Introduction to Quantum Theory and Atomic Structure (Oxford Chemistry Primers, Volume 37)


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And, as we will see, several attempts at a UV-completion of gravity, discussed in Sections 3. For a more exhaustive coverage of the history of the minimal length, the interested reader is referred to [ ]. This poverty might be an actual one of lacking experimental equipment, or it might be one of practical impossibility. But even if an experiment is not experimentally realizable in the near future, thought experiments serve two important purposes. First, by allowing the thinker to test ranges of parameter space that are inaccessible to experiment, they may reveal inconsistencies or paradoxes and thereby open doors to an improvement in the fundamentals of the theory.

The complete evaporation of a black hole and the question of information loss in that process is a good example for this. Second, thought experiments tie the theory to reality by the necessity to investigate in detail what constitutes a measurable entity. The thought experiments discussed in the following are examples of this. But the photon used to measure the position of the particle has a recoil when it scatters and transfers a momentum to the particle.

It does not, strictly speaking, even make sense to consider the position and momentum of the particle at the same time. Consequently, instead of speaking about the photon scattering off the particle as if that would happen in one particular point, we should speak of the photon having a strong interaction with the particle in some region of size R. Now we will include gravity in the picture, following the treatment of Mead [ ]. The photon carries an energy that, though in general tiny, exerts a gravitational pull on the particle whose position we wish to measure.

The gravitational acceleration acting on the particle is at least on the order of. Projection on the x -axis then yields the additional uncertainty of. Combining 8 with 2 , one obtains. Then one finds. Assuming that the normal uncertainty and the gravitational uncertainties add linearly, one arrives at. Adler and Santiago make the interesting observation that the GUP 13 is invariant under the replacement.

These limitations, refinements of which we will discuss in the following Sections 3. At the high energies necessary to reach the Planckian limit, the scattering is unlikely to be elastic, but the same considerations apply to inelastic scattering events. The question that the GUP then raises is what modification of quantum mechanics would give rise to the generalized uncertainty, a question we will return to in Section 4. Another related argument has been put forward by Scardigli [ ], who employs the idea that once one arrives at energies of about the Planck mass and concentrates them to within a volume of radius of the Planck length, one creates tiny black holes, which subsequently evaporate.

This effects scales in the same way as the one discussed here, and one arrives again at The above result makes use of Newtonian gravity, and has to be refined when one takes into account general relativity. Before we look into the details, let us start with a heuristic but instructive argument. One of the most general features of general relativity is the formation of black holes under certain circumstances, roughly speaking when the energy density in some region of spacetime becomes too high.

Once matter becomes very dense, its gravitational pull leads to a total collapse that ends in the formation of a horizon. The Hoop conjecture is unproven, but we know from both analytical and numerical studies that it holds to very good precision [ , ].

Thus, the larger the energy, the better the particle can be focused. The important point to notice here is that the extension of the black hole grows linearly with the energy, and therefore one can achieve a minimal possible extension, which is on the order of. For the more detailed argument, we follow Mead [ ] with the general relativistic version of the Heisenberg microscope that was discussed in Section 3. Again, we have a particle whose position we want to measure by help of a test particle.

As before, the test particle moves in the x direction. The task is now to compute the gravitational field of the test particle and the motion it causes on the measured particle. To obtain the metric that the test particle creates, we first change into the rest frame of the particle by boosting into x -direction. Because of Eq. For this, we note that the world line of the measured particle must be timelike. We denote the velocity in the x -direction with u , then we need.

Now we insert Eq. We simplify the requirement of Eq. One arrives at this estimate with reduced effort if one makes it clear to oneself what we want to estimate. We want to know, as previously, how much the particle, whose position we are trying to measure, will move due to the gravitational attraction of the particle we are using for the measurement. The faster the particles pass by each other, the shorter the interaction time and, all other things being equal, the less the particle we want to measure will move.

Now we can continue as before in the non-relativistic case. In this background, one can then compute the motion of the measured particle by using the Newtonian limit of the geodesic equation, provided the particle remains non-relativistic. In the longitudinal direction, along the motion of the test particle one finds. The derivative of f gives two delta-functions at the front and back of the cylinder with equal momentum transfer but of opposite direction.

The change in velocity to the measured particle is. Wigner and Salecker [ ] proposed the following thought experiment to show that the precision of length measurements is limited. Consider that we try to measure a length by help of a clock that detects photons, which are reflected by a mirror at distance D and return to the clock. Knowing the speed of light is universal, from the travel-time of the photon we can then extract the distance it has traveled.

How precisely can we measure the distance in this way? This means, according to the Heisenberg uncertainty principle, we cannot know its velocity to better than. We will consider the clock synchronization to be performed by the passing of light signals from some standard clock to the clock under question. So we see that the precision by which clocks can be synchronized is also bound by the Planck scale.

If the clock has a velocity u , then the proper time it records is more generally given by. Therefore, taking into account that the clock does not remain stationary, one still arrives at The above microscope experiment investigates how precisely one can measure the location of a particle, and finds the precision bounded by the inevitable formation of a black hole. However, this position uncertainty is for the location of the measured particle however and not for the size of the black hole or its radius.

There is a simple argument why one would expect there to also be a limit to the precision by which the size of a black hole can be measured, first put forward in [ 91 ]. Then, quantum fluctuations in the position of the black hole should affect the definition of the horizon. In Boyer-Lindquist coordinates, the horizon is located at the radius. In an argument similar to that of Adler and Santiago discussed in Section 3. It is clear that the uncertainty Maggiore considered is of a different kind than the one considered by Mead, though both have the same origin.

The smaller the wavelength of the emitted particle, the larger the so-caused distortion. To fill in this gap, Calmet, Graesser and Hsu [ 72 , 73 ] put forward an elegant device-independent argument. They first consider a discrete spacetime with a sub-Planckian spacing and then show that no experiment is able to rule out this possibility. The point of the argument is not the particular spacetime discreteness they consider, but that it cannot be ruled out in principle. We want to measure the expectation value of position at two subsequent times in order to attempt to measure a spacing smaller than the Planck length.

The spectra of any two Hermitian operators have to fulfill the inequality. From 54 one has. Since one needs to measure two positions to determine a distance, the minimal uncertainty to the distance measurement is. This is the same bound as previously discussed in Section 3. We use an apparatus of size R. To get the spacing as precise as possible, we would use a test particle of high mass.

But then we will run into the, by now familiar, problem of black-hole formation when the mass becomes too large, so we have to require. Thus, we cannot make the detector arbitrarily small. Taken together, one finds. A similar argument was made by Ng and van Dam [ ], who also pointed out that with this thought experiment one can obtain a scaling for the uncertainty with the third root of the size of the detector. If one adds the position uncertainty 58 from the non-vanishing commutator to the gravitational one, one finds. Optimizing this expression with respect to the mass that yields a minimal uncertainty, one finds up to factors of order one and, inserting this value of M in 61 , thus.

Since R too should be larger than the Planck scale this is, of course, consistent with the previously-found minimal uncertainty. Ng and van Dam further argue that this uncertainty induces a minimum error in measurements of energy and momenta. Then its uncertainty would be on the order of. However, note that the scaling found by Ng and van Dam only follows if one works with the masses that minimize the uncertainty With such a mass one has to worry about very different uncertainties.

For particles with masses below the Planck mass on the other hand, the size of the detector would have to be below the Planck length, which makes no sense since its extension too has to be subject to the minimal position uncertainty. The observant reader will have noticed that almost all of the above estimates have explicitly or implicitly made use of spherical symmetry.

The one exception is the argument by Adler and Santiago in Section 3. However, it was also assumed there that the length and the radius of the cylinder are of comparable size. In the general case, when the dimensions of the test particle in different directions are very unequal, the Hoop conjecture does not forbid any one direction to be smaller than the Schwarzschild radius to prevent collapse of some matter distribution, as long as at least one other direction is larger than the Schwarzschild radius.

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The question then arises what limits that rely on black-hole formation can still be derived in the general case. A heuristic motivation of the following argument can be found in [ ], but here we will follow the more detailed argument by Tomassini and Viaggiu [ ]. Taking into account this uncertainty on the energy, one has. Now we have to make some assumption for the geometry of the object, which will inevitably be a crude estimate.

While an exact bound will depend on the shape of the matter distribution, we will here just be interested in obtaining a bound that depends on the three different spatial extensions, and is qualitatively correct. With Eq. Thus, as anticipated, taking into account that a black hole must not necessarily form if the spatial extension of a matter distribution is smaller than the Schwarzschild radius in only one direction, the uncertainty we arrive at here depends on the extension in all three directions, rather than applying separately to each of them.

Since the bound on the volumes 71 follows from the bounds on spatial and temporal intervals we found above, the relevant question here is not whether?? The step in which one studies the motion of the measured particle that is induced by the gravitational field of the test particle is missing in this argument.

Thus, while the above estimate correctly points out the relevance of non-spherical symmetries, the argument does not support the conclusion that it is possible to test spatial distances to arbitrary precision. The main obstacle to completion of this argument is that in the context of quantum field theory we are eventually dealing with particles probing particles.

To avoid spherical symmetry, we would need different objects as probes, which would require more information about the fundamental nature of matter. We will come back to this point in Section 3. String theory is one of the leading candidates for a theory of quantum gravity. Many textbooks have been dedicated to the topic, and the interested reader can also find excellent resources online [ , , , ]. For the following we will not need many details. Most importantly, we need to know that a string is described by a 2-dimensional surface swept out in a higher-dimensional spacetime.

The total number of spatial dimensions that supersymmetric string theory requires for consistency is nine, i. In the following we will denote the total number of dimensions, both time and space-like, with D. In this Subsection, Greek indices run from 0 to D. A string has discrete excitations, and its state can be expanded in a series of these excitations plus the motion of the center of mass. Due to conformal invariance, the worldsheet carries a complex structure and thus becomes a Riemann surface, whose complex coordinates we will denote with z and.

Scattering amplitudes in string theory are a sum over such surfaces. In the following l s is the string scale, and. The string scale is related to the Planck scale by , where g s is the string coupling constant. Contrary to what the name suggests, the string coupling constant is not constant, but depends on the value of a scalar field known as the dilaton.

To avoid conflict with observation, the additional spatial dimensions of string theory have to be compactified. The compactification scale is usually thought to be about the Planck length, and far below experimental accessibility. The possibility that the extensions of the extra dimensions or at least some of them might be much larger than the Planck length and thus possibly experimentally accessible, has been studied in models with a large compactification volume and lowered Planck scale, see, e. That such possibilities exist means, whether or not the model with extra dimensions are realized in nature, that we should, in principle, consider the minimal length a free parameter that has to be constrained by experiment.

String theory is also one of the motivations to look into non-commutative geometries. Non-commutative geometry will be discussed separately in Section 3. A section on matrix models will be included in a future update. The following argument, put forward by Susskind [ , ], will provide us with an insightful examination that illustrates how a string is different from a point particle and what consequences this difference has for our ability to resolve structures at shortest distances.

The normal mode decomposition of the transverse coordinates has the form. The coefficients and are normalized to , and. This sum is logarithmically divergent because modes with arbitrarily high frequency are being summed over. Then, for large n , the sum becomes approximately. Thus, the transverse extension of the string grows with the energy that the string is tested by, though only very slowly so. Then one finds for large n approximately. Thus, this heuristic argument suggests that the longitudinal spread of the string grows linearly with the energy at which it is probed.

The above heuristic argument is supported by many rigorous calculations. That string scattering leads to a modification of the Heisenberg uncertainty relation has been shown in several studies of string scattering at high energies performed in the late s [ , , ]. Gross and Mende [ ] put forward a now well-known analysis of the classic solution for the trajectories of a string worldsheet describing a scattering event with external momenta. In the lowest tree approximation they found for the extension of the string. Here, z i are the positions of the vertex operators on the Riemann surface corresponding to the asymptotic states with momenta.

Thus, as previously, the extension grows linearly with the energy. One can interpret this spread of the string in terms of a GUP by taking into account that at high energies the spread grows linearly with the energy.

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Together with the normal uncertainty, one obtains. However, the exponential fall-off of the tree amplitude depends on the genus of the expansion, and is dominated by the large N contributions because these decrease slower. The Borel resummation of the series has been calculated in [ ] and it was found that the tree level approximation is valid only for an intermediate range of energies, and for the amplitude decreases much slower than the tree-level result would lead one to expect.

Yoneya [ ] has furthermore argued that this behavior does not properly take into account non-perturbative effects, and thus the generalized uncertainty should not be regarded as generally valid in string theory. We will discuss this in Section 3. It has been proposed that the resistance of the string to attempts to localize it plays a role in resolving the black-hole information-loss paradox [ ]. In fact, one can wonder if the high energy behavior of the string acts against and eventually prevents the formation of black holes in elementary particle collisions.

It has been suggested in [ 10 , 9 , 11 ] that string effects might become important at impact parameters far greater than those required to form black holes, opening up the possibility that black holes might not form. The completely opposite point of view, that high energy scattering is ultimately entirely dominated by black-hole production, has also been put forward [ 48 , ]. A recent study of string scattering at high energies [ ] found no evidence that the extendedness of the string interferes with black-hole formation.

The subject of string scattering in the trans-Planckian regime is subject of ongoing research, see, e. Let us also briefly mention that the spread of the string just discussed should not be confused with the length of the string. The length of a string in the transverse direction is. In this study, it has been shown that when one increases the cut-off on the modes, the string becomes space-filling, and fills space densely i. The length of a string is not the same as its average extension.

The lengths of strings in the groundstate were studied in [ ]. Yoneya [ ] argued that the GUP in string theory is not generally valid. To begin with, it is not clear whether the Borel resummation of the perturbative expansion leads to correct non-perturbative results. And, after the original works on the generalized uncertainty in string theory, it has become understood that string theory gives rise to higher-dimensional membranes that are dynamical objects in their own right. These higher-dimensional membranes significantly change the picture painted by high energy string scattering, as we will see in 3.

However, even if the GUP is not generally valid, there might be a different uncertainty principle that string theory conforms to, that is a spacetime uncertainty of the form. This spacetime uncertainty has been motivated by Yoneya to arise from conformal symmetry [ , ] as follows. Suppose we are dealing with a Riemann surface with metric that parameterizes the string. In string theory, these surfaces appear in all path integrals and thus amplitudes, and they are thus of central importance for all possible processes.

However, this length that we are used to from differential geometry is not conformally invariant. The so-constructed length is dimensionless and conformally invariant. Any more complicated shape can be assembled from such polygons. With a Minkowski metric, one of these directions would be timelike and one spacelike. Then the extremal lengths are [ , ]. Equal indices are summed over. As before, X are the target space coordinates of the string worldsheet. Thus, the width of these contributions is given by the extremal length times the string scale, which quantifies the variance of A and B by.

Thus, probing short distances along the spatial and temporal directions simultaneously is not possible to arbitrary precision, lending support to the existence of a spacetime uncertainty of the form Yoneya notes [ ] that this argument cannot in this simple fashion be carried over to more complicated shapes. Thus, at present the spacetime uncertainty has the status of a conjecture. However, the power of this argument rests in it only relying on conformal invariance, which makes it plausible that, in contrast to the GUP, it is universally and non-perturbatively valid.

The endpoints of open strings obey boundary conditions, either of the Neumann type or of the Dirichlet type or a mixture of both. For Dirichlet boundary conditions, the submanifold on which open strings end is called a Dirichlet brane, or Dp-brane for short, where p is an integer denoting the dimension of the submanifold. A D0-brane is a point, sometimes called a D-particle; a D1-brane is a one-dimensional object, also called a D-string; and so on, all the way up to D9-branes.

These higher-dimensional objects that arise in string theory have a dynamics in their own right, and have given rise to a great many insights, especially with respect to dualities between different sectors of the theory, and the study of higher-dimensional black holes [ , 45 ]. Dp-branes have a tension of ; that is, in the weak coupling limit, they become very rigid. Thus, one might suspect D-particles to show evidence for structure on distances at least down to l s g s. Taking into account the scattering of Dp-branes indeed changes the conclusions we could draw from the earlier-discussed thought experiments.

We have seen that this was already the case for strings, but we can expect that Dp-branes change the picture even more dramatically. At high energies, strings can convert energy into potential energy, thereby increasing their extension and counteracting the attempt to probe small distances. Therefore, strings do not make good candidates to probe small structures, and to probe the structures of Dp-branes, one would best scatter them off each other.

That with Dp-branes new scaling behaviors enter the physics of shortest distances has been pointed out by Shenker [ ], and in particular the D-particle scattering has been studied in great detail by Douglas et al. It was shown there that indeed slow moving D-particles can probe distances below the ten-dimensional Planck scale and even below the string scale.

For these D-particles, it has been found that structures exist down to. To get a feeling for the scales involved here, let us first reconsider the scaling arguments on black-hole formation, now in a higher-dimensional spacetime. Thus, the horizon or the zero of g 00 is located at. We see that in the weak coupling limit, this lower bound can be small, in particular it can be much below the string scale.

This relation between spatial and temporal resolution can now be contrasted with the spacetime uncertainty 82 , that sets the limits below which the classical notion of spacetime ceases to make sense. The curves meet at. Below the spacetime uncertainty limit, it would actually become meaningless to talk about black holes that resemble any classical object.

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Below the bound from spacetime uncertainty yet above the black-hole bound that hides short-distance physics shaded region , the concept of classical geometry becomes meaningless. At first sight, this argument seems to suffer from the same problem as the previously examined argument for volumes in Section 3. However, here the situation is very different because fundamentally the objects we are dealing with are not particles but strings, and the interaction between Dp-branes is mediated by strings stretched between them.

It is an inherently different behavior than what we can expect from the classical gravitational attraction between point particles. At low string coupling, the coupling of gravity is weak and in this limit then, the backreaction of the branes on the background becomes negligible.

For these reasons, the D-particles distort each other less than point particles in a quantum field theory would, and this is what allows one to use them to probe very short distances. The following estimate from [ ] sheds light on the scales that we can test with D-particles in particular. But if the D-particle is slow, then its wavefunction behaves like that of a massive non-relativistic particle, so we have to take into account that the width spreads with time. For this, we can use the earlier-discussed bound Eq.

If we add the uncertainties 96 and 98 and minimize the sum with respect to v , we find that the spatial uncertainty is minimal for. Thus, we see that the D-particles saturate the spacetime uncertainty bound and they can be used to test these short distances. D-particle scattering has been studied in [ ] by use of a quantum mechanical toy model in which the two particles are interacting by unexcited open strings stretched between them. The open strings create a linear potential between the branes. At moderate velocities, repeated collisions can take place, since the probability for all the open strings to annihilate between one collision and the next is small.

By considering the conversion of kinetic energy into the potential of the strings, one sees that the particles reach a maximal separation of , realizing a test of the scales found above. Douglas et al. For the D-particles, this corresponds to the maximal separation in the repeated collisions. The analogy may be carried further than that in that higher-order corrections should lead to energy shifts. Analogy between scales involved in D -particle scattering and the hydrogen atom.

After [ ]. The possibility to resolve such short distances with D-branes have been studied in many more calculations; for a summary, see, for example, [ 45 ] and references therein. For our purposes, this estimate of scales will be sufficient. We take away that D-branes, should they exist, would allow us to probe distances down to. In the presence of compactified spacelike dimensions, a string can acquire an entirely new property: It can wrap around the compactified dimension.

Then, in the direction of this coordinate, the string has to obey the boundary condition. The momentum is then. After renormalization, the energy is. In addition to the normal contribution from the linear momentum, the string energy thus has a geometrically-quantized contribution from the momentum into the extra dimension s , labeled with n , an energy from the winding more winding stretches the string and thus needs energy , labeled with w , and a renormalized contribution from the Casimir energy. The important thing to note here is that this expression is invariant under the exchange.

This symmetry is known as target-space duality, or T-duality for short. It carries over to multiples extra dimensions, and can be shown to hold not only for the free string but also during interactions. We have seen in Section 3. In this approach it is assumed that the elementary constituents of matter are fundamentally strings that propagate in a higher dimensional spacetime with compactified additional dimensions, so that the strings can have excitations and winding numbers.

By taking into account the excitations and winding numbers, Fontanini et al. Note that this discards all massless modes as one sees from Eq. The Fourier transform of this limit of the momentum space propagator is.

This claim has not been supported by independent studies. However, this argument has been used as one of the motivations for the model with path integral duality that we will discuss in Section 4. The interesting thing to note here is that the minimal length that appears in this model is not determined by the Planck length, but by the radius of the compactified dimensions. It is worth emphasizing that this approach is manifestly Lorentz invariant. Loop Quantum Gravity LQG is a quantization of gravity by help of carefully constructed suitable variables for quantization, variables that have become known as the Ashtekar variables [ 39 ].

While LQG theory still lacks experimental confirmation, during the last two decades it has blossomed into an established research area. Here we will only roughly sketch the main idea to see how it entails a minimal length scale. For technical details, the interested reader is referred to the more specialized reviews [ 42 , , , , ]. Then, the metric can be parameterized with the lapse-function N and the shift vector N i. The three metric by itself does not suffice to completely describe the four dimensional spacetime.

So far, one is used to that from general relativity. Next we introduce the triad or dreibein, , which is a set of three vector fields. The triad converts the spatial indices a , b small, Latin, from the beginning of the alphabet to a locally-flat metric with indices i , j small, Latin, from the middle of the alphabet.

The densitized triad. The other set of variables is an su 2 connection , which is related to the connection on the manifold and the external curvature by. Its value can be fixed by requiring the black-hole entropy to match with the semi-classical case, and comes out to be of order one. From the triads one can reconstruct the internal metric, and from A and the triad, one can reconstruct the extrinsic curvature and thus one has a full description of spacetime.

The reason for this somewhat cumbersome reformulation of general relativity is that these variables do not only recast gravity as a gauge theory, but are also canonically conjugated in the classical theory. The Lagrangian of general relativity can then be rewritten in terms of the new variables, and the constraint equations can be derived. In the so-quantized theory one can then work with different representations, like one works in quantum mechanics with the coordinate or momentum representation, just more complicated. One such representation is the loop representation, an expansion of a state in a basis of traces of holonomies around all possible closed loops.

However, this basis is overcomplete. Each such spin network is a graph with vertices and edges that carry labels of the respective su 2 representation. In this basis, the states of LQG are then closed graphs, the edges of which are labeled by irreducible su 2 representations and the vertices by su 2 intertwiners. In terms of the triad, this can be written as. This area can be promoted to an operator, essentially by making the triads operators, though to deal with the square root of a product of these operators one has to average the operators over smearing functions and take the limit of these smearing functions to delta functions.

One can then act with the so-constructed operator on the states of the spin network and obtain the eigenvalues. This way, one finds that LQG has a minimum area of. A similar argument can be made for the volume operator, which also has a finite smallest-possible eigenvalue on the order of the cube of the Planck length [ , , 41 ]. These properties then lead to the following interpretation of the spin network: the edges of the graph represent quanta of area with area , and the vertices of the graph represent quanta of 3-volume.

The main simplification is that, rather than using the full quantized theory of gravity and then studying models with suitable symmetries, one first reduces the symmetries and then quantizes the few remaining degrees of freedom. For the quantization of the degrees of freedom one uses techniques similar to those of the full theory. Here we will only pick out one aspect that is particularly interesting for our theme of the minimal length. In principle, one works in LQC with operators for the triad and the connection, yet the semi-classical treatment captures the most essential features and will be sufficient for our purposes.

The ansatz for the metric is. With this, Equation can be written in the more familiar form. Inserting , this equation can be integrated to get. Together with the energy conservation this fully determines the time evolution. Now to find the Hamiltonian of LQC, one considers an elementary cell that is repeated in all spatial directions because space is homogeneous. With that in mind, one can construct an effective Hamiltonian constraint from the classical Eq. This replacement makes sense because the so-introduced operator can be expressed and interpreted in terms of holonomies.

For this, one does not have to use the sinus function in particular; any almost-periodic function would do [ 40 ], but the sinus is the easiest to deal with. This yields. The semi-classical limit is clearly inappropriate when energy densities reach the Planckian regime, but the key feature of the bounce and removal of the singularity survives in the quantized case [ 56 , 44 , 58 , 57 ]. We take away from here that the canonical quantization of gravity leads to the existence of minimal areas and three-volumes, and that there are strong indications for a Planckian bound on the maximally-possible value of energy density and curvature.

The following argument for the existence of a minimal length scale has been put forward by Padmanabhan [ , ] in the context of conformally-quantized gravity. That is, we consider fluctuations of the conformal factor only and quantize them. The metric is of the form. The two-point function of the scalar fluctuation diverges and thereby counteracts the attempt to obtain a spacetime distance of length zero; instead one has a finite length on the order of the Planck length.

This argument has recently been criticized by Cunliff in [ 92 ] on the grounds that the conformal factor is not a dynamical degree of freedom in the pure Einstein-Hilbert gravity that was used in this argument. However, while the classical constraints fix the conformal fluctuations in terms of matter sources, for gravity coupled to quantized matter this does not hold. Cunliff reexamined the argument, and found that the scaling behavior of the Greens function at short distances then depends on the matter content; for normal matter content, the limit still goes to zero.

String theory and LQG have in common the aim to provide a fundamental theory for space and time different from general relativity; a theory based on strings or spin networks respectively. Asymptotically Safe Gravity ASG , on the other hand, is an attempt to make sense of gravity as a quantum field theory by addressing the perturbative non-renormalizability of the Einstein-Hilbert action coupled to matter [ ].

In ASG, one considers general relativity merely as an effective theory valid in the low energy regime that has to be suitably extended to high energies in order for the theory to be renormalizable and make physical sense. The Einstein-Hilbert action is then not the fundamental action that can be applied up to arbitrarily-high energy scales, but just a low-energy approximation and its perturbative non-renormalizability need not worry us. What describes gravity at energies close by and beyond the Planck scale possibly in terms of non-metric degrees of freedom is instead dictated by the non-perturbatively-defined renormalization flow of the theory.

To see how that works, consider a generic Lagrangian of a local field theory. The terms can be ordered by mass dimension and will come with, generally dimensionful, coupling constants g i. One redefines these to dimensionless quantities , where k is an energy scale. It is a feature of quantum field theory that the couplings will depend on the scale at which one applies the theory; this is described by the Renormalization Group RG flow of the theory.

To make sense of the theory fundamentally, none of the dimensionless couplings should diverge. In more detail, one postulates that the RG flow of the theory, described by a vector-field in the infinite dimensional space of all possible functionals of the metric, has a fixed point with finitely many ultra-violet UV attractive directions. The requirement that the theory holds up to arbitrarily-high energies then implies that the natural world must be described by an RG trajectory lying in this surface, and originating in the UV from the immediate vicinity of the fixed point.

In ASG the fundamental gravitational interaction is then considered asymptotically safe. This necessitates a modification of general relativity, whose exact nature is so far unknown. Importantly, this scenario does not necessarily imply that the fundamental degrees of freedom remain those of the metric at all energies.

Also in ASG, the metric itself might turn out to be emergent from more fundamental degrees of freedom [ ]. It is beyond the scope of this review to discuss how good this evidence for the asymptotic safety of gravity really is. The interested reader is referred to reviews specifically dedicated to the topic, for example [ , , ]. For our purposes, in the following we will just assume that asymptotic safety is realized for general relativity. In particular, we define with. Here and in the rest of this subsection, a tilde indicates a dimensionless quantity.

Then the beta function of takes the form. This beta function has an IR attractive fixed point at and also a UV attractive nontrivial fixed point at. The solution of the RG equation is. Therefore, the Planck length, , becomes energy dependent. If we are in the regime of sub-Planckian energies, and the first term on the right side of Eq. The solution of the flow equation is.

This is the regime that we are all familiar with. One naturally expects the threshold separating these two regimes to be near the Planck scale. With the running of the RG scale, must go from its fixed point value at the Planck scale to very nearly zero at macroscopic scales. At first look it might seem like ASG does not contain a minimal length scale because there is no limit to the energy by which structures can be tested. In addition, towards the fixed point regime, the gravitational interaction becomes weaker, and with it weakening the argument from thought experiments in Section 3.

It has, in fact, been argued [ 51 , ] that in ASG the formation of a black-hole horizon must not necessarily occur, and we recall that the formation of a horizon was the main spoiler for increasing the resolution in the earlier-discussed thought experiments. However, to get the right picture one has to identify physically-meaningful quantities and a procedure to measure them, which leads to the following general argument for the occurrence of a minimal length in ASG [ 74 , ]. Energies have to be measured in some unit system, otherwise they are physically meaningless.

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In general, the unit itself will depend on the scale that is probed in any one particular experiment. In fact, since , an energy measured in units of will be bounded by the Planck energy; it will go to one in units of the Planck energy. One may think that one could just use some system of units other than Planck units to circumvent the conclusion, but if one takes any other dimensionful coupling as a unit, one will arrive at the same conclusion if the theory is asymptotically safe.

As Percacci and Vacca pointed out in [ ], it is essentially a tautology that an asymptotically-safe theory comes with this upper bound when measured in appropriate units. Consider a scattering process with in and outgoing particles in a space, which, in the infinite distance from the scattering region, is flat. However, since the metric depends on the scale that is being tested, the physically-relevant quantities in the collision region have to be evaluated with the metric. With that one finds that the effective Mandelstam variables, and thus also the momentum transfer in the collision region, actually go to , and are bounded by the Planck scale.

This behavior can be further illuminated by considering in more detail the scattering process in an asymptotically-flat spacetime [ ]. In particular, we will consider the scattering of two particles, scalars or fermions, by exchange of a graviton. In the s -channel, the squared amplitude for the scattering of two scalars is. As one expects, the cross sections scale with the fourth power of energy over the Planck mass. In particular, if the Planck mass was a constant, the perturbative expansion would break down at energies comparable to the Planck mass.

However, we now take into account that in ASG the Planck mass becomes energy dependent. For the annihilation process in the s -channel, it is , the total energy in the center-of-mass system, that encodes what scale can be probed. Thus, we replace m Pl with. One proceeds similarly for the other channels. From the above amplitudes the total cross section is found to be [ ]. Cross section for scattering of two scalar particles by graviton exchange with and without running Planck mass, in units of the low-energy Planck mass.

The dot-dashed purple line depicts the case without asymptotic safety; the continuous blue and dashed grey line take into account the running of the Planck mass, for two different values of the fixed point, and 0. Figure from [ ]; reproduced with permission from IOP. If we follow our earlier argument and use units of the running Planck mass, then the cross section as well as the physically-relevant energy, in terms of the asymptotic quantities , become constant at the Planck scale.

These indications for the existence of a minimal length scale in ASG are intriguing, in particular because the dependence of the cross section on the energy offers a clean way to define a minimal length scale from observable quantities, for example through the square root of the cross section at its maximum value. However, it is not obvious how the above argument should be extended to interactions in which no graviton exchange takes place.

It has been argued on general grounds in [ 74 ], that even in these cases the dependence of the background on the energy of the exchange particle reduces the momentum transfer so that the interaction would not probe distances below the Planck length and cross sections would stagnate once the fixed-point regime has been reached, but the details require more study.

Recently, in [ 30 ] it has been argued that it is difficult to universally define the running of the gravitational coupling because of the multitude of kinematic factors present at higher order. In the simple example that we discussed here, the dependence of G on the seems like a reasonable guess, but a cautionary note that this argument might not be possible to generalize is in order.

Non-commutative geometry is both a modification of quantum mechanics and quantum field theory that arises within certain approaches towards quantum gravity, and a class of theories in its own right. Thus, it could rightfully claim a place both in this section with motivations for a minimal length scale, and in Section 4 with applications. We will discuss the general idea of non-commutative geometries in the motivation because there is a large amount of excellent literature that covers the applications and phenomenology of non-commutative geometry.

Thus, our treatment here will be very brief. For details, the interested reader is referred to [ , ] and the many references therein. String theory and M-theory are among the motivations to look at non-commutative geometries see, e. This approach has been very fruitful and will be discussed in more detail later in Section 4.

The simplest way to do this is of the form. In this type of non-commutative geometry, the Poisson tensor is not a dynamical field and defines a preferred frame and thereby breaks Lorentz invariance. The above commutation relation leads to a minimal uncertainty among spacial coordinates of the form. Quantization under the assumption of a non-commutative geometry can be extended from the coordinates themselves to the algebra of functions f x by using Weyl quantization. What one looks for is a procedure W that assigns to each element f x in the algebra of functions a Hermitian operator in the algebra of operators.

One does that by choosing a suitable basis for elements of each algebra and then identifies them with each other. The most common choice 9 is to use a Fourier decomposition of the function f x. From Eqs. The star product is a particularly useful way to handle non-commutative geometries, because one can continue to work with ordinary functions, one just has to keep in mind that they obey a modified product rule in the algebra.

With that, one can build non-commutative quantum field theories by replacing normal products of fields in the Lagrangian with the star products. To gain some insight into the way this product modifies the physics, it is useful to compute the star product with a delta function. For that, we rewrite Eq. In contrast to the normal product of functions, this describes a highly non-local operation.

This non-locality, which is a characteristic property of the star product, is the most relevant feature of non-commutative geometry. It is clear that the non-vanishing commutator by itself already introduces some notion of fundamentally-finite resolution, but there is another way to see how a minimal length comes into play in non-commutative geometry. To see that, we look at a Gaussian centered around zero. Gaussian distributions are of interest not only because they are widely used field configurations, but, for example, also because they may describe solitonic solutions in a potential [ ].

For simplicity, we will consider only two spatial dimensions and spatial commutativity, so then we have. This is a greatly simplified scenario, but it will suffice here. Besides the candidate theories for quantum gravity so far discussed, there are also discrete approaches, reviewed, for example, in [ ]. For these approaches, no general statement can be made with respect to the notion of a minimal length scale.

Though one has lattice parameters that play the role of regulators, the goal is to eventually let the lattice spacing go to zero, leaving open the question of whether observables in this limit allow an arbitrarily good resolution of structures or whether the resolution remains bounded. One example of a discrete approach, where a minimal length appears, is the lattice approach by Greensite [ ] discussed also in Garay [ ] , in which the minimal length scale appears for much the same reason as it appears in the case of quantized conformal metric fluctuations discussed in Section 3.

Even if the lattice spacing does not go to zero, it has been argued on general grounds in [ 60 ] that discreteness does not necessarily imply a lower bound on the resolution of spatial distances. One discrete approach in which a minimal length scale makes itself noticeable in yet another way are Causal Sets [ ]. In this approach, one considers as fundamental the causal structure of spacetime, as realized by a partially-ordered, locally-finite set of points.

This set, represented by a discrete sprinkling of points, replaces the smooth background manifold of general relativity. In full generality, this conjecture is so far unproven, though it has been proven in a limiting case [ 63 ]. Intriguingly, the causal sets approach to a discrete spacetime can preserve Lorentz invariance. This can be achieved by using not a regular but a random sprinkling of points; there is thus no meaningful lattice parameter in the ordinary sense. What the initial conditions and the laws of nature do not do, however, is to cause our actions—simply because they do not cause anything.

They are grounds for everything, but causes for nothing. If, then, we want to hold onto the idea that there are causal relations in the world, relations between us and our actions among them, then we are committed to limit our inspection on proper subsystems of the universe. What we need to do, we could say, is to move from the Block Universe into a bubble within: from the idea of an unchanging four dimensional, all encompassing block of spacetime to a view from within such a block, a bubble where causes and effects can take place, surrounded by a plethora of equally possible, alternative bubbles.

From within our bubble, therefore, we are presented with various options, and when we act on our free choices, we are always able to behave in more than one way. Figure 1. Bubbled Block Universe, where S is the complete set of initial conditions the low-entropy Big Bang at the time-coordinate t 0 , L is the complete set of all the fundamental laws of physics, A is a complete state of spacetime at the time-coordinate t 1 , and a 1 is a specific action performed at t 1. The action a 1 is a macrophysical event, and there is therefore a set of equiprobable mutually exclusive microphysical states p 1 — p 4 that are consistent with the occurrence of a 1.

Supposing that w 1 represents the actual circumstances that a 1 occurs and is realized by p 1 , the subsystems w 2 — w 4 represent the possible, non-actual alternatives to w 1. In other words, from the perspective of a 1 each of the systems w 1 — w 4 are equally possible. Figure not to scale. Disarming the consequence argument comes therefore in two stages.

First, the asymmetry of determinism that the argument rests on is rejected: from the perspective of the fundamental laws of physics the present, or any future state of the universe entails any past state of it as much as vice versa. Second, once we turn our focus on the asymmetries we perceive, most notably to the temporal and causal asymmetries that constantly surround us, the symmetric determinism imposed by the fundamental laws of physics evaporates, and we are actually faced with overwhelming degrees of microphysical freedom.

Our decisions and the actions resulting from them are thus macrophysical, emergent phenomena for the simple reason that causation is a macrophysical, emergent phenomenon. There is no causation at the fundamental physical level, but that does not make causation any less real. Elzein et al. So what this account does not commit us to is some sort of backward causation, anymore than—and exactly to the same extent as—the Second Law commits us to water spontaneously boiling at a room temperature.

That is, the total entropy of the universe practically never decreases to any significant extent , although there exists a small likelihood that that might actually happen. In fact, as the fluctuation theorem indicates [ 84 — 86 ], and the subsequent experimental evidence shows [ 87 — 89 ], if we are inspecting a small enough system or in principle any isolated system in a state of maximum entropy , the total entropy of the system will fluctuate, and in such systems time—and causation—will actually run backwards or back and forth.

Local, momentary backward influences are therefore a physical fact, and although our actions will always be accompanied by such minute rippling backward influences, they will evaporate before we are able to perceive them, and they are of no practical use to us. So although we can—and do—exert backward causal influence, we posses only forward causal, pivotal control: we can only bring about changes in the future, although our actions influence also the past.

If this sounds absurd, it is only because our intuition is holding onto an absolute and objective distinction between the past and the future, which is something that has no ground in current physics. Can we now rest assured that the correct understanding of physics will deliver us from all our worries concerning free will and agency? Unfortunately not.

The most burning question is this: can we really defeat the consequence argument by relying on symmetric determinism at the microphysical level while at the same time holding onto temporal and causal asymmetries at the macrophysical level? Macrophysical asymmetries are of course consistent with microphysical symmetries, and this relationship can be seen to ground the temporal and causal asymmetries that surround us, as Carroll [ 81 ] notes.

However, is noting this connection really enough to undermine the consequence argument? More specifically: why wouldn't the incompatibilist be now entitled to insist that it was determination in the macrophysical, asymmetric sense all along that has been keeping her awake at nights, and that nothing that has been said above has challenged that sort of determinism—on the contrary, hasn't it now simply been given a solid footing in current physics?

The initial charge was that the incompatibilist has been relying on causal determinism when in fact there is no such thing in fundamental symmetric physics. However, once an explanation is given to the perceived macrophysical asymmetries, and causal determinism is thus brought back in—albeit in terms of probability distributions—the incompatibilist could simply now restate her case in macrophysical terms.

If any backward influences that might accompany our actions are negligible compared to their forward influences, as the argument went, then surely the incompatibilist could simply point out that now every decision and every action is preceded by events that exert stronger influence on those decisions and actions than vice versa? We cannot reverse the growth of entropy, and we cannot reverse the direction of causation, and we cannot change the events in the past that determine our behavior—and hence we are not free to choose the courses of our actions, the incompatibilist could now insist.

It is important to understand, however, that we are not back to square one. It is true that the consequence argument is typically phrased, explicitly or implicitly, in terms of microphysical determinism. The traditional, Laplacean, worry is that if we only knew the exact, detailed state of the universe at some moment in time, we could, if the fundamental laws of nature are deterministic, and we would posses complete knowledge of them too, calculate all the future events in perfect precision and absolute certainty. The incompatibilist argumentation based on this traditional worry has now been shown to be misguided: it is not absurd to think that you acting differently than you actually did would change the past simply because at the scale of the entire universe the past and the future are symmetrically related, and hence all future changes would, as a matter of physical fact, result in changes in the past.

This is an important observation, and a real advancement in the debate. However, the compatibilists have been too quick, it would now seem, to conclude that the consequence argument is invalid and that we can therefore have such a thing as free will in a deterministic universe. The initial conditions or the state of the universe at any arbitrary point in time in the past , S , could now be interpreted in macrophysical terms, encompassing all the microphysical variation that is compatible with the macrophysical evolution of the universe between t 0 and t 1.

In other words, all the microphysical leeway that there might be is simply irrelevant, since all these microphysical states will converge into the same macrophysical states that constitute our particular decisions and actions, and you would have to be able to change the macrophysical, low-entropy state of the universe in order to have free will and be able to act otherwise. But such an entropic reversal is practically impossible at the scale of our interest, as those exploiting the microphysical symmetries to defeat the consequence argument are eager to stress.

There is another, perhaps a more tangible way of phrasing the problem we are now faced with. What the argumentation reviewed in here may have been able to achieve, is to show how we can be genuine causal agents in a thoroughly physical world—how we, by making conscious decisions, can initiate actions and exert control on future courses of events. This would be a major achievement, of course. It would show that the widespread worries about our mental states being causally inert are misguided.

But that, however, as it has already been stressed, would only address a part of the issue we are facing with the problem of free will. Another, and arguably more fundamental issue has been left unaddressed: the question of whether we are truly free to make the decisions that give rise to our actions. So the core problem is not really that the rest of the universe determines our actions; the problem is that it determines our decisions. And in fact, as the preceding discussion arguably shows, the first worry can now be dispensed with: we, rather than the rest of the universe, might indeed function as genuine, pivotal difference-makers with regard to our actions—we, and our decisions and actions might actually matter.

We might be, as it were, indispensable cogs in the vast clockwork of the universe, the removals of which would make things run differently. There remains the worry, however, that every movement of the cogs would still be determined by the prior movements of some of the other cogs in the clockwork; that our decisions would be fully determined by factors beyond our sphere of influence. To address that worry, and to successfully soothe it, one would need to show that our decisions are underdetermined by prior events, or, to put it in current parlance, that there are no prior events one can point to as pivotal difference-makers with regard to our decisions.

But such an argument, it now becomes apparent, would seem to be missing. There is a gap, therefore, that would need to be closed to complete the argument sketched here. We can offer a consistent physical story on how the macroscopic irreversibility emerges from the microscopic reversibility, and maybe we can, in the way outlined here, ground causal relations, genuine mental causation among them, on such macroscopic asymmetries.

But that would only explain how our will can make a difference. To fully solve the problem of free will—to tackle the issue of the freedom of the will—one would also need to explain how that will could be free. It might be true that the past years or so has not brought much progress in the debate on free will, as some have claimed; all involved in the debate, philosophers, and scientists, may have been talking more past each other than to—or with—each other. At the same time, it is clear that major advancements have been made in closely related issues in philosophy of science and in philosophy of mind, and, most obviously, in physics and other natural sciences.

It might thus not be too audacious to claim that we might well soon be witnessing some progress in this issue as well—maybe it is just the question of putting all the pieces together the right way. Although the recent developments reviewed here leave many questions in the air, and often the discussion could benefit from a more systematic framing of the most fundamental issues in the debate, the general framework of the argumentation is very enticing indeed, and fits quite nicely into the more encompassing story that the naturalistic philosophy and the empirical sciences have been putting together for half a century or so.

Even if these developments would not turn out to constitute the event that made us turn the leaf of history on this matter, they most certainly will be something that the future will hold as pivotal in the chain of events that resulted in that turn. The author confirms being the sole contributor of this work and approved it for publication. The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

I would like to thank Dr. Nadine Elzein, Ms. Sophie Kikkert, Mr.

Chris Oldfield, Mr. Otto Overkamp, Prof. David Papineau, Dr. Michael Ridley, Mr. Athamos Stradis, Mr. Kalle Timperi, Prof. However, neither of these assumptions can be taken for granted. First, are we empirically justified in thinking that the laws of nature are constant? Einstein [ 65 , 66 ] and Dirac [ 67 ] already suggested the possibility that some of the fundamental physical constants might not be time-invariant, and a number of recent studies have indicated that that might actually be the case cf.

The ramifications of such a conclusion would of course be radical, and it would entail prima facie that determinism does not hold at any scale or that the whole notion would collapse. Second, if we are relying on a Humean as opposed to non-Humean or Aristotelian account of the laws of nature as those proposing this line of argumentation typically are , then we cannot, in principle, separate L from S —what we actually have is only S encompassing everything.

The laws of nature are simply theorems of the best systematization of this Humean mosaic. What this would then seem to entail is that any changes anywhere in the spacetime would result in changes in the laws since chains of events are not driven by the laws of nature but exactly the opposite. This would again radically transform the basic elements of the debate, and maybe puts Ismael's [ 6 ], p. Carroll S. New York, NY: Dutton Dennett DC.

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Introduction to Quantum Theory and Atomic Structure (Oxford Chemistry Primers, Volume 37) Introduction to Quantum Theory and Atomic Structure (Oxford Chemistry Primers, Volume 37)
Introduction to Quantum Theory and Atomic Structure (Oxford Chemistry Primers, Volume 37) Introduction to Quantum Theory and Atomic Structure (Oxford Chemistry Primers, Volume 37)
Introduction to Quantum Theory and Atomic Structure (Oxford Chemistry Primers, Volume 37) Introduction to Quantum Theory and Atomic Structure (Oxford Chemistry Primers, Volume 37)
Introduction to Quantum Theory and Atomic Structure (Oxford Chemistry Primers, Volume 37) Introduction to Quantum Theory and Atomic Structure (Oxford Chemistry Primers, Volume 37)
Introduction to Quantum Theory and Atomic Structure (Oxford Chemistry Primers, Volume 37) Introduction to Quantum Theory and Atomic Structure (Oxford Chemistry Primers, Volume 37)
Introduction to Quantum Theory and Atomic Structure (Oxford Chemistry Primers, Volume 37) Introduction to Quantum Theory and Atomic Structure (Oxford Chemistry Primers, Volume 37)
Introduction to Quantum Theory and Atomic Structure (Oxford Chemistry Primers, Volume 37) Introduction to Quantum Theory and Atomic Structure (Oxford Chemistry Primers, Volume 37)

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