A general definition of symmetry: If a certain change does not change a property, then that property is symmetric with respect to that part...
A general definition of symmetry: If a certain change does not change a property, then that property is symmetric with respect to that particular change.
Actually, things can have properties, and anything that has properties is a thing. A particular thing is something that can be distinguished from other things in some way. A particular thing can be identified with itself in some way, so it has at least one unique property that can be identified with the thing. Since symmetry is related to property, therefore symmetry can be interpreted for things, symmetry can be for things, objects.
In order to specify a property, it is also necessary to specify the measure of the property, i.e. the amount by which the given property can be quantitatively characterized. According to the definition of symmetry, which states that a symmetry exists when a given change does not change the property, the inevitable consequence of an existing symmetry - that is, the preservation of the property - must also mean the conservation of the metrics associated with that property. In the case of symmetry, the conservation of the quantity represented by the preserved property during the change is a trivial consequence of symmetry, a definite consequence of the definition of symmetry.
It is worth emphasizing the relationship between symmetry and conserved quantity: if symmetry is present on an object, the symmetry is coupled to some conserved quantity related to that object, and if during changes of an object a conserved quantity appears among the properties of the object, then a symmetry related to those changes can be found. The consequence of symmetry is the conserved quantity, and the conserved quantity is a sign of existing symmetry. The two things go together, but only one is the cause of the other.
For example, if a change does not change the distance and relative position of the parts of a planar object, i.e. the symmetry is congruence, then the quantity that is conserved is the quantity related to the distance and relative position of the parts of the object, e.g. the area and perimeter of the object, which is necessarily conserved during the change.
A change of the planar object as a symmetry is also the operation of the projection, which does not change the relative position of the connected parts of the planar object. The characteristic quantity related to the position interpreted on a planar object is, for example, the direction of the parts relative to each other, so the conserved quantity related to the projection as symmetry is, for example, the sum of the angles of the planar object.
As can be seen, conservation of the quantity of a property related to symmetry is a trivial consequence of symmetry, but the conserved quantity related to symmetry is not necessarily obvious, and if there is a conserved quantity, it is not necessarily apparent what kind of symmetry creates it.
In the case of the existence of certain symmetries, Lagrangian mechanics can be used to mathematically deduce which conserved quantity belongs to a particular symmetry. A typical example of a symmetry-conserved quantity relationship that can be mathematically derived using Lagrangian mechanics is when, for example, the nature of the operation of an object does not change in the case of displacement or rotation of the object, so that the operation of displacement and rotation is a symmetry regarding the operational state of the object. In these cases, the non-obvious but mathematically derived appropriate conserved quantities regarding that object as a whole or any isolated parts are the conservation of linear momentum and angular momentum, or if the change of time is the symmetry of the operation of the object, then the conserved quantity of the object is energy. These seemingly surprising symmetry-conserved quantity pairs were discovered by Emmy Noether.
Since Lagrangian mechanics and the mathematics of the proof can be interpreted in the case of continuous changes, the necessity of continuity of symmetry seems to be a necessary condition, but in fact the condition of continuity comes from the nature of the tools used for the proof, not came from the associated symmetry, as it would be a generally required law regarding the symmetry.
It can be noticed that conserved quantity related to symmetry does not exist only in the case of continuous symmetry. For example, if mirroring (which is characterized as a discrete symmetry) of a planar object is a symmetry, then conserved quantities also exist in this case of discrete symmetry, such as perimeter, area, and sum of angles of the planar object.
A counterargument is that the discrete symmetry of mirroring in two dimensions can be replaced by a continuous symmetry with rotation in three dimensions. However, this argument is debatable, since a two-dimensional system is fundamentally different and typically cannot be identified with a system existing in three dimensions.
Still, it is an interesting question, a connection to consider, whether all existing discrete symmetries can somehow be replaced by continuous symmetries? (For example, the discrete symmetry experienced in particle interactions during the creation of matter-antimatter pairs - where, for example, the remaining quantity is the rest mass of the particles - in what way can it be replaced by continuous symmetry?) As a conjecture, one could assume that all discrete physical symmetries can be replaced by continuous symmetries in some way. If the conjecture can be verified, then the condition of Noether's theorem about the requirement of apparent continuous symmetry is actually a triviality.
As another example of a symmetry conserved quantity, if a change in a multi-part system does not change the relative position of the components of that system, then the structure of the system is symmetric with respect to that change. For example, in solid state materials, the interactions between the constituent parts of the material determine the relative position of those parts, and depending on the strength of the interactions between the constituent parts, up to a certain limit, the relative position of the constituent parts does not change when the kinetic energy of the constituent parts changes.
The relative positions of the constituent parts of solid bodies are symmetrical to the change in kinetic energy of the constituent parts up to a well-defined limit. In solids, the arrangement of the constituent parts as a whole determines the shape of the given assemblage of solids as a physical property. The arrangement of the constituents of a given solid is symmetric to the temperature change up to a limit. The conserved physical quantities related to this symmetry are measures of the physical form of the solid.
When generalizing symmetry, it is necessary to consider that objects can differ according to their internal complexity, and therefore changes can affect complex objects in a more complex way, which also affects the property of apparent symmetry.
A non-complex object consists of a single thing. (Single thing means that the thing exists without internal structure, or a thing can even be non-complex having internal structure, but with a structure in which the constituent parts of the structure have constant relative relations to each other). Acting on a non-complex object, a change does not create a relative difference on the object, the change is necessarily global regarding the object as a whole, and therefore if the change is symmetry, the symmetry and the conserved quantity related to the symmetry are also global properties of that object.
We can consider an object to be complex if the object has constituent parts, the constituent parts belong together in some way, and the relationship of the constituent parts to each other is not fixed. For example, a solid can behave as a non-complex object or as a complex object with respect to a change, depending on the nature and extent of the change, depending on whether the relative relationship of the components of the solid is maintained or changed as a result of the change.
When acting on a complex object, a change can create relative differences on the object, the change created by an action can be not only global but also local, and therefore, if the change is symmetry, the symmetry and the conserved quantity related to the symmetry apply not only to the system as a whole, but the change can even be represented on a limited part of that object.
From the point of view of investigating the rules of symmetry-related properties connection regarding complex objects, the examination of complex objects consisting of structural elements with the same properties is particularly interesting. An object consisting of structural elements with identical properties can be complex if the relative relation of the constituent parts is not fixed for all properties. Such an object can be called a field, which is a similar concept known from modern physics.
In this sense, the field is a composite object that has small building parts having properties of the same kind, which are the constituents that make up the field. The constituents of the field are related to each other in a way determined by their properties, but the degrees of actual relatedness are not necessarily the same for all constituents of the field. Note that, by this definition, the space consisting of parts without properties (which can thus be considered as empty space) is not considered to be a field.
The aggregated magnitude of the properties of an isolated field, free from external interactions, is trivially fixed, it is the property of the given system, and necessarily constant, thus globally conserved for the system as a whole. Isolation itself is an abstract symmetry. If all changes applied to an object do not change the object in any way, symmetry exists, the object is symmetrical with respect to all changes. This symmetry may be called isolation. The conserved quantity related to isolation as symmetry is the global and aggregated metrics of this object.
Since, by definition, the metrics of the properties of a field are not necessarily the same in all parts of the field, but in the case of isolation the global quantity of the metrics of the properties of the field must be constant, the trivial symmetry and necessary state of the isolated field is a gauge-like symmetry, and all changes can be characterized by the distribution-like variations, oscillations, vibrations around the mean of the metrics of the properties of the field resulting from the interrelationships of its constituent parts.
The gauge-like symmetry of such an isolated field means that, due to the global conservation of the field's properties and the possible local differences in the metrics of the field, local symmetric structures with opposite metrics to each other necessarily arise and must exist within the field. The components of these structures continue to oscillate around the mean properties of the structure according to the internal functioning of the field.
These locally existing vibrational structures might also result in stable remaining resonances determined by the properties of the field components and the nature of their relationship to each other, which can also move on the field according to the properties of the field. The formation of self-contained oscillations can also form stationary resonances. When stationary resonances are formed from self-contained forms of vibration, their interconnection through synchronized vibration can create locally existing, complex, stable structures.
It is worth noting that these resulting resonances, based on the inevitable symmetry of changes in a field, can be strikingly related to the building blocks of our material world. The grid model discussed in these thoughts is a model that interprets the characteristics of our material world and uses the symmetries of changes in the kind of field defined here.
As discussed earlier, the global symmetry of isolated systems is a trivial symmetry. We can define symmetry as trivial if the symmetry is not the consequence of the unique specificity of the components that make up the system, but the necessity of the system itself. Empty space has trivial symmetry. Empty space is globally and locally symmetric, changes do not alter the properties of empty space, and these symmetries are trivially unrelated to the - non-existent - constituents of empty space. Chaos, a chaotic system, also has trivial symmetry. Every chaotic system is locally trivially symmetric with respect to changes. Chaos absorbs the specific local effects of changes on the components of a chaotic system.
A particular change in symmetry - what the scientific literature calls a symmetry breaking - typically occurs when a system transitions from chaos to order. Both chaos and order have symmetry, but the origin of the symmetries in the two cases is fundamentally different. Chaos erases all local differences, it is trivially symmetric. The symmetry of chaos is not specific, it is not a consequence of the unique properties of the components that make up the system.
Orderedness, on the other hand, can form specific symmetries related to the properties of the constituent parts of the system, and orderedness can also preserve local differences, and the resulting symmetry is unique to the system.
The origin of the symmetries of chaos and order are fundamentally different, therefore the connection of the two different symmetries during the transition from one to the other, the transformation associated with this change, called symmetry breaking, requires caution in comparing the degree of the two types of symmetries.
In the comparison between the symmetries of chaos and orderedness, it is worth considering that symmetries that are not of the same origin or of a similar nature play a role. The remaining quantities derived from the symmetry of order originate from the specific characteristics of the components of the system. The symmetry of chaos, on the other hand, is typically not specific to the components of the system, and results in conserved quantities that come from the system as a whole.
The symmetry-related role of chaos and order in the synthesis of organic molecules is clearly visible. Organic molecules are typically chiral structures that can exist in different mirror symmetric forms. The inorganic synthesis of organic molecules typically produces different structures in terms of chirality to the same extent, typically because inorganic synthesis is a chemical reaction that takes place during the chaotic movement of molecules. In contrast, biological synthesis is chirality-specific and supports only one chiral form, typically because organic synthesis is a chemical reaction using a template, a chirality-specific model for synthesis. The natural selection of the type of biological template is one of the mysteries of the origin of life, but it is probably more a random contingency than a necessity based on natural laws, because there seems to be no natural reason why biological reactions could somehow work exclusively according to one form of chirality.
In general, since the global symmetry of an isolated field is trivial, if an isolated field has a global asymmetry, the asymmetry is either a consequence of the asymmetric effect on the field as a whole, or it is the result of the asymmetry of the constituent parts of the field during the transition of the field from chaos to order, or it is also possible that the metrics of an actual global asymmetry are the conserved quantity of a once-existing global symmetry.
Our universe has been asymmetric with respect to the generally existing type of matter since its early stages, even though the laws of nature have symmetry for the matter-antimatter pair. Considering the possible causes mentioned above, the origin of this asymmetry can be, for example, the consequence of some kind of asymmetry present in the initial state of our universe as a whole. Matter-antimatter global asymmetry can also be caused by a kind of characteristic asymmetry of certain particle properties, as it is represented and as it can be seen in experiments, for example by showing specific differences of matter-antimatter behavior in the case of the neutral Kaon composite particles. The resulting matter-antimatter asymmetry that follows and is derived from the asymmetry of the components that make up the field, and what we can see as a characteristic asymmetry of certain particle properties, could then be possible to create the matter-antimatter asymmetry under the existing conditions of the initial period of our universe, and this asymmetry could show up, even if limited, in today's conditions.
In the search for the cause of the matter-antimatter asymmetry, it might also be possible and worth investigating that the origin of the dominance of matter in the early stages of the universe is not the result of some kind of asymmetry, but that the dominant existence of matter is a conserved quantity related to a symmetry of the beginning of our physical universe, or the consequence of a once conserved quantity belonging to a once prevailing symmetry. This idea perhaps not need to be obviously rejected if we consider that there is a similar symmetry, albeit a constant one - the symmetry of the operation of the universe in relation to time and the conservation of the global energy of our universe. The origin of the matter-antimatter asymmetry requires further considerations.
The fact that our universe appears to be CPT symmetric can provide a clue to understanding reality. CPT symmetry means that if we replaced our matter particles with antimatter particles, and all processes would take place in a mirror image in all dimensions of space, and we would also reverse the course of the processes in the direction of time, we would have a universe with the same properties as we have now.
Since our universe seems to have complete symmetry for this common transformation, these possible changes necessarily refer to all possible internal degrees of freedom of the fundamental constituent parts of the universe, because complete symmetry for the whole can only exist with symmetry of all degrees of freedom of the constituent parts.
The meaning of these changes in particle physics, the physical interpretation of CPT transformations is not obvious, but the fact that there is some kind of general combined symmetry in the universe supports the idea that the basic building blocks of our universe are and their behavior, at least for this complex transformation, can be symmetrical. It can mean that, according to the proposed definition of the field and the behavior of its constituent parts, the vibration of the basic constituents of the field has no asymmetry leading to specificity, at least above a certain existing vibrational energy. The conclusion could therefore be drawn that the matter-antimatter asymmetry of our universe could have been caused by a global asymmetry existing in the initial state, or somehow similar, it is a conserved quantity with respect to a once existing symmetry.
The grid model may offer a similar origin of this matter-antimatter asymmetry. According to that, our material universe started from a state of global resonance of the field representing the universe, whose orderliness ended globally and the vibration of the field transformed into a disordered, chaotic state at the birth of our universe. The actual state of the global resonance at the moment of termination is a specific symmetry, which can create a global conserved quantity, specific to the actually formed system during the transition to the chaotic state. If the formed chaotic vibrating field is suitable for the creation of local resonances due to its specificity, the termination of the global resonance could also create a specific state capable of forming complex ordered structures, for example our experienced universe, which can exist until the locally existing resonance structures cease to exist in the chaotic environment, whereupon the resulting chaotic system can return to the state of global resonance by necessary contingency, and a new cycle begins. If the chaotic system created from global resonance is not suitable for the formation of locally ordered resonant structures, if no stable structures can be formed, the system remains in the chaotic state and eventually returns to the state of global resonance due to the necessary contingency, from which a new cycle can also begin. In this way, the cyclical universe offers a possible solution to the specific state of our experienced universe, which science calls the problem of the universe tuned to life.
It is also worth investigating the relationship between classical physical fields and the field discussed in this thought as a complex system consisting of identical structural parts. According to the current physical approach, our physical world consists of different particle fields, such as photon field, electron field, quark field, etc., whose excited states are the manifestations of material particles. Based on the field concept discussed in this thought, it is not necessary for different particle fields to exist. According to this model, the different material particles are created by the different combined vibrational patterns and resonances of the same constituent parts that make up the field. According to this view, there are not different physical fields that exist, but a single field, and the different types of possible local resonance patterns determined by the possible different vibrational states of its constituent parts of this single field are the physical manifestations of the particles of our material world. The grid model is also a physical model of this theory.
The symmetries of our world related to changes and the conserved quantities related to existing symmetries are general defining attributes of the physical operation that creates our material world. Their study is a principal guideline for learning and understanding the functioning of our material world. Our material world seems convincingly of the structure of a complex system consisting of identical building blocks with simple functioning behavior that creates complexity through their interacting functions. The emergent mode of operation through the transition from simplicity to complexity is demonstrated to us by the symmetries and conserved quantities that apply to our universe, and knowing them can help us understand the origin of the world.
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