolar structure but metallic gray in its crystalline structure.

Bromine

Bromine (Br) is an element in group 17 (halogen). It does not exist in elemental form in nature. Bromine is barely liquid at room temperature, boiling at about 330 kelvins. Bromine is also quite toxic and corrosive, but bromide ions, which are relatively inert, can be found in halite, or table salt. Bromine is often used as a fire retardant because many compounds can be made to release free bromine atoms.

Krypton

Krypton (Kr) is a noble gas, placed under argon and over xenon. Being a noble gas, krypton rarely interacts with itself or other elements; although compounds have been detected, they are all unstable and decay rapidly, and as such, krypton is often used in fluorescent lights. Krypton, like most noble gases, is also used in lighting because of its many spectral lines and the aforementioned reasons.

Biological role

Many period 4 elements find roles in controlling protein function as secondary messengers, structural components, or enzyme cofactors. A gradient of potassium is used by cells to maintain a membrane potential which enables neurotransmitter firing and facilitated diffusion among other processes. Calcium is a common signaling molecule for proteins such as calmodulin and plays a critical role in triggering skeletal muscle contraction in vertebrates. Selenium is a component of the noncanonical amino acid, selenocysteine; proteins which contain selenocysteine are known as selenoproteins. Manganese enzymes are utilized by both eukaryotes and prokaryotes, and may play a role in the virulence of some pathogenic bacteria. Vanabins, also known as vanadium-associated proteins, are found in the blood cells of some species of sea squirts. The role of these proteins is disputed, although there is some speculation that they function as oxygen carriers. Zinc ions are used to stabilize the zinc finger milieu of many DNA-binding proteins.

Period 4 elements can also be found complexed with organic small molecules to form cofactors. The most famous example of this is heme: an iron-containing porphyrin compound responsible for the oxygen-carrying function of myoglobin and hemoglobin as well as the catalytic activity of cytochrome enzymes. Hemocyanin replaces hemoglobin as the oxygen carrier of choice in the blood of certain invertebrates, including horseshoe crabs, tarantulas, and octopuses. Vitamin B₁₂ represents one of the few biochemical applications for cobalt.